*238*
*56*
*31MB*

*English*
*Pages 298*
*Year 2010*

- Author / Uploaded
- Jorge Arvesu (editor)
- Francisco Marcellan (editor)
- Andrei Martinez-finkelshtein (editor)

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CONTEMPORARY MATHEMATICS 507

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http://dx.doi.org/10.1090/conm/507

Recent Trends in Orthogonal Polynomials and Approximation Theory

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CONTEMPORARY MATHEMATICS 507

Recent Trends in Orthogonal Polynomials and Approximation Theory International Workshop in Honor of Guillermo López Lagomasino's 60th Birthday September 8–12, 2008 Universidad Carlos III de Madrid Leganés, Spain

Jorge Arvesú Francisco Marcellán Andrei Martínez-Finkelshtein Editors

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Editorial Board Dennis DeTurck, managing editor George Andrews

Abel Klein

Martin J. Strauss

2000 Mathematics Subject Classiﬁcation. Primary 15A52, 26C10, 30E10, 31A15, 31C15, 33C47, 41A20, 42C05, 44A60, 65D32.

Library of Congress Cataloging-in-Publication Data International Workshop on Orthogonal Polynomials and Approximation Theory (2008 : Universidad Carlos III de Madrid) Recent trends in orthogonal polynomials and approximation theory : conference in honor of Guillermo L´ opez Lagomasino’s 60th birthday, September 8–12, 2008, Universidad Carlos III de Madrid, Legan´es, Spain / Jorge Arves´ u, Francisco Marcell´ an, Andrei Mart´ınez-Finkelshtein, editors. p. cm. — (Contemporary mathematics ; v. 507) Includes bibliographical references. ISBN 978-0-8218-4803-6 (alk. paper) 1. Orthogonal polynomials—Congresses. 2. Approximation theory—Congresses. I. L´ opez Lagomasino, Guillermo, 1948– II. Arves´ u, Jorge, 1968– III. Marcell´ an, Francisco. IV. Mart´ınezFinkelshtein, Andrei. V. Title. QA404.5.I587 2010 515.55—dc22 2009040384

Copying and reprinting. Material in this book may be reproduced by any means for educational and scientiﬁc purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the ﬁrst page of each article.) c 2010 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Copyright of individual articles may revert to the public domain 28 years after publication. Contact the AMS for copyright status of individual articles. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines

established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1

15 14 13 12 11 10

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With our best wishes to Guillermo L´ opez Lagomasino on the occasion of his 60th birthday.

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Contents Preface

ix

Guillermo L´opez Lagomasino: Mathematical life ´ n and Andrei Mart´ınez-Finkelshtein Francisco Marcella

1

A walk through approximation theory Bernardo de la Calle Ysern

25

Asymptotic uniqueness of best rational approximants to complex Cauchy transforms in L2 of the circle Laurent Baratchart and Maxim Yattselev

87

Quadrature rules on the unit circle. A survey. ´n Luis Garza and Francisco Marcella

113

On the multilinear trigonometric problem of moments Alberto Ibort, Pablo Linares, and Jose G. LLavona

141

Multiple orthogonal polynomial ensembles Arno B. J. Kuijlaars

155

Some equivalent formulations of universality limits in the bulk Eli Levin and Doron S. Lubinsky

177

Greedy energy points with external ﬁelds ´ pez Garc´ıa Abey Lo

189

On asymptotic behavior of Heine-Stieltjes and Van Vleck polynomials Andrei Mart´ınez-Finkelshtein and Evgenii A. Rakhmanov

209

Remarks on relative asymptotics for general orthogonal polynomials Edward B. Saff

233

Fine structure of the zeros of orthogonal polynomials: A progress report Barry Simon

241

A potential-theoretic problem connected with complex orthogonality Herbert Stahl

255

Orthogonal polynomials and approximation theory: Some open problems Walter Van Assche

287

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Preface This volume contains articles by speakers in the International Workshop on Orthogonal Polynomials and Approximation Theory, held on September 8-12, 2008, at the Legan´es campus of the Universidad Carlos III de Madrid, in honor of the 60th birthday of Professor Guillermo L´opez Lagomasino. The ﬁve-day conference attracted 140 participants from 22 countries. The program of the conference, video, and pdf ﬁles of most of the talks can be found at http://gama.uc3m.es. This volume’s articles cover a wide range of topics in the theory of Orthogonal Polynomials as well as in Rational Approximation with a special emphasis on their applications in random matrices, integrable systems, and numerical quadrature. The contributors are celebrated researchers in these domains. A brief account of the scientiﬁc contributions by Professor Guillermo L´opez Lagomasino constitutes one of the features of this volume. His activity in the former Soviet Union, in the framework of the group lead by A. A. Gonchar, as well as their connections with the Western groups working in Orthogonal Polynomials and Approximation Theory, fulﬁls a good historical perspective of the internationalization of these domains in the last 50 years. This is an example of how mathematicians are able to open physical and political barriers and contribute to the improved links between individuals and institutions. New results and methods are presented in the contributed papers as well as a careful choice of open problems to foster an interest in research in these mathematical areas in the coming years. The reader will ﬁnd survey presentations, an account of recent developments, and the exposition of new trends in the areas of Orthogonal Polynomials, Special Functions, and Approximation Theory, from a theoretical and an applied perspectives. As co-organizers of the workshop and editors of this volume it is our happy task to thank those individuals and institutions whose eﬀorts made it possible. First, we acknowledge Ministerio de Educaci´on y Ciencia of Spain (grant MTM2007-31221E), Proyecto Ingenio Mathematica (i-MATH)(grant C3-0116), Universidad Carlos III de Madrid, Universidad Polit´ecnica de Madrid, Real Sociedad Matem´atica Espa˜ nola, Sociedad Espa˜ nola de Matem´atica Aplicada, Ayuntamiento de Legan´es, and International Association for the promotion and cooperation with scientists (INTAS) for their ﬁnancial support. Second, it is a pleasure to thank all the members of the local organization Committee of the Universidad Carlos III de Madrid for the excellent organization of this meeting. Last, but certainly not least, we express our gratitude to the participants of the workshop who made it a memorable

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PREFACE

event, to the contributors to this volume, and to Christine Thivierge of the AMS staﬀ for her eﬃcient support in the production of these proceedings.

Jorge Arves´ u, Universidad Carlos III de Madrid Francisco Marcell´an, Universidad Carlos III de Madrid Andrei Mart´ınez Finkelshtein, Universidad de Almer´ıa and Instituto Carlos I de F´ısica Te´ orica y Computacional, Universidad de Granada.

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http://dx.doi.org/10.1090/conm/507/09953

Contemporary Mathematics Volume 507, 2010

Guillermo L´ opez Lagomasino: mathematical life Francisco Marcell´ an and Andrei Mart´ınez-Finkelshtein This paper is obviously dedicated to the 60th Birthday of our friend, Guillermo (Bill) L´ opez Lagomasino

Abstract. This is a very brief account of the biography of Guillermo L´ opez Lagomasino, up to the date of publication. Along with a description of the major events in Guillermo’s life, we try to describe both the social events and mathematical environment within which his mathematical contributions, analyzed in an accompanying paper by B. de La Calle, were obtained.

1. Introduction This paper, based on a presentation of the second author at the International Workshop on Orthogonal Polynomials and Approximation Theory 2008 (IWOPA’08), and available at http://gama.uc3m.es/iwopa_live, is a very brief account of Guillermo’s biography, with several digressions. This article will play a supporting role for the paper in this volume by B. de la Calle, where some of Guillermo’s mathematical contributions are discussed. We think that it is valuable to put them into the framework in which they were obtained. Moreover, Guillermo’s life has been anything but boring, and explaining brieﬂy the social contexts where he moved can also be illuminating. 2. First years Guillermo L´opez Lagomasino was born in Havana, Cuba, on December 21, 1948, a single child of an electrician (his father) and a schoolteacher (his mother). At the age of eight (in March, 1957, a few months after Fidel Castro with a bunch of people returned secretly from an exile in Mexico on a small yacht named Granma and disembarked in Cuba, starting a guerrilla war), his family emigrated to the United States. These were tough times, and Guillermo’s father, as so many people, 1991 Mathematics Subject Classiﬁcation. Primary 01A70. Secondary 01A60. The ﬁrst author was partially supported by the Direcci´ on General de Investigaci´ on, Ministerio de Educaci´ on y Ciencias of Spain, research project MTM-13000-C03-02. The second author was partially supported by Junta de Andaluc´ıa, grants FQM-229, FQM481, and P06-FQM-01735, as well as by the research project MTM2008-06689-C02-01 from the Ministry of Science and Innovation of Spain and the European Regional Development Fund (ERDF). c 2010 American c Mathematical 0000 (copyright Society holder)

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decided to look for a better life in the USA. They moved to Cleveland, Ohio, where Guillermo attended St. Stephen’s Primary School. There, he also learned to play the violin. During 7th and 8th grade he was the concert master of the school’s orchestra and fourth in the line-up of its baseball team. It is easy to deduce that mathematics was not his top priority yet, although Guillermo remembers himself as always interested in math, computation, and related issues.

Figure 1. From St. Stephen’s year book. Guillermo is the second from the left in the last row. In January 1959 the Cuban Revolution won. It was a nationalist and anticorruption movement, supported by a vast majority of Cuban population, sick and tired of General Fulgencio Batista’s regime. In its ﬁrst year, the new revolutionary government carried out measures such as the expropriation of private property with no or minimal compensation, the nationalization of public utilities, and began a campaign to institute tighter controls on the private sector such as the closing down of the gambling industry. The government also evicted from the island many Americans, including mobsters, who, in collusion with Batista’s administration, ran the gambling casinos in Havana, and whose presence in the city was quite notorious. However, the process became more radical along the next few years, in part by the escalation of the conﬂict with the Eisenhower administration, attempting to overthrow Cuba’s government by diﬀerent means, such as the Bay of Pigs Invasion in 1961 (at that time already sponsored by John F. Kennedy). Starting from 1959 many Cubans emigrated to the United States. The exodus that occurred immediately after the Cuban Revolution was primarily of the upper and middle classes. It is estimated that during the early years of the revolutionary period about 215.000 Cubans moved to the United States. All in all, at the end of Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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the academic year of 1962 Guillermo’s family decided to take an opposite direction and to return to Cuba. That turned out to be a diﬃcult task. First attempt was by ship, using the Spanish company “Compa˜ n´ıa Transatl´antica Espa˜ nola” operating two ships, Covadonga and Guadalupe, from New York. But a severe longshoremen strike created a dilemma whether to travel separately from the baggage, or to wait for a better occasion. Since Guillermo’s father didn’t ﬁnd the ﬁrst choice too attractive, they looked for Plan B. And here came the second surprise: the Cuban Missile Crisis, in October 1962, and the US naval blockade of Cuba. Traveling to Cuba in these conditions was out of question, and Guillermo’s family was obliged to extend their stay in the USA for another six months. With no job, no place to live, they moved to a relative’s place in Paterson, New Jersey. Still, regardless all the personal troubles that this situation meant, it could have been a lucky strike for Guillermo and his studentsto-be. Guillermo began Central High School in Paterson, where mathematics was taught at a much higher level. With all this hassle of moving he started school late, when classes were going on for a few weeks. It was a kind of shock for Guillermo, since he barely could understand anything, failing the ﬁrst exam that was given on a weekly basis. His pride was hurt, and he promised his Math professor, Mr. White, to get the maximum score next time. He studied the textbook the whole week and managed to get the maximum. In three weeks he was already helping his mates in Math. These months that Guillermo and his family were forced to stay in the United States probably laid the ﬁrst ground of his mathematical vocation. In March 1963, Guillermo returned to Cuba. As a curious anecdote, he ﬂew back on a direct ﬂight from Miami to Havana on a Pan American carrier with only four passengers (Guillermo’s parents, himself, and a son of a former army colonel of the deposed regime). The ﬂight was carrying food and medicine as part of the compensation the USA agreed to pay the Cuban government for the release of prisoners seized during the attack of Playa Gir´ on1. The trip was arranged by the International Red Cross for humanitarian reasons. Back in Cuba, he continued his studies at Secundaria B´ asica (Secondary School) “Carlos de la Torre”. He fell in love with proofs, especially in geometry (or better to say, planimetry), that he learned from a course that was taught by Luis Triana. He was dazzled by auxiliary constructions allowing to establish statements out of the hat. He also loved solving systems of equations and correlating this with geometric interpretations. At the end of the 9th grade Guillermo won a scholarship for talented students to study at Instituto Preuniversitario (High School) Cepero Bonilla where he obtained top marks. Instituto Pre-Universitario Especial “Ra´ ul Cepero Bonilla” (or “El Cepero”) was founded in 1963. Its foundation has a goal to form a new professional and scientiﬁc elite in Cuba. Its students were recruited among young people of outstanding performance, accepting to go through a much more advanced curricula in comparison with other high schools in Cuba. It had this special character until 1974. “El Cepero” was diﬀerent in many senses. First, its curriculum was much more advanced. For instance, Physics of the 11th grade was taught by [12], and occasionally included diﬀerential equations, so the Math course had to correspond. 1 Also known as the Bay of Pigs; in fact, 1113 prisoners were exchanged for US $53 million in food and medicine.

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Figure 2. Instituto Pre-Universitario Especial “Ra´ ul Cepero Bonilla” Actually, the last two years of the High School included many topics from the undergraduate university curricula. It was not only Physics and Math: advanced courses were taught in Biology, Chemistry, and even Literature2. Another feature was the quality of the students, really strong and motivated, so the environment was great. Cepero was a boarding school (this type of schools became very popular in Cuba), and cohabitation of so many talented and diﬀerent young people was a breeding ground for an interesting experience. Last but not least it was the quality of teachers, almost in all topics. However, according to Guillermo, in Math they were not the best (unlike in Chemistry or Physics), students used to engage in heated discussion with teachers who at the end let them do things their way. The diﬀerence was so notorious, that at some point Guillermo had doubts whether to study Mathematics or Physics. Nevertheless, this was deﬁnitely a good school for Guillermo. 3. Havana University: a student So, in 1967 Guillermo started his undergraduate studies at the Havana University, Faculty of Sciences, getting a degree in Mathematics in 1972. The School of Mathematics of the Havana University was created in 1962. Many mathematicians from diﬀerent countries took part in its development. Probably the ﬁrst important mathematician arriving in Havana after the revolution was Celiar Silva Rehermann, born in Uruguay and specialized in general topology, who stayed in Cuba initially from 1961 to 1969, and later from 1976 until he died in 2000. He played an important pedagogical role and was the Director of School of Mathematics in the beginning (from 1963 to 1968), when the exodus of a few existing professional mathematicians to the United States was especially important. One of the ﬁrst mathematicians (at least, as far as we are aware of) connected with the Russian (or Soviet) mathematical school was Carlos Vega, a Soviet-Spanish mathematician, well known in Spain for his translations of mathematical books from Mir Publishers; he came in the mid-sixties, and spent a long time in Cuba 2 Actually, instead of standard subjects like “Spanish” or “Literature” they had the “Literary anthology of culture”.

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teaching classical diﬀerential geometry and other topics. He was a graduate student of another Soviet mathematician, A. G. Kostyuchenko, from the Moscow State University, who was a specialist in Functional Analysis of Diﬀerential Operators. Kostyuchenko spent 3 months in Havana in the 1966–67 academic year, and remained connected to the Cuban Mathematical School for many years more. He had a very big inﬂuence on many Cuban mathematicians, such as Miguel Jim´enez Pozo, who at that time was already professor at the Havana University, and started teaching measure theory to his own mates being undergraduate (there was no one else able to do it). In 1968 several prominent French mathematicians attended a Cultural Congress in Havana, among them Didier Dacunha-Castelle, Marie Duﬂo (Professeur Emerite at the Universit´e Paris-Est, Marne-la-Vall´ee, France, now retired) and the Fields medallist and political activist Laurent Schwartz (professor from Universit´e Par´ıs 7). They launched the idea of a Liaison Committee whose goal was to support higher education in Cuba; Dacunha-Castelle became its ﬁrst president. Particularly in Mathematics (one of the most active branches) this Committee started by organizing several intensive Summer schools in Havana in 1968 and 69. Some French mathematicians came to Havana for a 2 year stay; among them were Claude George (in 1968, who taught analysis) and Claude Mutaﬁan (a mathematician and historian who specializes now in Armenian history, but who published at that time several books on algebra). Both had a great impact on the curricula of the Mathematics students at all levels. In particular, Mutaﬁan had the greatest inﬂuence on Guillermo’s instruction. According to Guillermo, Mutaﬁan was a wonderful educator. His style consisted in giving ﬁrst the heuristic and intuitive arguments related to a theorem, and once the audience was convinced, he gave the formal proof. Guillermo took from him both complex analysis (falling in love with the topic, something that can deﬁnitely be appreciated from Guillermo’s papers) and multilinear algebra. Since they had no textbook available, in the middle of the school year Mutaﬁan divided the topic in chapters and distributed them among his students as an assignment to prepare a text. He coordinated closely the work, and this experience became also very formative for Guillermo. Other French mathematicians had important inﬂuence in other branches of Math or in diﬀerent universities, such as above mentioned Marie Duﬂo (Statistics) and Marie Cottrell (formerly, Coret), lecturing at the Pedagogical Institute “Enrique Jos´e Varona” in Havana (who is now Professeur at Universit´e Paris 1), and who inﬂuenced applied mathematics and probability. The Liaison Committee brought to Cuba not only French mathematicians. Diederich Hinrichsen, from the Federal Republic of Germany (FRG), was contacted by Dacunha-Castelle, who was looking for somebody to teach general topology. He arrived in Havana in 1970 for a one year stay, starting a regular and fruitful relationship with Cuban mathematicians. Among other activities, Hinrichsen gave lectures and run regular seminars both at Havana University and Pedagogical Institute, wrote lecture notes, later published as textbooks (that became very popular and relevant, especially in the situation of scarce bibliography), and was involved in the reform of mathematical education3. German mathematicians also brought 3 Hinrichsen played also a relevant role in improving computer skills of Guillermo in the late eighties, helping him in smuggling Guillermo’s ﬁrst personal computer across the West-East Germany border.

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a new air. In a clear contrast with the abstract Bourbaki-like teaching of Claude George and some others, they introduced a diﬀerent didactical style, emphasizing intuitive ideas, illustrative examples and applications of the general theory. In July 1970 an organization similar to the French Liaison Committee was founded in FRG by scientists and students who sympathized with the Cuban revolution. It was called “Kowizuku” (Komitee fuer die wissenschaftliche Zusammenarbeit mit Kuba = Comittee for scientiﬁc collaboration with Cuba). Its ﬁrst Secretary General was Klaus Krickeberg (at that time, professor at the Univeresity of Bielefeld), replaced by Diederich Hinrichsen shortly after his return from Cuba. The aim of the committee was two-fold: a scientiﬁc one (send qualiﬁed personel for long and short term stays according to the requests of Cuban universities, sending books, journals and laboratory equipment to Cuba) and a political one (provide authentic information about Cuba and counteract the strong anti-Cuban propaganda in the west-german media). The committee had more than 100 members and was organized in local groups at 12 universities in the FRG. Scientiﬁc cooperation was mainly with the University of Havana, but to a lesser degree also with the universities of Las Villas and Oriente. The main areas of collaboration were Mathematics, Human Genetics, Bioquemistry, and Geography/Oceanography. The activity of both committees helped many Cubans to ﬁnish their graduate studies in Western Europe, or even to receive some specialized medical help not available at that time in the country. But the intense relation of mathematicians from France and FRG with Cuba did not last too long. On the French side, practically after leaving Mutaﬁan and George nobody replaced them. In fact, Mutaﬁan and others visited Cuba again several times, but for short periods. No more Summer Schools were organized. Also Kowizuku stopped its activities around 1975; after that, the scientiﬁc collaboration with Cuba continued mainly on an individual basis, partially supported by DAAD (German service for academic exchange). There are possibly several reasons explaining this change, such as the ideological evolution of the Cuban process and some other political issues that created tension, in particular with FRG. Now the basic help came from the Eastern Europe, especially Germany (GDR), with mathematicians like Olaf Bunke, Bernd Bank and J¨ urgen Guddat, and the Soviet mathematical school. Among Cuban mathematicians, Miguel Jim´enez played a major role. He taught measure theory and analysis at the Havana university, being one of the most prominent educators in the School of Mathematics, exerting great inﬂuence on his students (both Guillermo and the second author). But let us get back to Guillermo at the Havana University; he had an intense life during his undergraduate studies, devoting his time not only to mathematics. Along with his studies, he was very active in the University Student’s Federation (FEU) where he was elected member of the Universities’ Bureau and President of the FEU of the Faculty of Sciences during his last two academic years. Before the Cuban Revolution of 1959, students joined diﬀerent organizations, aligning themselves directly or indirectly with some political party. The strongest of all these organizations was the FEU (Federaci´on Estudiant´ıl Universitaria or University students federation) created by Julio Antonio Mella, a co-founder of the Cuban Communist Party in the 1920s. The European tradition of college-based political activism and the alleged corruption of Cuban political parties at the time

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turned the FEU into the most inﬂuential of Cuban political organizations before 1959. After the coup d’´etat by Fulgencio Batista in 1952 when democratic elections were suspended, the violent clashes between university students and Cuban police reached its extremes. Students known to be members of the FEU were tortured and killed, and the organization reacted with an irregular war in Havana, aiming mainly (and eventually succeeding) to assassinate police oﬃcers of high rank. After the assault on the Moncada barracks by Fidel Castro (who graduated from Havana University School of Law, and who had contacts in the FEU), this organization became an ally of Castro’s new July 26th Movement, though there were discrepancies between the leaders in the form that the forthcoming revolution should be carried out. While Fidel Castro was leading the guerrilla war in the Sierra Maestra mountains, the FEU, lead by Jose Antonio Echeverria, attempted to kill Fulgencio Batista in an armed assault at the Cuban Presidential Palace on March 13, 1957. Batista escaped by pure chance, and many student assaultants died in the action, as did Echeverria himself. During the months that followed, the police executed many of the students that lead the failed coup. President Batista ordered the university to be closed, and it remained so until Batista ﬂed the country and Fidel Castro entered Havana on January 1st, 1959.

Figure 3. Havana University Since the triumph of the Cuban Revolution all other student political organizations in the university were suppressed, except FEU. However, although ideologically dependent from the Youth Communist organization of Cuba, FEU could preserve its own traditions and goals, centering its activity on education, sports, culture and other aspects of the university life. Guillermo ﬁnished his undergraduate studies in 1972 being the President of FEU of the Faculty of Science of the Havana University and member of its Board. He achieved excellent grades which allowed him, after graduation, to stay at the Havana University, obtaining a position at the Department of Mathematics of the Faculty of Sciences. Since no elections of FEU were scheduled for that time nor Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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they could ﬁnd replacement for Guillermo in his duties, he had to stay another 6 months after graduation being President of this student organization at his place of work. During his undergraduate studies, besides mathematics and his student organization activities, there was another issue that kept him busy: in 1968 he decided to marry Maria Elena Garc´ıa. No doubt this wise decision had the greatest impact on the rest of his mathematical life. 4. Ph.D. in Moscow After two years working in that faculty, he embarked for the Soviet Union to carry out postgraduate studies. Guillermo was very lucky to start to work under the supervision of a really outstanding mathematician: Andrei A. Gonchar, in that time already head of the section on the theory of functions of one complex variable at the Steklov Mathematical Institute and a corresponding member of the Soviet Academy of Sciences.

Figure 4. A.A. Gonchar It deﬁnitely was luck. In 1972 a mathematician from Bielorussia named V. N. Rusak visited Havana and gave a one month course in approximation theory, leaving a list of research problems. Next year, when he visited Havana again, Guillermo had the problems solved, which granted him a Master degree. On the other hand, two of his colleagues from the Havana University (Concepci´on Vald´es and Carlos S´ anchez) started their Ph.D. in Moscow before Guillermo; they heard about Gonchar and mentioned his name in a conversation with Guillermo. All the ingredients were there. Guillermo arrived in Moscow in 1974. One of the ﬁrst things that stroke Guillermo (and the same happened to the second author when he followed his Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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steps years later) was the research seminar (“The Seminar”) on current problems in complex analysis at the Steklov Institute. Gonchar led (and still leads) the seminar on a weekly basis, every Monday afternoon. He organized it in 1972 after a trip where he met Hans Wallin and learned from him about the Pad´e approximation construction. Gonchar immediately recognized it as a very promising subject and just after he returned to Moscow he founded his seminar. So, its birth has been intrinsically associated with Pad´e and rational approximation. The seminar has been going on without interruption all these years (although the time schedule has changed: at the beginning it started at 3 p.m. and now it starts at 6 p.m. and ﬁnishes at 9 p.m. with a break of 1/2 hour).

Figure 5. Talking at the blackboard in Moscow It is a highly competitive, very professional and extremely broad seminar. Many mathematicians regard it as an honor to give a talk there. Typically, either some external speakers were invited (from other parts of the Soviet Union or from abroad), not necessarily specialists in approximation theory, or regular members of the seminar who volunteered to speak could do it in a rather informal way. But informal did not mean sloppy. On the contrary, the speaker usually was interrupted with questions digging deeper in the statements, proofs or antecedents, and you rarely could get away with just general sentences. Although from a diﬀerent period, the second author remembers his ﬁrst visit to Gonchar’s seminar: somebody was presenting on the blackboard his idea of the connection of the recurrence relation satisﬁed by the general orthogonal polynomials and the discretization of a diﬀerential equation, and how to deduce a Mehler-Heine asymptotics from it. He was submitted to a detailed cross-examination, far from a much gentler Cuban tradition this author was accustomed to. It turned out later that the speaker was A. Aptekarev, the audience so insistently trying to ﬁnd a mistake in his arguments were A. Gonchar, E. Chirka and others, and all were on perfectly good terms. The list of active participants is always impressive, and should have been even more impressive then. Take note that at that time the following mathematicians were Gonchar’s Ph.D. students, and used to take active part in the seminar: Serguei Suetin, Ralitza Kovacheva, Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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Evgenii Rakhmanov, Valery Vavilov, and Vasily Prokhorov. Actually, this seminar also played an additional important role: it was a mandatory ﬁlter for any Ph. D. student. If you were able to present your results and survive the seminar, you were ready for the defense. But let us go back to Guillermo’s Ph. D.; in order to get the feeling of the environment found by Guillermo upon his arrival in Moscow we need to make some mathematical digression and explain some concepts. In the second half of the seventies there were several centers in the Soviet Union active in approximation theory: Moscow, Leningrad, Armenia, Georgia, Ukraine, Bielorussia, Azerbaidzhan. Moscow was obviously a very important one, and Gonchar and his group were prominent players in rational approximation, developing mostly the analytic theory. In particular, they were interested in a fundamental problem of the constructive theory of functions: the connection between analytic properties of a function and the rate of its approximation. What were the main tools and methods in a standard toolbox of any constructive approximator of that time? We can ﬁnd part of the answer in the book of Walsh [15]. The logarithmic potential theory, which is a blend of real and complex analysis, was already playing a role, but not to the extent it gained later. It was perfectly known that logarithmic potential has a direct connection with polynomials and rational functions, and the following electrostatic problem has its origins in the works of Gauss (like almost all problems): let K be a compact set in C, and let M(K) denote the set of all probability (i.e., positive unit) measures supported on K. The logarithmic potential associated with µ ∈ M(K) is 1 V µ (z) = log dµ(t), |z − t| which is harmonic outside the support S(µ) of µ and is superharmonic in C. The energy of such a potential is deﬁned by 1 µ = I(µ) = V (z) dµ(z) dµ(t)dµ(z). log |z − t| The electrostatics problem involves the determination of def

ρK = inf {I(µ) : µ ∈ M(K)} , which is called the Robin constant for K. Then, the logarithmic capacity of K, denoted by cap(K), is deﬁned by cap(K) = e−ρK . def

If ρK = +∞, we set cap(K) = 0. Such sets are called polar and they are very “thin”. A fundamental theorem of Frostman [4], 1935, asserts that if cap(K) > 0, there exists a unique measure µK ∈ M(K) such that I(µK ) = ρK . This extremal measure is called the equilibrium measure (or Robin measure) of K. The potential V µK associated with µK is the equilibrium potential (or conductor potential ) of K. Some basic facts about cap(K) and V µK are: • µK is supported on the outer boundary ∂∞ K of K and practically ﬁlls it: cap (∂∞ K \ S(µK )) = 0; • V µK (z) = ρK quasi-everywhere (i.e., everywhere except possibly a polar set) on S(µK ). Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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11

S(µK )

Ω

K

Figure 6. Support of the equilibrium measure. There is an important relation between the equilibrium potential and the notion of Green function, that is, the function g(·, ∞) that is harmonic in the unbounded component Ω of C \ K, such that limz→∞ (g(z) − log |z|) exists and is ﬁnite, and such that g is continuous in the closed domain Ω and equals zero on its boundary. Such function, if it exists, is unique. In such a case, V µK (z) = ρK − g(z, ∞). A related problem is to determine the minimal sup-norm on K for monic polynomials of degree n: τn (K) := min pK , n p(z)=z +...

where f K = maxz∈K |f (z)|. Let K contain inﬁnitely many points (which is always the case if cap(K) > 0). Then for every n there is a unique monic polynomial Tn (z) = z n + . . . such that Tn K = τn (K). It is called the n-th Chebyshev polynomial for K. The Chebyshev constant of K is def

τ (K) = lim τn (K)1/n n n (this limit always exists). If we write Tn (z) = k=1 (z − ζk ), then (4.1)

|Tn (z)|

1/n

= exp (−V νn (z)) ,

where 1 δζk n n

(4.2)

νn =

k=1

is the normalized zero counting measure associated with the polynomial Tn . A fundamental theorem of Classical Potential Theory states that cap(K) = τ (K), and if cap(K) > 0, then lim |Tn (z)|1/n = exp (−V µK (z)) n

locally uniformly in Ω. Moreover, the normalized zero counting measures for Tn converge in the weak-* sense to µK . Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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An example of an application of the classical potential theory to polynomials is the following Bernstein-Walsh Lemma: if a polynomial p of degree n satisﬁes |p(z)| ≤ M for z ∈ K, then |p(z)| ≤ M eng(z,∞) , z ∈ Ω. These tools allow also to study the polynomial approximation of analytic functions. Let C \ K be connected; if f ∈ C(K) and def

en (f ; K) = min f − pK , p∈Pn

where Pn is the class of all algebraic polynomials of degree ≤ n, then en (f ; K) → 0 as n → ∞. However, we can characterize the rate of convergence in terms of the analytic continuation of f (the so called direct theorems). Namely, if for R > def 0, ΓR = {z : g(z, ∞) = R} and R(f ) denotes sup R such that f admits analytic continuation to the interior of ΓR , then, according to Walsh, lim sup en (f )1/n = e−R(f ) . n

The world of rational approximation, at the time comparatively new to mathematicians, turned out to be considerably diﬀerent from the world of polynomial approximation. Rational approximation is much richer and much more diﬃcult. Besides deciding where to approximate and with which degree (as in the polynomial case) we must say something about poles. Should we ﬁx them somewhere and prescribe their behavior? Or should we leave them free and allow them to ﬁnd their places according to some external rule? Gonchar knew that rational approximation with free poles leads to essentially new results in the study of global properties of analytic functions through the properties of their best approximations. In the early 1970s, coinciding with Guillermo’s arrival in Moscow, the attention of Gonchar and his students was drawn to rational analogues of power series, namely, Pad´e approximations and their various generalizations, in the development of whose analytic theory Guillermo’s Ph.D. makes a contribution. The need for this research was dictated by the numerous successful applications of such approximations in various applied problems of mathematical physics and other areas of science; the number of similar applications was enormous, but it was without any rigorous theoretical justiﬁcation, which essentially detracted from the reliability of the results obtained in this vein. We mentioned already that Gonchar learned about Pad´e approximation from Hans Wallin (at least, according to Sergey Suetin) in 1972. His ﬁrst paper on this subject was [5] and appeared in 1973, a year before Guillermo started his graduate studies in Moscow. Extending methods of potential theory to rational approximation requires generalizing several concepts. Observe that if R(z) = P1 (z)/P2 (z) is a rational function, where P1 and P2 are monic polynomials of degree n, then one can write 1 log |R(z)| = V ν1 (z) − V ν2 (z), n where ν1 and ν2 are the normalized zero counting measures for P1 , P2 , respectively. The right-hand side represents the logarithmic potential of the signed measure ν = ν1 − ν2 , and we need to develop a theory of such potentials. It was probably T. Bagby who introduced ﬁrst [1] in 1967 the concept of capacity of a condenser. Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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µ∗1

13

µ∗2

K1 K2

Figure 7. A condenser. Let K1 , K2 be two closed sets that are a positive distance apart. The pair (K1 , K2 ) is said to be a condenser and the sets Ki are the plates. Consider the signed measure µ = µ1 − µ2 such that µi ∈ M(Ki ). Then µ(C) = 0. A nontrivial fact is that in this case the energy I(µ) > 0, and we may consider the minimization problem def ρ(K1 , K2 ) = inf I(µ) over all neutral signed measures of the above form. Again, the inﬁmum is attained at a unique signed measure µ∗ = µ∗1 − µ∗2 (called the equilibrium measure of the condenser) and the condenser capacity is 1 def . cap(K1 , K2 ) = ρ(K1 , K2 ) ∗

The condenser potential V µ has the feature that it is constant on each plate: ∗

V µ = c1 on K1 ,

∗

V µ = c2 on K2

(at least quasi everywhere), so that ρ(K1 , K2 ) = c1 − c2 (diﬀerence of the potential on the plates). We now turn to rational approximation. Let K ⊂ C be compact. We denote by Rn the collection of all rational functions of the form R = P/Q, where P , Q are polynomials of degree at most n, and Q has no zeros in K. For f analytic in a domain containing K, let def

rn (f ; K) = inf f − rK r∈Rn

be the error in best approximation of f by rational functions from Rn . Clearly, since polynomials are rational functions, we have rn (f ) ≤ en (f ). A basic theorem regarding the rate of rational approximation was proved by Walsh. The following statement is a special case of this theorem: let K be a single Jordan arc or curve and let f be analytic on a simply connected domain D ⊃ K. Then lim sup rn (f )1/n ≤ exp {−1/cap(K, ∂D)} . n

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This bound is sharp. However, this bound is not totally satisfactory: it can be proved that in many situations it is still valid if we replace rn (f ) by en (f ). But the family Rn contains twice the number of free parameters than Pn . This motivates the following conjecture of Gonchar: under the assumptions above, lim inf rn (f )1/n ≤ exp {−2/cap(K, ∂D)} . n

Actually, this rate of convergence is achieved by suitable sequences of Pad´e approximants. Remember that the classical Pad´e approximation generalizes the concept of the Taylor expansion. Given a formal power series (4.3) f (z) = fk z k , k≥0

for any pair of indices m, n ∈ N ∪ {0} we may ﬁnd polynomials Pn,m ∈ Pn and Qn,m ∈ Pm , Qn,m ≡ 0, such that Qn,m (z)f (z) − Pn,m = O z n+m+1 . The rational def

function πn,m = [n, m]f = Pn,m /Qn,m is uniquely deﬁned and is the [n, m] Pad´e approximant to f . The deﬁnition goes back to 1892, and basic algebraic properties of the Pad´e table [n, m], m, n ≥ 0, have been studied by many mathematicians along the XIX-th century (Frobenius, Cauchy, Jacobi, Chebyshev). If we want to study the convergence of this construction to the approximated function f , we need to decide ﬁrst how our parameters m, n tend to inﬁnity. We have the following obvious choices: Row sequences

Ray sequences

[n, m]f Diagonal Figure 8. Pad´e table. • Rows: we ﬁx m and make n → ∞. In particular, taking m = 0 yields the Taylor expansion. • Rays: we assume that both m and n tend to inﬁnity in such a way that the limit lim m/n = λ exists. • Diagonal: we take m = n, or at least λ = 1 in the previous situation. Each scheme has its own features. For instance, the rows should work to approximate in disks, when we know a priori the number of poles our function f . The classical result here is: Theorem (R. de Montessus de Ballore, [2]): let Dm be the largest disk centered at the origin where f can be continued meromorphically with ≤ m poles. It f has in Dm precisely m poles (counting their multiplicity), then πn,m converge to Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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15

f (as n → ∞) on each compact subset K of Dm not containing the poles of f . Furthermore, under these assumptions we can even estimate the rate of convergence: 1/n

lim sup πn,m − f K ≤ max |z|/Rm , z∈K

n

where Rm is the radius of Dm . This statement is known as a direct result, where we assert something about convergence knowing the properties of functions to be approximated. In practice a very big role play the so-called inverse results. By construction, Pad´e approximants are locally the best rational approximations to a given power series. We have seen that they are built directly in terms of its coeﬃcients and enable us to realize an eﬃcient analytic continuation of the series beyond its circle of convergence. In a sense the poles of the approximants localize the singular points (including the poles and their multiplicities) of the extended function in the corresponding domain of convergence and on its boundary. The last property of the Pad´e approximants is based on the fact that all their poles are “free” and are determined only by the condition that the tangency to the given power series be maximal. For this reason, the Pad´e approximants diﬀer substantially from rational approximants whose poles are ﬁxed (completely or partially), and in particular from polynomial approximations, in which case all poles are ﬁxed at inﬁnity. A typical inverse result is Theorem (Fabry, [3]): if the coeﬃcients of the power series (4.3) satisfy lim n

fn fn+1

= a,

then z = a is a singular point of f . What is the relation to Pad´e approximants? It is quite easy to show that the ratio in the left hand side above is the pole of the Pad´e approximant [n, 1]f . So, this is an inverse result about the ﬁrst row of the Pad´e table that was substantially generalized later by Gonchar and his school (with contributions from V. Buslaev, V. Prokhorov, S. Suetin, V. Vavilov, and Guillermo himself, see e.g. [14]. As it was said above, the row sequences of Pad´e approximants are primarily suitable for the description of meromorphic continuations of a series (4.3) to the corresponding circles. If the boundary of the circle of convergence of this series contains some singularity of another character, for instance, a branch point, then no row sequence is eﬃcient in solving the problem of analytic continuation. It is quite another matter when treating diagonal sequences of Pad´e approximants. One of the ﬁrst results of general nature on convergence of these rational approximants of analytic functions is the classical Markov theorem [7] obtained in terms of Chebyshev continued fractions for functions of the form (Markov function) dµ(t) def (4.4) µ (z) = t−z where µ is a positive Borel measure with S(µ) R: Theorem: for a function µ of the form (4.4) with compact support S(µ) R consisting of inﬁnitely many points, the diagonal Pad´e approximants [n/n]µ constructed from the coeﬃcients of the expansion of µ in a Laurent series at the point z = ∞ converge to µ uniformly on compact subsets of the domain C \ [a, b], where [a, b] is the convex hull of S(µ). Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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´ AND A. MART´INEZ-FINKELSHTEIN F. MARCELLAN

It is a natural extension to this problem to consider a rational perturbation to a Markov function, that is, the case f = µ + r, where r is a rational function. It is well known that the construction of Pad´e approximations is essentially non-linear, and therefore the study of the convergence of these approximations for the “perturbed” function is an extremely complicated problem. Gonchar himself, and later Rakhmanov, worked on this problem, and it is completely natural that Guillermo got interested in these questions, in particular, related to the so called multipoint Pad´e approximants, when you interpolate not at a single point (origin or inﬁnity), but at a table of nodes. All in all, in March 1978 Guillermo defended his Ph. D. thesis under the supervision of Academician Andrei A. Gonchar. The title of his Memoir was “On the Convergence of Multipoint Pad´e Approximants”. 5. Back in Havana

Figure 9. Havana University, Faculty of Mathematics On his return to Havana, Guillermo competed and won successively positions as Assistant Professor (1978–1983), and later, Full Professor (1983–1995). For several years, he was head of the Department of Function Theory. During these years Guillermo developed an intensive work leading the mathematical research in Cuba. This period of time was very fruitful from the scientiﬁc point of view and also for his international mathematical contacts. Guillermo devoted an important time to editorial activities. In 1980, taking advantage of the support of the Havana University, together with Miguel Jim´enez and some other colleagues, he started the journal Revista Ciencias Matem´ aticas, that is still active. Guillermo had to devote time to university administration too. He was the viceDean for research of the Faculty of Mathematics. He deﬁnitely was very dedicated, and did a lot of good job, but he was peculiar. First, he was doing active research. Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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17

He managed to save at least one day a weak when he stayed home and worked. That wasn’t an easy task, since usually meetings were scheduled every day. On the other hand, everybody knew that Guillermo is absent-minded. They took it as a feature of a typical mathematician; because of that many missed meetings were forgiven, after Guillermo accepting that he just forgot. Still now we have some doubts if he is as absent minded as he pretends to be, or it was a trick to skip some unpleasant meetings and duties. As it was mentioned, in that post-doctoral period Guillermo had also intense international contacts. Just after returning from Moscow with his Ph.D. he met for the ﬁrst time Ed Saﬀ at the International Congress of Mathematicians in Helsinki (1978). In 1981 Guillermo attends the International Conference on Constructive Approximation celebrated in Varna, Bulgaria, where he meets Paul Nevai, who at that time was struggling with the extension of the Szeg˝o theory. The series of works of Mat´e–Nevai–Totik had an important impact on Guillermo’s own work. In 1983 Guillermo attends also the International Congress of Mathematics in Warsaw, Poland, where he met some Spanish mathematicians, such as the ﬁrst author (FM), Jaime Vinuesa and Jos´e Javier (Chicho) Guadalupe. There is a curious story about how Guillermo and FM met: both attended Jenya Rakhmanov’s talk and both wanted to discuss something with the speaker afterwards. FM was faster than Guillermo, so that Guillermo had to wait for his turn. Meanwhile he realized that Jenya was not too ﬂuent in English and that they were having some communication problems. So he volunteers to translate from Russian into English. After a while FM looked at Guillermo’s badge, saw “Cuba” on it and asked bewildered “Why are we speaking English?”. That was the way they met. The same year, 1983, Guillermo traveled for the ﬁrst time to the USA (after so many years), to the “International Conference on Rational Approximation and Interpolation” organized in Tampa by Ed Saﬀ. Among other important events that Guillermo attended there were also the International Congress on Orthogonal Polynomials and their Applications in Bar le Duc, France (1984) and the International Congress on Orthogonal Polynomials and their Applications in Segovia, Spain (1986). Actually, Guillermo played an important role in arranging the participation of E. Rakhmanov at this conference. The diﬃculties to plan the invitation and organizing his trip can be a subject of a thriller. But probably one of the most visible international achievements of Guillermo and his colleagues at the Havana University at that time was the organization of the International Seminar “Approximation and Optimization”. The idea of an international conference on constructive approximation to be held in Havana was initially proposed by the Bulgarian mathematician Vasil Popov to Guillermo L´ opez Lagomasino and Miguel Jim´enez in 1979 during his visit to Cuba. It took a long period of time to organize it, since the plan was ambitious, and the diﬃculties enormous. The ﬁrst conference on Approximation and Optimization was held in Havana, Cuba, in January, 1987. The conference was jointly organized by the Havana University, the Academy of Sciences of Cuba, and the Cuban Society of Mathematics and Computer Science, with the collaboration of various institutions of the former socialist countries and the support of the International Mathematical Union. It

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´ AND A. MART´INEZ-FINKELSHTEIN F. MARCELLAN

was a great success, gathering about 100 researchers; the Soviet Union, the USA, German Democratic Republic and Hungary were particularly well represented. In the list of participants we ﬁnd many outstanding mathematicians (some of them attended this workshop also), such as B. Bank and J. Guddat (GDR), F. Marcell´an, J. Vinuesa (Spain), W. Dahmen, D. Hinrichsen (FRG), T. Erdelyi, J. Szabados, P. Vertesi (Hungary), A. Gonchar, S. Nikolski, B. Pobedria, E. Rakhmanov, S. Stechkin, V. Temliakov, V. Tikhomirov (USSR), W. Fuchs, W. Gragg, E. Saﬀ, Ch. A. Micchelli, P. Nevai (USA), L. Lorch (Canada), P. Petrushev (Bulgaria), A. Ronveaux, A. Draux (Belgium), apart from obviously a lot of Cuban mathematicians (the list is far from complete). The second author (AMF) also took part as an undergraduate student, and it was his ﬁrst international meeting. It was not trivial at all to organize an event of this scale in the conditions of Cuba and embargo. A good example is the nice story Ed Saﬀ told us during his talk at the conference about the aﬃdavit he had to sign in order to get permission to travel, assuring the Department of States that his research has “no practical value”. The proceedings of this International Conference were published by Springer Verlag in the series “Lecture Notes in Mathematics”, Vol. 1354. Later on, the mathematicians J¨ urgen Guddat and Hubertus Th. Jongen promoted the idea of organizing a series of international conferences in the Caribbean region as a continuation of the ﬁrst one held in 1987. An Executive Committee was founded and the series had received its present title of International Conferences “Approximation and Optimization in the Caribbean”. The second conference in 1993 took place also in Havana. The third, fourth and ﬁfth conferences were held in Puebla (Mexico, 1995), Caracas (Venezuela, 1997) and Point-a-Pitre (Guadeloupe, 1999), respectively. The sixth, seventh and eighth conferences took place in Guatemala City (Guatemala, 2001), Leon (Nicaragua, 2004), and Santo Domingo (Dominican Republic, 2006). The last one, the Ninth International Conference, was celebrated in March 2-7, 2008 in San Andr´es Island, Colombia. The next one, unless changes, will be in 2010 in Costa Rica. In 1987 Guillermo and Miguel Jim´enez took an active part in the organization of the International Mathematical Olympiad (IMO) in Havana. It was attended by 237 participants from 42 countries (among them, Terry Tao). These years Guillermo was honored with several distinctions, such as the Award Pablo Miquel (1984) given by the Cuban Mathematical Society for his overall work; and a Special Distinction (1986) given by the Minister of Education to the best researchers among university professors. Guillermo also was adviser for his ﬁrst 3 Ph.Ds: • Ren´e Piedra de la Torre, 1986. • Jes´ us Ill´an Gonz´alez, 1987. • Ren´e Hern´ andez Herrera, 1987. 6. His second doctorate Guillermo took leave from his responsibilities to visit during 1987–1989 the Steklov Mathematical Institute, USSR, as a Visiting Research Fellow. During his stay he defended a second doctoral thesis entitled “Rational Approximation of Meromorphic Functions of Stieltjes Type”.

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19

There were really big changes in comparison with Guillermo’s ﬁrst visit, both in the social and mathematical sense. First, it was time of “perestroika” and “glasnost”, and there is no need to explain all its social implications and all positive and negative impact it had on the population in general. In the mathematical framework there were very visible changes in the ﬁeld of constructive approximation. One of the most relevant was the maturity of the potential theory as an analytic tool for rational approximation and orthogonal polynomials. The major breakthrough occurred in the ﬁrst half of the eighties in the works of Gonchar and Rakhmanov in the Soviet Union, and Mhaskar and Saﬀ in the USA. Both teams came independently to the notion of equilibrium in the presence of an external ﬁeld that allowed to tackle multiple problems related with the so called varying orthogonality and multipoint Pad´e approximation. Let again K be a compact set in C, M(K) the set of all probability measures supported on K, and assume now that we have a (nice) real-valued function Q deﬁned on K, that we call the external ﬁeld. Consider the modiﬁed energy integral for µ ∈ M(K), 1 IQ (µ) = dµ(t)dµ(z) + 2 Q(t) dµ(t) = (V µ + 2Q) dµ. log |z − t| The electrostatic problem involves again the determination of def

ρK (Q) = inf {IQ (µ) : µ ∈ M(K)} ; then the weighted capacity is deﬁned by cap(K; Q) = e−ρK (Q) . def

Under suitable restrictions on Q, there exists a unique measure µQ ∈ M(K), called the weighted equilibrium measure (or equilibrium measure in the presence of the external ﬁeld Q), such that IQ (µQ ) = ρK (Q). It can be characterized by several properties. For instance, in the “regular” case the equilibrium potential V µQ satisﬁes = const, z ∈ S(µQ ), µQ V (z) ≥ const, z ∈ K. Alternatively, µQ satisﬁes the following min-max property: (6.1)

min (V µQ + Q) (x) = max min (V µ + Q) (x).

x∈K

µ∈M(K) x∈K

Unlike the unweighted case, the support of µQ not necessarily coincides with ∂∞ K and, in fact, it can be quite an arbitrary closed subset of K. Determining this set is typically one of the most important aspects of weighted potential theory. An important tool for that is the so-called F -functional of Mhaskar and Saﬀ, F (E) = log cap(E) − Q dµE , E

where µE is the standard (Robin) equilibrium measure for E. Mhaskar and Saﬀ showed that max F (E) = F (∂∞ S(µQ )) . E⊂K

This fact was established ﬁrst in the paper [9] from 1985, with a fancy name Where does the sup norm of a weighted polynomial live? For K ⊂ R this functional yields Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

´ AND A. MART´INEZ-FINKELSHTEIN F. MARCELLAN

20

formulas for the endpoint of the support of µQ that appeared earlier in the work [6] of Gonchar and Rakhmanov in 1984. Both teams showed how to use this modiﬁed equilibrium in the analysis of deep problems from approximation theory and orthogonal polynomials. Typically, the external ﬁeld appears where a varying weight occurs. For instance, imagine we want to analyze the asymptotic property of the sequence Pn (z) = z n + . . . of monic polynomials satisfying the following orthogonality property: Pn (x)xj wn (x) dx = 0, j = 0, 1, . . . , n − 1. K⊂R

Here w > 0 is a certain positive function (the weight), but due to the power n the weight varies together with the degree of the polynomial Pn . We can guess the answer by following heuristic arguments: if Pn (x) = nj=1 (z − ζj ), then like in (4.1)–(4.2), 1/n = exp (−V νn (z)) , |Pn (z)| where νn is the normalized zero counting measure associated with the polynomial Pn . We know that orthogonality implies minimization in the L2 norm, so that we are interested in the problem νn minn P wn/2 2 = min e−n(V +Q) 2 P (x)=x +...

L

ν

L

where we have denoted Q = − 12 log w. If we replace the L2 norm by the L∞ norm, and consider the problem not only for the discrete zero counting measures, but for all probability measures on K, then we arrive at our min-max problem (6.1). One of the key ideas in this ﬁeld is that even when we don’t have a varying weight, we can “manufacture” it, and this, instead of making the problem more complicated, solves it. One of the ﬁrst examples was the solution of several conjectures of Freud (in 1982 by Rakhmanov [11], and in 1984 by Mhaskar and Saﬀ [8], with earlier contributions by Nevai-Dehesa [10] and Ullman [13]): if for c > 0 and γ > 1, Pn satisﬁes γ Pn (x)xj e−cx dx = 0, j = 0, 1, . . . , n − 1, def

R

then a simple rescaling x → tn1/γ yields an equivalent condition

γ n Pn tn1/γ tj e−ct dt = 0, j = 0, 1, . . . , n − 1, R

and we are in business. There are much more sophisticated tricks allowing to “generate” varying weights and reduce the problem to a weighted logarithmic equilibrium that are mentioned in the contribution by B. de la Calle in this volume. Obviously, once the ice of the classical potential theory was broken nothing could stop people from ﬁnding further useful generalizations of the extremal problems in potential theory. Gonchar and Rakhmanov formulated the vector equilibrium problem, that is a key for the asymptotic analysis of the Hermite-Pad´e approximants, Stahl showed how to carry out this machinery to the complex plane and the non-hermitian orthogonality, Rakhmanov found a connection between discrete orthogonality and equilibrium with constraints, and so on. Summarizing, although the ﬁrst visit of Guillermo to Moscow (for his Ph.D.) was during the beginning of this rebirth of the rational approximation theory, the second stay, with the purpose Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

´ GUILLERMO LOPEZ LAGOMASINO: MATHEMATICAL LIFE

21

to get his second doctorate, was in the middle of the maturity of the potential theory and coincided with a hectic activity in this ﬁeld. In 1989 Guillermo and his family returned to Havana, where he continued his important mathematical activity and international contacts. Despite of ﬁnancial diﬃculties, he managed to attend the US-USSR International Conference on Approximation Theory, Tampa, 1990, organized again by tireless Ed Saﬀ. That conference gathered teams on rational approximation and orthogonal polynomials from both countries almost in complete, with some additional invited participants, such as Guillermo, who on the other hand was representing in justice the Soviet school. 7. Spanish connection Guillermo visited Spain during the eighties in the framework of an agreement between the Universidad Complutense de Madrid and the Havana University. He gave several seminars in the Department of Mathematical Analysis and met some prominent Spanish analysts like F. Bombal, J. L. Gonz´ alez-Llavona, J. Carrillo, B. Rubio, among others. In 1991–1992 he spent a sabbatical year at Escuela T´ecnica Superior de Ingenieros Industriales, Universidad Polit´ecnica de Madrid, Spain, sponsored by the Spanish Ministry of Education, and one semester at Universidad Carlos III de Madrid, 1992–1993. His ﬁrst long visit to Spain was very productive, starting a fruitful scientiﬁc collaboration with the ﬁrst author (who was his counterpart in the grant) and the Spanish school of orthogonal polynomials. During his sabbatical in Spain he co-organized with Jaime Vinuesa, Chicho Guadalupe and the ﬁrst author a Summer School in Laredo, in September of 1992. We managed somehow to get enough funds to be able to invite (paying all expenses) a bunch of Cuban mathematicians.

Figure 10. The Cuban team in Laredo, Spain, in 1992.

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´ AND A. MART´INEZ-FINKELSHTEIN F. MARCELLAN

Guillermo and his family left Cuba permanently in 1995, not so much for political, but mostly for personal reasons. The procedures in Cuba were very strict, and after insisting for a while in his return, the Havana University authorities made a sad decision to formally expel Guillermo from the list of their faculty members, despite all his contributions to Cuban mathematics. In 1994 Guillermo obtained the National Order “Carlos J. Finlay”, awarded by the State Council of Cuba to the best scientists of the country. But he was unable to attend the ceremony, and it physically never reached his hands, either due to a series of unfortunate coincidences, or later, being already “banned”. During the next six years Guillermo was a “permanent” Visiting Scholar (almost an oxymoron) at the Department of Mathematics of the Universidad Carlos III de Madrid. Having obtained the Spanish citizenship in 2000 he applied and obtained, ﬁrst, a permanent position as Associate Professor in 2001 and a Full Professorship in 2002. Besides that, Guillermo kept traveling. In 2000 he was a Visiting Scholar at the University of South Florida, USA, and in 2004 a Visiting Scholar at Vanderbilt University, USA, each time for one semester. During his ﬁrst visit to Tampa, he rented a room from Maria Carvalho, a former secretary to Ed Saﬀ, having Vilmos Totik as a roommate. Vilmos remembers from that time: No particular story, but we have lots of memories, for we used to go for some weekends to Marco Island to Guillermo’s cousin house. We have shared lots of meals together - I recall that every 2 weeks or so Guillermo came home with excitement from Publix that he found another wonderful tropic root - to me all of them looked liked potato. Guillermo also has a long standing collaboration with the Portuguese-Spanish team (Amilcar Branquinho and Ana Foulqui´e), visiting in several occasions the Universities of Coimbra and Aveiro for periods ranging between 3 to 5 months. In fact, this collaboration started during his ﬁrst long visit to Spain, when both Amilcar and Ana were working on their Ph.D. under the ﬁrst author’s supervision. We asked Amilcar about his experience: Guillermo has been in the genesis of the Portuguese olympic team (Project Delfos), and he recommended Valeri Vavilov as a very experienced trainer for math competitions. This was a great success, according to the results that Portugal has achieved lately. Another story is that in one occasion I had to leave for a short trip, asking Guillermo to cover my classes. When the head of the Math Department approached me concerned with this decision, I told him that I wish I could attend these classes. Guillermo has had many other activities on the international scene, organizing conferences and meetings, inviting scholars, and probably the most notorious one, being an active editor of the Journal of Approximation Theory. Being already in Spain he supervised (or co-advised) the rest of his Ph. D. students, some of them from Cuba. We count 7, beside the other 3 mentioned above: • Manuel Bello Hern´andez, 1996, • Francisco Cala Rodr´ıguez, 1997, • H´ector Pijeira Cabrera, 1998, Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

´ GUILLERMO LOPEZ LAGOMASINO: MATHEMATICAL LIFE

23

• Abel Fern´ andez Infante, 1998, • Angel Ribalta, 2000, • Bernardo de la Calle Ysern, 2000, • Ulises Fidalgo Prieto, 2004, although he was also involved indirectly in the doctoral dessertations of many other mathematicians. Guillermo has also found time for university administration: during the last three years he has been Head of the Department of Mathematics at Univ. Carlos III de Madrid. Recently, he was reelected for another two years term. 8. Conclusions It is a diﬃcult and unfair task to write a mathematical biography of somebody in the middle of his scientiﬁc life, announcing regularly new papers and new great results (if you are an approximator, keep track of the advances in the normality of the Nikishin systems, this is almost a scoop). Nevertheless, we wanted to outline some major stages that were clearly decisive in his professional activity, those related with Cuba, the place where he was born and grew up, and Russia (former USSR), his second homeland, at least from the mathematical point of view. Being Guillermo a friend, a colleague and a collaborator, we want to ﬁnish this note wishing him a long and productive life, for his friends’, his family’s, and our personal beneﬁt. Acknowledgments and disclaimer We are grateful to our colleagues Miguel A. Jim´enez Pozo, Diederich Hinrichsen, and Sergey P. Suetin, who provided many historical details and helped us to close some gaps in this sense; the former two also read the ﬁrst draft of this paper and made a number of corrections and remarks, improving the readability. This article should not be regarded as a serious study in the History of Mathematics; obviously, we take the whole responsibility for possible omissions and mistakes. References [1] T. Bagby. The modulus of a plane condenser. J. Math. Mech., 17:315–329, 1967. [2] R. de Montessus de Ballore. Sur les fractions continues alg´ebriques. Bull. Soc. Math. France, 30:28–36, 1902. [3] E. Fabry. Sur les points singuliers d’une fonction donn´ ee par son d’eveloppement de Taylor. ´ Ann. Ecole Norm. Sup. Paris, 13(3):367–399, 1896. [4] O. Frostman. Potentiel d’´equilibre et capacit´ e des ensembles avec quelques applications ` a la th´ eorie des fonctions (Thesis). Meddel. Lunds Univ. Mat. Sem., 3:1–118, 1935. [5] A. A. Gonˇcar. The convergence of Pad´e approximations. Mat. Sb. (N.S.), 92(134):152–164, 167, 1973. [6] A. A. Gonchar and E. A. Rakhmanov. Equilibrium measure and the distribution of zeros of extremal polynomials. Mat. Sbornik, 125(2):117–127, 1984. translation from Mat. Sb., Nov. Ser. 134(176), No.3(11), 306-352 (1987). [7] A. A. Markov. Deux d´emonstrations de la convergence de certaines fractions continues. Acta Math., 19:93–104, 1895. [8] H. N. Mhaskar and E. B. Saﬀ. Extremal problems for polynomials with exponential weights. Trans. Amer. Math. Soc., 285:204–234, 1984. [9] H. N. Mhaskar and E. B. Saﬀ. Where does the sup norm of a weighted polynomial live? (A generalization of incomplete polynomials). Constr. Approx., 1:71–91, 1985. [10] P. Nevai and J. S. Dehesa. On asymptotic average properties of zeros of orthogonal polynomials. SIAM J. Math. Anal., 10:1184–1192, 1979.

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´ AND A. MART´INEZ-FINKELSHTEIN F. MARCELLAN

[11] E. A. Rakhmanov. On asymptotic properties of polynomials orthogonal on the real axis. Math. USSR Sb., 47:155–193, 1984. [12] R. Resnick and D.Halliday. Physics for Students of Science and Engineering. John Wiley and Sons, 1960. [13] J. L. Ullman. Orthogonal polynomials associated with an inﬁnite interval. Michigan Math. J., 27:353–363, 1980. [14] V. V. Vavilov, G. Lopes, and V. A. Prohorov. An inverse problem for the rows of a Pad´ e table. Mat. Sb. (N.S.), 110 (152)(1):117–127, 160, 1979. [15] J. L. Walsh. Interpolation and Approximation by Rational Functions in the Complex Domain, volume 20 of Colloquim Publications. Amer. Math. Soc., Providence, 1960. Department of Mathematics, University Carlos III de Madrid, Spain E-mail address: [email protected] Department of Statistics and Applied Mathematics University of Almer´ıa, SPAIN, ´ rica y Computacional, Granada University, SPAIN and Instituto Carlos I de F´ısica Teo E-mail address: [email protected]

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http://dx.doi.org/10.1090/conm/507/09954 Contemporary Mathematics Contemporary Mathematics Volume 507, 2010

A Walk through Approximation Theory Bernardo de la Calle Ysern Dedicated to Guillermo L´ opez Lagomasino on the occasion of his sixtieth birthday

Abstract. A survey of the work of Professor Guillermo L´ opez Lagomasino is presented. His thirty years of research on approximation theory, orthogonal polynomials, and related topics let us highlight some of the most interesting subjects in the ﬁeld.

1. Introduction Guillermo L´opez Lagomasino celebrated his sixtieth birthday on December 21, 2008. He was born in Havana, Cuba, where he spent the ﬁrst part of his childhood, living later in the United States for six years. Back in Cuba, he obtained a degree in Mathematics at Havana University in 1971 and, after that, moved to the former Soviet Union in order to continue his studies, presenting his Ph. D. thesis, with Gonchar as advisor, at Moscow State University in 1978. Then, G. L´ opez carried out an intense academic and research work in Cuba as well as several research visits to Madrid and Moscow. In 1995 he moved to Spain, obtaining Spanish citizenship in 2000. Since 1995 he has worked at Carlos III University in Madrid. His almost 90 research papers published until now deal with a rich variety of problems and greatly reﬂect his life and work since, in such papers, one can notice a strong inﬂuence from the Russian school on rational approximation and orthogonal polynomials and, in the second part of his career, many connections with the Spanish school on orthogonal polynomials. The results obtained by G. L´ opez can be basically classiﬁed in one (or more) of the following subjects: • Pad´e approximation of Cauchy transforms. • Orthogonal polynomials. • Hermite-Pad´e approximation. • Quadrature rules. Such division corresponds with the sections of this paper. Clearly, all the subjects are interconnected and many cross-references will be made. 2000 Mathematics Subject Classiﬁcation. Primary 42C05, 41A21; Secondary 41A20, 65D32. Key words and phrases. Orthogonal polynomials, Pad´ e approximants, varying measures, Hermite-Pad´ e approximation, quadrature rules, Sobolev orthogonal polynomials. This work was supported by the Direcci´ on General de Investigaci´ on, Ministerio de Educaci´ on y Ciencia, under grants MTM2006-13000-C03-02 and MTM2006-07186 and by UPM through Grupo Consolidado “Teor´ıa de Aproximaci´ on Constructiva y Aplicaciones”. 1

c 2010 American Mathematical Society

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B. DE LA CALLE YSERN

L´ opez’ work is characterized by a strong use of certain techniques coming from orthogonal polynomials, classical analysis, potential theory, linear operator theory, etc... Except for a few brief comments about some basic facts of the theory of orthogonal polynomials, there is no intention here to explain such notions since they can be easily found elsewhere. In particular, any result on linear operator theory used in L´ opez’ work appears in [GoGo, Kat] whereas [ST, Ra] are excellent books on potential theory and its connection with approximation theory. For a short introduction on potential theory, see [Mar06]. As for information on orthogonal polynomials, the reader may consult the monographs [Sz75, N, Si05a]. On the other hand, L´ opez’ work presents many technical diﬃculties mostly related to the concept of varying measures. To overcome them, a number of conditions are used. This survey tries to explain those ideas and the connections between them, focusing on L´opez’ very own concepts. At the same time, an eﬀort has been made to write it in such a way that it appears as self-contained as possible. Results by other people are mentioned when they help to understand the development of the theory and appreciate the subsequent results. Naturally, it has not been possible to include each and every paper by L´ opez. In order to keep the survey within reasonable limits and due to its length, some interesting subjects have had to be left out. Among them, we can mention the study of the convergence of rows of Pad´e approximants [L81a, L99d] and an analog of a classical theorem by Hadamard [L03b]. Equally, other results that could have been included somewhere have not been commented on, like the extension of the simultaneous approximation to the setting of orthogonal expansions [L07b, L09b], the construction of quadrature rules for complex weight functions [L95f, L96a], and the application of simultaneous approximation to quadratures [L04b, L07c]. Our criteria and the space needed to present the results have both inﬂuenced the choices made. 2. Pad´ e approximation of Cauchy transforms We are concerned with the approximation of a formal power series of the type (2.1)

f (z) =

∞ cm m+1 z m=0

which will usually represent a function holomorphic in an unbounded region of the complex plane C. For each given nonnegative integer n there are polynomials pn and qn of degree at most n such that qn ≡ 0 and (2.2)

qn (z) f (z) − pn (z) = O(1/z n+1 ),

z → ∞.

The ratio pn /qn of any two such polynomials deﬁnes a unique rational function πn which is called the n-th (diagonal) Pad´e approximant of f developed at z = ∞. The choice of the point z = ∞ in the deﬁnition above is made for technical reasons and no essential diﬀerences appear when taking instead any ﬁnite point of the domain of f . The function πn can also be deﬁned as the rational function of order at the most n which has maximal order of contact (within the class of all such functions) with the function f at the point z = ∞. In this respect, Pad´e approximants are the rational counterparts of Taylor polynomials. Unlike them, the convergence of the sequence of Pad´e approximants to f is a subtle and diﬃcult matter (see [Bak]). For instance, there exist entire functions whose sequence of Pad´e approximants diverges Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

A WALK THROUGH APPROXIMATION THEORY

27 3

at every point of the complex plane. On the other hand, Pad´e approximants may converge to the function f in regions much larger than Taylor polynomials do. The following theorem obtained by Markov [Mark] is one of the ﬁrst results of general character on the convergence of such approximants. Let µ be a ﬁnite positive Borel measure whose support, denoted by Σ, is a compact subset of the real line R. Let ∆ be the convex hull of Σ. The so-called Markov function µ is the Cauchy transform deﬁned by dµ(x) µ (z) = , z ∈ C \ Σ. Σ z−x It is clear that the function µ is of the type (2.1) since it may be expressed as ∞ cm µ (z) = , cm = xm dµ(x), m = 0, 1, . . . m+1 z Σ m=0 Theorem 2.1 (Markov Theorem). The sequence {πn }n∈N of Pad´e approximants of µ at z = ∞ converges uniformly to µ on compact subsets of the domain C \ ∆. The key step in proving Markov’s Theorem is the fact that the denominators qn of the Pad´e approximants turn out to be polynomials orthogonal with respect to the measure µ, that is, it holds (2.3) qn (x) xν dµ(x) = 0, ν = 0, 1, . . . , n − 1. Σ

In particular, the poles x1 , . . . , xn of the Pad´e approximant πn belong to ∆. Additionally, the approximant πn admits the decomposition πn (z) =

(2.4)

n i=1

λn,i , z − xn,i

where λn,i , i = 1, . . . , n, are the Christoﬀel numbers of the Gauss-Jacobi quadrature corresponding to the measure µ. Hence, λn,i > 0, i = 1, . . . , n, and the family {πn }n∈N is normal in C \ ∆ whence the proof follows. Let gK stand for the maximum absolute value of the continuous function g on the compact set K. A straightforward application of the Maximum Principle gives the estimate 1/2n (2.5) lim sup µ − πn K ≤ e−τ , τ = inf gC\Σ (z, ∞), z ∈ K , n→∞

where K is any compact subset of C\ ∆ and gΩ (z, ∞) stands for the Green function of the region Ω. When Σ is an unbounded set of R the corresponding Cauchy transform is called a Stieltjes function for historical reasons. In that case, for convenience, we will denote the measure involved by σ instead of µ. Now, in order to prove convergence of {πn }, deﬁned in terms of the moments {cm }, the measure σ needs to be determined by the moments, which does not hold in general. Under that assumption, the rest of the proof runs along the same lines as that of Markov’s Theorem except for the fact that σ is not analytic on a neighborhood of z = ∞ which provokes some technical diﬃculties (see [Sti]). Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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Theorem 2.2 (Stieltjes Theorem). Suppose that the moment problem for the measure σ is determinate. Then, the sequence {πn }n∈N of Pad´e approximants of σ at z = ∞ converges uniformly to σ on compact subsets of the domain C \ ∆. There implicitly appear two leitmotifs in this type of results. First, Pad´e approximation of Cauchy transforms leads to dealing with some kind of orthogonality. Second, convergence of the Pad´e approximants is possible provided that we are able to maintain the poles of the approximants under control (see, for instance, Theorem 3.5 and Lemma 2.4 below). 2.1. Meromorphic Stieltjes functions. In the classical theorems mentioned above we ﬁnd the simplest possible situation. Let us describe a much more sophisticated one whose solution was given by L´opez in [L89b]. We have opted to begin with that work because it is possible to ﬁnd in it the most common ingredients and diﬃculties in Pad´e approximation and shows how varying measures come into the game and help to solve it. Let σ be a measure supported on Σ = [0, +∞) such that all its moments cm are ﬁnite numbers. Consider the Pad´e approximation of the function ∞ dσ(x) + r(z), f (z) = z−x 0 where r is a rational function whose poles lie in C \ [0, +∞) and r(∞) = 0. The situation is then as follows: it is supposed that we know the structure of the function, that is, f is a meromorphic Stieltjes function on C \ [0, +∞) with a ﬁnite number of poles. However, we do not know either the measure σ or the rational function r. We would like the approximants πn to look for and ﬁnd the poles of r and converge to the function f . For that, a ﬁnite number of poles of the Pad´e approximant should converge to the poles of r whereas the rest would draw the positive semi-axis. Is that possible? We already know that the measure σ must be required to be determined by its moments cm , m = 0, 1, . . . . A suﬃcient condition for that, given by Carleman, is (2.6)

∞

1 = +∞. √ cm

2m

m=1

Roughly speaking, the Carleman condition means that the measure σ does not have much weight at inﬁnity and it behaves as a measure with a compact support. What is ﬁrst done in [L89b] is a change of variable given by the M¨obius transformation ζ = (z − 1)/(z + 1), z ∈ C \ [0, +∞), in order to pass to the equivalent problem of approximating the function 1 d σ (x) + r(z), f(z) = z −x −1 by means of Pad´e approximants π n developed at z = 1. The perturbation r is again a rational function with poles outside [−1, 1]. The new measure σ is supported on [−1, 1] and veriﬁes that the transformed moments 1 d σ (x) cm = (1 − x)m −1 Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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are ﬁnite numbers, m = 0, 1, . . . . The Carleman condition (2.6) is transformed into ∞

(2.7)

1 √ = +∞, cm

2m

m=1

which analogously says that the measure σ does not have much weight at z = 1. Notice that the interpolation is carried out at a point in the support of the measure σ . So, it remains to be a pure Stieltjes situation. It turns out that the denominator of π n , the polynomial qn , satisﬁes orthogonality relations with respect to the measure σ but they are far from being standard. Namely, if the polynomial td of degree d is the denominator of r, it holds 1 d σ (x) (2.8) xj qn (x) td (x) = 0, j = 0, 1, . . . , n − 1 − d. (1 − x)2n −1 The orthogonality relations are not complete due to the term −d, they depend on the degree of the polynomial qn , that is, they are varying, and, ﬁnally, they are complex because of the presence of the polynomial td . So, we have a complex varying measure with incomplete orthogonality relations. The trick to overcome all these problems is to consider the orthonormal polynomials with respect to the positive part of the varying measure. Thus, let ln,m (x) = κn,m xm + . . . , κn,m > 0, be the m-th orthonormal polynomial with respect to d σ (x)/(1 − x)2n . Due to the orthogonality relations (2.8), we have (2.9)

qn (x) td (x) =

d k=−d

cn,k ln,n+k (x) ⇐⇒

d ln,n+k (x) qn (x) td (x) = . cn,k ln,n+d (x) ln,n+d (x) k=−d

In the above formula the number of terms in the linear combination is independent of n. This is important since information about { qn } may be obtained provided we know the ratio asymptotics of the varying orthogonal polynomials {ln,m }. The next result is the key step. Theorem 2.3 ([L89b]). Suppose that σ > 0 almost everywhere on [−1, 1] and Carleman condition (2.7) holds. Then ln,n+j+1 (z) = z + z 2 − 1, j ∈ Z, lim n→∞ ln,n+j (z) on √ compact subsets of the domain C \ [−1, 1], where the square root is taken so that 1 = 1. Once ratio asymptotics of {ln,m } is obtained, it can be proved that, given any compact set K ⊂ C\[−1, 1], at most d zeros of qn lie on K as n tends to inﬁnity. This and Theorem 2.3 are enough to prove convergence in capacity of the approximants { πn } to the function f inside the region C \ [−1, 1]. Convergence in capacity is an analog of convergence in measure but using the logarithmic capacity to measure the size of the sets. To be more precise, the sequence of functions {gn }n∈N is said to converge in capacity to the function g inside the region G if, for any ε > 0 and any compact K ⊂ G, it holds lim cap{z ∈ K : |gn (z) − g(z)| > ε} = 0,

n→∞

where capA stands for the logarithmic capacity of the set A. Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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A number of results in Pad´e approximation show that convergence in capacity may be considered as the natural convergence for Pad´e approximants (see, for instance, [Pom] and [St]). The ﬁnal step in the proof needs the following result on convergence in capacity proved by Gonchar in [G75b]. Lemma 2.4 (Gonchar’s Lemma). Suppose that the sequence {gn }n∈N of functions deﬁned on the domain G converges in capacity to a function g inside G. Then the following assertions hold true: - If the functions gn , n ∈ N, are holomorphic in G, then the sequence {gn } converges uniformly on compact subsets of G. - If each function gn is meromorphic and has no more than d < +∞ poles in G and the function g is meromorphic and has exactly d poles in G, then all gn , n ≥ N , also have d poles in G; the poles of gn tend to the poles z1 , , . . . , zd of g (taking account of their orders) and the sequence {gn } tends to g uniformly on compact subsets of the domain G \ {z1 , , . . . , zd }. The Gonchar Lemma basically says that convergence in capacity plus control over the poles gives uniform convergence. With the aid of that lemma, we can conclude that the sequence of Pad´e approximants π n converge to the function f uniformly on compact subsets of C \ ([−1, 1] ∪ { r = ∞}), which in turn proves the following result. Theorem 2.5 ([L89b]). Suppose that σ > 0 almost everywhere on [0, ∞) and the Carleman condition (2.6) holds. Then - Each pole of the rational function r attracts, as n → ∞, as many poles of πn as its multiplicity. - The sequence of Pad´e approximants πn converges to the function σ +r uniformly on compact subsets of C \ ([0, ∞) ∪ {r = ∞}). The analogous problem for meromorphic Markov functions was solved by Gonchar in [G75a]. The essential diﬀerence there is that, under the hypothesis µ > 0 a. e. on [−1, 1], ratio asymptotics of orthogonal polynomials is obtained whereas a similar property for orthogonal polynomials on the positive semi-axis does not hold; this is what makes the problem for non-compact supports so diﬃcult. L´ opez’ idea was to transform the problem in order to keep using ratio asymptotics of orthogonal polynomials in a compact support but related to a varying measure. Rakhmanov [R77b] showed that convergence does not hold for arbitrary positive measures µ and general rational function r; this is due to the possible bad behaviour of the poles of the approximants. If the coeﬃcients of r are required to be real then we can assume that td (x) > 0, x ∈ [−1, 1]. The resulting orthogonality conditions are 1 xj qn (x) td (x) dµ(x) = 0, j = 0, 1, . . . , n − 1 − d. −1

Hence, n − d of the poles of the rational approximants πn are in [−1, 1] and it is possible to obtain convergence of the sequence of πn to the function f without any restriction on the measure µ. This result is also due to Rakhmanov [R77b]. Later, L´opez [L81b] extended Rakhmanov’s result to meromorphic Stieltjes functions only assuming that Carleman’s condition (2.6) is satisﬁed. The proof loosely follows the classical proofs of the Markov and Stieltjes Theorems. As in [R77b], if the coeﬃcients of r are real, the denominator qn of the approximant Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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can be represented as qn = qn,1 qn,2 , where deg qn,2 ≤ d and the zeros of qn,1 are simple and belong to [0, +∞). Now, the following decomposition of the type (2.4) is obtained n

(2.10)

pn,1 (z) λn,i = , qn,1 (z) z − xn,i i=1

n ≥ n − d,

where pn,1 is a certain polynomial associated to qn,1 . The number of positive λn,i is at least n − d, since, as in (2.4), the decomposition (2.10) induces a nearly Gaussian rule. The fraction pn,1 /qn,1 is closely related to the error of the approximants πn to the function σ + r. So, it is necessary to deal with the problem of the zeros of qn,1 having negative coeﬃcients λn,i as well as that of the poles of πn (zeros of qn,2 ) lying in C \ [0, +∞). Fortunately, the number of points with these undesired properties does not depend on n which allows the author to obtain some estimates of the error which, because of Carleman’s condition, are suﬃcient to prove convergence in capacity and, therefore, uniform convergence. Theorem 2.6 ([L81b]). Let σ be a measure with support [0, +∞) for which the Carleman condition (2.6) holds. Suppose that the coeﬃcients of the rational function r are real. Then - Each pole of r attracts, as n → ∞, as many poles of πn as its multiplicity. - The sequence of Pad´e approximants πn converges to the function σ +r uniformly on compact subsets of C \ ([0, ∞) ∪ {r = ∞}). A slightly stronger result than Theorem 2.6 may be found in [L83]. 2.2. Multi-point Pad´ e approximation and varying measures. We have seen how relevant varying measures and orthogonal polynomials may be for Pad´e approximation. Let us take a closer look at them. In [L89b] the varying measure arises from a change of variable which transforms an unbounded set into a compact one. We will see this became a common strategy to be used in a number of papers. The other source of varying measures is multi-point Pad´e approximation. To ﬁx ideas, consider a Markov function µ with compact support Σ ⊂ R and ∆ being the convex hull of Σ. Now, let us pose the problem of interpolating µ along a table of points A = {a2n,i , i = 1, . . . , 2n, n ∈ N} given by the zeros of the polynomials with real coeﬃcients

2n x (2.11) A2n (x) = 1− , a2n,i ∈ C \ ∆, i = 1, . . . , 2n, n ∈ N. a2n,i i=1 We can assume that A2n (x) > 0, x ∈ ∆. It is easy to verify that for each n ∈ N there exists a unique rational function πnA = pn /qn , where pn and qn satisfy the following conditions: (i) deg qn ≤ n, deg pn ≤ n − 1, and qn ≡ 0. (ii)

− pn qn µ is an analytic function on C \ ∆. A2n

(iii)

(z) − pn (z) qn (z) µ = O(1/z n+1 ), A2n (z)

z → ∞.

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The rational function πnA is called the multi-point Pad´e approximant of µ which interpolates the function µ at the zeros of the polynomial A2n . Now, the denominator qn of the approximant πnA is orthogonal with respect to the varying measure dµ(x)/A2n (x), that is, dµ(x) = 0, ν = 0, 1, . . . , n − 1. qn (x) xν (2.12) A 2n (x) Σ When a2n,i = ∞ for all i = 1, . . . , 2n, the polynomial A2n is equal to 1 and we regain the orthogonality (2.3) and the classical Pad´e approximation. A number of cases may be distinguished according to the location of the points {w2n,i }. Firstly, the interpolation may be carried out along a table of points A = {a2n,i , i = 1, . . . , 2n, n ∈ N} compactly contained in C \ ∆. That is the easiest case since we are interpolating well inside the region of analyticity of µ and no additional conditions are required to obtain convergence. If we put πnA (z)

=

n i=1

λn,i , z − xn,i

it follows from condition (iii) that n p(xn,i ) dµ(x) = , (2.13) p(x) λn,i A (x) A 2n 2n (xn,i ) Σ i=1 for any polynomial p of degree not greater than 2n − 1, whence the numbers {λn,i } are positive and, therefore, the family πnA is normal. The rest of the proof consists basically in the use of the Maximum Principle for subharmonic functions. Theorem 2.7 ([L78]). Suppose that the table of interpolation points A is contained in the compact set F ⊂ C \ ∆, then 1/2n µ − πnA K ≤ e−τ , τ = inf gC\Σ (z, ζ), z ∈ K, ζ ∈ F , lim sup n→∞

where K is any compact subset of C \ ∆ and gΩ (z, ζ) stands for the Green function of the region Ω with singularity at the point ζ. Let us pose the corresponding problem of multi-point Pad´e approximation of a Markov meromorphic function µ + r. The construction of the approximants πnA of the function µ + r is analogous to those of µ with obvious modiﬁcations. If the rational perturbation r has complex coeﬃcients, following Gonchar’s work [G75a], we need the denominators qn to verify ratio asymptotics. Construct the table of orthogonal polynomials with respect to the varying measures dµ(x)/A2n (x), i.e., let ln,m (x) = κn,m xm + . . . , κn,m > 0, be the m-th orthonormal polynomial with respect to the measure dµ(x)/A2n (x). It is clear, due to (2.12), that the polynomial qn is equal to ln,n except for a constant factor. Theorem 2.8 ([L87b]). Let µ be a positive measure supported on [−1, 1] such that µ > 0 a.e. on [−1, 1]. Suppose that the table of points A is contained in a compact subset of C \ [−1, 1]. Then ln,n+j+1 (z) = z + z 2 − 1, j ∈ Z, lim n→∞ ln,n+j (z) on compact subsets of the domain C \ [−1, 1]. Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

A WALK THROUGH APPROXIMATION THEORY

33 9

Although varying measures implicity appear since the ﬁrst works by L´ opez (cf. (2.13)), it is in [L87b] where he develops for the ﬁrst time an independent theory on orthogonal polynomials with respect to varying measures. The main result is precisely Theorem 2.8 from which the following result follows in a similar fashion as Theorem 2.5 follows from Theorem 2.3. Theorem 2.9 ([L87b]). Under the conditions of Theorem 2.8, it holds - Each pole of the rational function r attracts, as n → ∞, as many poles of πnA as its multiplicity. - The sequence of Pad´e approximants πnA converges to the function µ +r uniformly on compact subsets of C \ ([−1, 1] ∪ {r = ∞}). Secondly, let us consider the case when the points {a2n,i } approach the support of the measure or even lie on it. Put Σ = [−1, 1] and remove the condition a2n,i ∈ [−1, 1], i = 1, . . . , n, from the deﬁnition of the table A. Natural requirements are 1 dµ(x) < +∞, n ∈ N. (2.14) A2n (x) ≥ 0, x ∈ [−1, 1], and −1 A2n (x) Once a measure µ satisfying (2.14) is chosen, the essential condition is given by (2.15)

lim

n→∞

2n

1 − |Ψ(a2n,i )|−1 = +∞,

Ψ(z) = z +

z 2 − 1.

i=1

Limit (2.15) means that the zeros of A2n as a whole do not tend to [−1, 1] very quickly. If we keep thinking in terms of interpolation, condition (2.15) makes sense; we do not want a lot of interpolation points near the set where the function µ behaves bad. Finally, a technical condition is needed for positive integers k, namely −1 1 k 1 − x dµ(x) ≤ M < +∞, n ∈ N. (2.16) a2n,i −1 i=1 We will say that (µ, A, k) is admissible if conditions (2.14), (2.15), and (2.16) are satisﬁed. The following result is a generalization of Theorem 2.8. Theorem 2.10 ([L89a]). Let µ be a positive measure supported on [−1, 1] such that µ > 0 a.e. on [−1, 1]. Suppose that (µ, A, 2k) is admissible for all k ∈ N. Then ln,n+j+1 (z) = z + z 2 − 1, j ∈ Z, lim n→∞ ln,n+j (z) on compact subsets of the domain C \ [−1, 1]. The most diﬃcult case is when many of the points in A are on the support of the measure; see, for instance, (2.8). Then we need the measure not to put much weight where we are interpolating; that is what Carleman’s condition ensures and Theorem 2.3 is obtained. Actually, Theorem 2.3 was obtained under weaker assumptions. Assume that a2n,i ∈ [1, +∞] for all i = 1, . . . , 2n and n ∈ N. Set

−1 1 m x 1− cn,m = dµ(x), m = 1, . . . , 2n, n ∈ N. a2n,i −1 i=1 Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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The numbers {cn,m } may be considered as generalized moments which provide information of the measure µ and the table A at the same time. We will say that (µ, A, m) satisﬁes the generalized Carleman condition if it holds (2.17)

cn,m ≤ Mm < +∞,

n ∈ N,

and

lim

n→∞

2n

1 = +∞. √ cn,m

2m

m=1

Condition (2.17) has the nice property of being weaker than the Carleman and admissibility ones. In [L89b], Theorem 2.3 was proved requiring (2.17) for all m ∈ N (cf. Theorem 4 and Corollary 1 therein). In [L79] a generalization of the Stieltjes Theorem for multi-point Pad´e approximants is obtained. The interpolation points {a2n,i } belong to the interval [−∞, a], a < 0, whereas the measure σ is supported on [0, +∞). Obvious modiﬁcations are needed in order to properly deﬁne the rational interpolants πnA . Also, since the point of the measure’s support where interpolation may be carried out has changed from z = 1 to z = ∞, condition (2.17) must be transformed by deﬁning the moments ∞ xm dσ(x) (2.18) cn,m = , m = 0, 1, . . . , 2n − 1, n ∈ N. m+1 0 i=1 (1 − x/a2n,i ) ∞ Notice that cn,m ≤ 0 xm dσ(x) < +∞ for all n ∈ N. Theorem 2.11 ([L79]). Let σ be a measure with support [0, +∞). Suppose that lim

n→∞

2n−1

1 = +∞ √ cn,m

2m

m=1

with numbers {cn,m } given by (2.18). Then, the sequence of multi-point Pad´e uniformly on compact subsets of approximants πnA converges to the function σ C \ [0, ∞). For a recent extension of Theorem 2.11, see Theorem 3.3 in [BeMi]. Summing up, it is possible to obtain convergence results if either the speed of approach of the interpolation points to the singularities of the function is not very quick or the measure does not have much weight where we are interpolating. If we face a case where neither of these two conditions are fulﬁlled we still have hope of proving convergence using the mixed-type condition (2.17) or an analog of it. How are these hypotheses used to prove ratio asymptotics of varying orthogonal polynomials in Theorems 2.3, 2.8, and 2.10? The way of proceeding is similar in all cases. The problem is ﬁrstly solved for polynomials orthogonal with respect to a (varying) measure supported on the unit circle. Let us mention the main idea of the proof in the simplest situation, corresponding to Theorem 2.8. For each n ∈ N, let Wn (z) = z n + . . . be a monic polynomial whose zeros wn,1 , . . . , wn,n , lie on the unit disc D = {z ∈ C : |z| < 1}. Let ρ be a positive Borel measure on the unit circle T. If ρn is a sequence of positive Borel measures on T, ∗ by ρn −→ ρ, we denote the weak convergence of ρn to ρ as n tends to inﬁnity. This means that for every continuous 2π-periodic function f on [0, 2π] 2π 2π lim f (θ) dρn (θ) = f (θ) dρ(θ). n→∞

0

0

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A WALK THROUGH APPROXIMATION THEORY

35 11

Consider the polynomials ϕn,m (z) = αn,m z m + . . . , αn,m > 0, which are orthogonal with respect to the measure dρ/|Wn |2 normalized by 2π dρ(θ) 1 |ϕn,m (z)|2 = 1, z = eiθ . 2π 0 |Wn (z)|2 The ﬁrst and most important step in the proof is to obtain the following theorem on weak star convergence of varying measures, which generalizes a result very well known in the classical setting. Theorem 2.12 ([L87b]). Suppose that the zeros of the polynomials {Wn }n∈N lie in a compact subset of D. Then, for any integer j, it holds |Wn (z)|2 ∗ dθ −→ dρ(θ), |ϕn,n+j (z)|2

z = eiθ .

Obviously, Theorem 2.12 relies on Geronimus’ formula [Ger] 2π 2π dθ dρ k z = zk , k ∈ Z, |k| ≤ m, 2 2 |ϕ (z)| |W n,m n (z)| 0 0 where z = eiθ . Once this formula is applied, the problem can be reduced to proving that a certain sequence of analytic functions {en }n∈N tends to zero uniformly on compact subsets of D, where each en equals zero at the points wn,i , i = 1, . . . , n. This is done using that n z − wn,i (2.19) lim 1 − wn,i z = 0 n→∞ i=1 uniformly on compact subsets of D, which clearly follows from the fact that there exists r < 1 such that |wn,i | ≤ r, i = 1, . . . , n, for all n ∈ N. The analogous result of Theorem 2.12 under the conditions related to Theorem 2.10 uses the fact that the condition of admissibility (2.15) (with zeros {wn,i } playing the role of {a2n,i }) is suﬃcient for (2.19) to occur. The situation in the case of dealing with the Carleman or generalized Carleman conditions is much more delicate; formula (2.19) is no longer true in general and the sequence {en }n∈N is proved to converge to zero by considering an identity problem of analytic functions. After Theorem 2.12 is obtained, the proof of Theorem 2.8 follows arguments developed by M´at´e, Nevai, and Totik [MNT85] in their proof of Rakhmanov’s Theorem [R77a, R83], of which Theorem 2.8 is a generalization. Similar ideas apply in the other cases. Finally, let us mention the work [L95b] which constitutes a step forward concerning the use of potential theory in L´opez’ papers on Pad´e approximation. There, the exact rate of convergence of two-point Pad´e approximants to a meromorphic Stieltjes function is obtained. The interpolation is proportionally distributed at the points z = 0 and z = ∞. That is, for each n ∈ N, the interpolation points are (2.20)

{0, . . . , 0, ∞, . . . , ∞} 2n − λn λn

with

lim

n→∞

λn = θ ∈ [0, 1]. 2n

The measure σ is taken to be a Freud-type weight of the form (2.21)

dσ(x) = xα e−τ (x) dx,

x > 0,

α ∈ R,

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36 12

B. DE LA CALLE YSERN

with τ a continuous function on (0, +∞) satisfying lim (sx)γ τ (x) = lim (sx)−γ τ (x) = 1,

x→0+

x→+∞

for some γ > 1/2 and s > 0. By combining some ideas of Rakhmanov [R84], Mhaskar, and Saﬀ [MhS], the authors obtain logarithmic asymptotics for the polynomials orthogonal with respect to the varying measures dσ(x)/xλn , whence the following result follows. Let Γ be Euler’s gamma function. Set

1/(2γ) −4γ Γ(γ + 1/2) √ ν = 1 − 1/(2γ), H(γ) = . 2γ − 1 πΓ(γ) Theorem 2.13 ([L95b]). Let σ be a measure of the kind (2.21) and suppose that the table of interpolation A veriﬁes (2.20). Then, on each compact subset K of C \ ([0, +∞) ∪ {r = ∞}), it holds

√ log σ + r − πnA K ν ν lim (1 − θ) = H(γ) inf sz + θ 1/(sz) , n→∞ z∈K (2n)ν √ where the branch of the root is taken so that −1 = i. 2.3. Pad´ e-type approximants. The Pad´e and multi-point Pad´e approximants admit a further generalization. Since the set of singular points of a Cauchy transform is contained in the support Σ of the measure, we can take advantage of this fact ﬁxing all or part of the poles of the approximant precisely on the set Σ. These approximants are commonly called in recent years Pad´e-type approximants, and Markov-type results involving them have been proved. Additionally, Pad´e-type and multi-point approximation may be combined to give multi-point Pad´e-type approximants. This kind of problems was studied by L´opez and Cala in a series of papers [L95e, L98c, L01a]. Let us describe the latter work, which represents the most evolved solution with some ideas borrowed from [AW98]. Take µ, Σ, ∆, A2n , and A with the same meanings as in (2.11). Let us ﬁx another family of monic polynomials {Bn }n∈N whose zeros lie in ∆ such that deg Bn = k(n) ≤ n. The zeros of the polynomials Bn2 , n ∈ N, form the table B. The multi-point Pad´e-type approximant of µ with preassigned poles at the zeros of the polynomial Bn2 , which interpolates the Markov function at the zeros of the polynomial A2n is the unique rational function pn , (2.22) πnA,B = qn Bn2 such that pn and qn satisfy the following conditions: (i) deg qn ≤ n − k(n), deg pn ≤ n + k(n) − 1, and qn ≡ 0. (ii)

− pn qn Bn2 µ is an analytic function on C \ ∆. A2n

(iii)

(z) − pn (z) qn (z) Bn2 (z) µ = O(1/z n−k(n)+1 ), A2n (z)

z → ∞.

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A WALK THROUGH APPROXIMATION THEORY

When k(n) = 0 we regain the multi-point Pad´e approximant. The numerator of πnA,B has degree n+k(n)−1 whereas the denominator has degree n+k(n). This way of choosing the degrees of pn and qn allows the results to be displayed independently of the ratio k(n)/n as we will see below. The reason for taking Bn squared in the deﬁnition above is that the polynomial qn satisﬁes the orthogonality relations B 2 (x) dµ(x) = 0, ν = 0, 1, . . . , n − k(n) − 1, (2.23) qn (x) xν n A2n (x) Σ resulting in a polynomial orthogonal with respect to the varying measure Bn2 (x) dµ(x) , A2n (x) which is positive due to the squaring of Bn . The Hermite formula Bn2 (x) qn2 (x) dµ(x) A2n (z) (2.24) µ (z) − πnA,B (z) = 2 , 2 Bn (z) qn (z) Σ A2n (x) z−x

z ∈ C \ Σ,

is essential in order to express the error of the Pad´e approximation in terms of the asymptotic behavior of the polynomials involved. Other ingredients needed for the statement and proof of the main result of [L01a] are laid out in the following deﬁnitions and conditions. For a given polynomial T , we denote by ΘT the normalized zero counting measure of T . That is 1 (2.25) ΘT = δξ . deg T ξ: T (ξ)=0

The sum is taken over all the zeros of T and δξ denotes the Dirac measure concentrated at ξ. Any subsequence of the probability measures {ΘA2n } has a weak star limit. If we are interested in describing the exact rate of convergence of the approximants πnA,B it is natural to require that limit to be made explicit. Thus, it is said that the sequence of polynomials {A2n }n∈N has the measure α as its asymptotic ∗ zero distribution if ΘA2n −→ α, n → ∞. By λΣ,α we will denote the equilibrium measure of the set Σ in the presence of the external ﬁeld given by (minus) the logarithmic potential of α. In the sequel, we will suppose that (1) Table A is contained in a compact set F of C \ ∆. ∗ (2) ΘA2n −→ α, n → ∞. Obviously, the measure α is supported on F . (3) The support Σ is a regular set, that is, the domain C \ Σ is regular with respect to the Dirichlet problem. (4) The measure µ is regular in the sense of [StT], which will be denoted by µ ∈ Reg. Condition (4), as well as (3), is necessary in general for obtaining the exact rate of convergence of the approximants since regular measures are precisely the class of measures for which the associated orthogonal polynomials {qn (µ; z)}n∈N have logarithmic asymptotics. Namely, µ ∈ Reg if and only if (2.26) lim |qn (µ; z)|1/n = exp gC\Σ (z, ∞) , n→∞

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B. DE LA CALLE YSERN

locally uniformly on C \ ∆. In earlier papers the more restrictive condition µ > 0 a.e. on [−1, 1] plays the same role as µ ∈ Reg here. When k(n) = 0 for all n ∈ N, i.e., if we do not ﬁx beforehand any pole of the approximant, the denominator qn of the approximant is orthogonal with respect to the varying measure dµ(x)/A2n (x) and, by using standard arguments in potential theory , it can be proved that the sequence of polynomials {qn } has the measure λΣ,α as its asymptotic zero distribution, see [GR86]. Now, when dealing with Pad´e-type approximants, the essential condition is (2.27)

lim sup n→∞

k(n) ΘBn ≤ λΣ,α , n

which means that the limit distribution of the ﬁxed poles must be smaller, in the weak star sense, than λΣ,α in order to allow the free nodes to ﬁll the gap between the two measures. Actually, under these conditions, the normalized zero counting measures of the polynomials qn Bn tend to the equilibrium measure λΣ,α , whence the following results follows. Theorem 2.14 ([L01a]). Suppose that conditions (1)–(4) and (2.27) are fulﬁlled. Then 1/2n lim µ (z) − πnA,B (z) = e−τ (z) , τ (z) = gC\Σ (z, ζ) dα(ζ), n→∞

F

uniformly on compact subsets of C \ (F ∪ ∆), and 1/2n (z) − πnA,B (z) lim sup µ = e−τ (z) , n→∞

uniformly on compact subsets of C \ ∆ of positive capacity. The asymptotic estimates given by this theorem should be compared to that of Theorem 2.7 and with (2.5) when a2n,i = ∞ for all i = 1, . . . , n, and n ∈ N. The work [L98c] essentially deals with the same problem. The conditions required are more restrictive (µ > 0 a.e. on [−1, 1]) while another choice of the degrees of the polynomials pn and qn makes the result to be displayed in a slightly diﬀerent fashion. Also, the rate k(n)/n must have a limit. The ﬁrst paper by L´opez on Pad´e-type approximants was [L95e] which has similar features to [L98c] but with all of the interpolation points placed at z = ∞. Pad´e-type approximants are especially suitable for solving the problem of approximating a meromorphic Markov function. Most of the poles can be ﬁxed on the support Σ (cutting the computational cost of the construction of the approximant) and a few free poles would ﬁnd the poles of the rational perturbation. The purpose of [L01b] was to obtain a result of this kind. Let us consider the multipoint Pad´e-type approximation problem for the function µ + r, where r is a rational function with real coeﬃcients. The construction of the approximants πnA,B for µ +r is analogous to that of the πnA,B for µ . As the coeﬃcients of r are real, in light of the results in [R77b], no condition should be put on the measure µ if we are not interested in obtaining the exact rate of convergence. Regarding the ﬁxed poles, we must require more restrictive hypothesis. Instead of (2.27), we will need the ﬁxed poles to tend to the equilibrium measure, that is, (2.28)

∗

ΘBn −→ λΣ,α .

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Table 1. Papers on Pad´e approximation of Cauchy transforms Function

Approximants

Authors

Conditions

µ

Pad´e

Markov [Mark]

σ

Pad´e

Stieltjes [Sti]

µ + complex r

Pad´e

Gonchar [G75a]

µ + real r

Pad´e

Rakhmanov [R77b]

µ

multi-point Pad´e

Gonchar-L´ opez [L78]

σ

multi-point Pad´e

L´ opez [L79]

σ + real r

Pad´e

L´ opez [L81b]

Carleman

µ + complex r

multi-point Pad´e

L´ opez [L87b]

(1), (5)

σ + complex r

Pad´e

L´ opez [L89b]

Carleman, σ > 0

σ + complex r

two-point Pad´e

Mart´ınez-L´ opez [L95b]

µ

Pad´e-type

Cala-L´ opez [L95e]

(2.29) , (5)

µ

mult. Pad´e-type

Cala-L´ opez [L98c]

(1)-(2), (2.27) , (5)

µ

Pad´e-type

Ambrol.-Wallin [AW98]

(3)-(4), (2.29)

µ

mult. Pad´e-type

Cala-L´ opez [L01a]

(1)-(4), (2.27)

µ + real r

mult. Pad´e-type

de la Calle-L´ opez [L01b]

(1)-(3), (2.28)

σ determined ratio asymptotics

(1) gener. Carleman

Freud-type σ

Theorem 2.15 ([L01b]). Suppose that the coeﬃcients of the rational function r are real and that conditions (1)–(3) and (2.28) are fulﬁlled. Then - Each pole of r attracts, as n → ∞, as many poles of πnA,B as its multiplicity. - On each compact subset K of C \ (∆ ∪ {r = ∞}), we have 1/2n + r − πnA,B K ≤ e−τ , τ = inf gC\Σ (z, ζ) dα(ζ) . lim µ n→∞

z∈K

F

Finally, all the papers discussed in this section are summarized in Table 1 which is not intended to be exhaustive by any means. In particular, there are relevant works by Gonchar [G75b], Karlberg and Wallin [KaW], and Ambroladze and Wallin [AW96, AW97], among others, that do not appear in it. The fourth column of Table 1 shows the most important conditions required in each paper for the main result on convergence. For short, the condition µ > 0 a.e. on [−1, 1] is denoted by (5) in the table and, additionally, we will need the analog of (2.27) when there is no external ﬁeld involved, i.e., (2.29)

lim sup n→∞

k(n) ΘBn ≤ λΣ , n

where λΣ is the equilibrium measure of Σ. The conditions (2.29) and (2.27) appear slightly modiﬁed in the papers [L95e] and [L98c], respectively, since there the ratio k(n)/n is required to have a limit as n tends to ∞, say l, and the limit l appears Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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in both sides of the inequality. This fact is denoted in Table 1 by adding a prime on those conditions. Overall it can be noticed how, over the years, the approximants become more and more general, the conditions get weaker and weaker, and the results become more and more precise. 3. Orthogonal polynomials 3.1. Relative asymptotics and varying measures. We will see now how varying measures are useful not only to prove results on rational approximation but to obtain theorems about polynomials orthogonal with respect to a ﬁxed measure as well. That is an interesting fact since, apparently, we complicate the setting in order to solve a standard problem. Let us see a couple of examples. The work [L90] deals with relative asymptotics of orthogonal polynomials. Let us say some preliminary words about this problem. Let ρ be a ﬁnite positive Borel measure on the unit circle T (or, equivalently, on [0, 2π]) and let h be a non-negative function on T such that h ∈ L1 (dρ). We will denote by ϕn (dρ; z) the n-th orthonormal polynomial with respect to the measure dρ and by ϕn (hdρ; z) the orthonormal polynomial with respect to hdρ. The relative asymptotics problem consists in ﬁnding conditions under which limn→∞ ϕn (hdρ; z)/ϕn (dρ; z) exists and give an explicit expression of that limit. If the measure ρ veriﬁes log ρ ∈ L1 [0, 2π] we will say that ρ belongs to Szeg˝o’s class and it holds (see [Sz75, Si05a]) (3.1)

lim

n→∞

ϕn (dρ; z) 1 = , n z S(ρ ; 1/z)

uniformly on compact subsets of E = {z ∈ C : |z| > 1}, where S(ρ ; z) is the Szeg˝o function associated to ρ deﬁned by 2π w+z 1 log ρ (θ) dθ , w = eiθ , |z| < 1. (3.2) S(ρ ; z) = exp 4π 0 w − z The relative asymptotics’ problem may be considered an extension of Szeg˝ o’s theory since if dρ = dθ/(2π) we have ϕn (dρ; z) = z n . In the case that both measures hdρ and dρ belong to Szeg˝ o’s class a straightforward application of formula (3.1) gives

ϕn (hdρ; z) 1 S(ρ ; 1/z) (3.3) lim = = , z ∈ E. n→∞ ϕn (dρ; z) S(hρ ; 1/z) S(h; 1/z) This result is not satisfactory because the much weaker condition log h ∈ L1 [0, 2π] is suﬃcient for the existence of the right-hand side of (3.3). Due to that, the research has focused on trying to ﬁnd the weakest possible assumptions for (3.3) to happen. This is an interesting open problem. Gonchar [G75a] was the ﬁrst to study a particular problem on relative asymptotics. In full generality, the problem was posed by P. Nevai [N] and studied by Rakhmanov, [R87], and M´ at´e, Nevai, and Totik, [MNT84, MNT87]. These works assure the existence of the limit (3.3) under mild conditions on the function h (see below) and with ρ such that ϕn (dρ; z) satisﬁes ratio asymptotics. Also, formulas on relative asymptotics on the support of the measure, requiring h to fulﬁll Lipschitz-type properties, are given. In [L90], L´ opez obtains an analog of (3.3) for measures supported on the real line R. As relative asymptotics is closely connected to ratio asymptotics, the Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

A WALK THROUGH APPROXIMATION THEORY

41 17

foremost diﬃculty is, again, the lack of ratio asymptotics for orthogonal polynomials on the real line. The method of the proof is similar to what we have already seen before. The problem is transformed by means of a change of variable and a formula on relative asymptotics for varying orthogonal polynomials on the unit circle is then proved. Due to that, let us ﬁrst consider a ﬁnite positive Borel measure ρ on the unit circle T. For each n ∈ N, set Wn (z) = (z − 1)n and put dρn = dρ/|Wn |2 . Let g be a function deﬁned on T. We will need to impose on g the following conditions. (c1 ) g ∈ L1 (ρ) and g ≥ 0 on T. (c2 ) There exists an algebraic polynomial Q such that |Q| g ±1 ∈ L∞ (ρ). Whereas condition (c1 ) is clearly necessary, perhaps condition (c2 ) could be weakened. Nevertheless, it already appears in results on relative asymptotics in the classical setting [MNT87]. The following result constitutes the key step. Theorem 3.1 ([L90]). Let ρ be a positive measure on T such that, for each j ∈ Z, it holds (3.4)

lim

n→∞

ϕn+j+1 (dρn ; z) = z, ϕn+j (dρn ; z)

uniformly on compact subsets of C \ D. Then, for each j ∈ Z, we have lim

n→∞

ϕn+j (gdρn ; z) 1 = , ϕn+j (dρn ; z) S(g; 1/z)

uniformly on compact subsets of E, provided that g veriﬁes (c1 ) and (c2 ). Thus, all we need is ratio asymptotics for the varying orthogonal polynomials ϕn (dρn ; z). But we have already mentioned this type of results (cf. Theorem 2.3); following similar arguments, ratio asymptotics (3.4) holds provided that ρ > 0 a.e. on T and ρ veriﬁes the Carleman condition ∞ 1 dρ(ζ) = +∞, sm = , m ∈ N. √ 2m s |ζ − 1|2m m T m=1 Now, consider the original problem on the real line R. Let ν be a positive Borel measure on R with all its moments cm = xm dν(x), m ∈ N, R

being ﬁnite numbers. The problem is carried to the exterior of the unit circle E through the change of variable z+1 (3.5) ζ(z) = i , z−1 which transforms E onto the upper half-plane H = {z ∈ C : z > 0}. The resulting varying measures are of the type considered in Theorem 3.1. The other conditions are also aﬀected by the transformation (3.5) so that Theorem 3.1 can be used. Thus, the Carleman condition for R is (3.6)

∞

1 = +∞. √ c2m

2m

m=1

Corresponding conditions for the perturbation h are (c3 ) h ∈ L1 (ν) and h ≥ 0 on R. Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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B. DE LA CALLE YSERN

|Q| h±1 ∈ L∞ (ν). (1 + x2 )N For the statement of the main theorem we need the Szeg˝o function associated to the function h and the region H deﬁned by 1 xz + 1 dx SH (h; z) = exp , z ∈ H. log h(x) 2πi R z − x 1 + x2 (c4 ) There exist a polynomial Q and N ∈ N such that

Theorem 3.2 ([L90]). Let ν be a measure such that ν > 0 a.e. on R and it satisﬁes the Carleman condition (3.6). Let h be a function on R verifying (c3 ) and (c4 ). Then ϕ n (hdν; z) SH (h; z) = , lim n→∞ ϕ n (dν; z) SH (h; i) uniformly on compact subsets of H, where ϕ n is the orthogonal polynomial of degree n normalized so that it takes the value 1 at z = i. Eight years later, we ﬁnd a similar result in [L98a] for an arc of the unit circle. Let γ = z = eiθ , θ1 ≤ θ ≤ θ2 , θ2 − θ1 ≤ 2π be an arc of the unit circle T and ρ a positive Borel measure deﬁned on it. Let h be a function deﬁned on γ such that (c5 ) h ∈ L1 (ρ) and h ≥ 0 on γ. (c6 ) There exists an algebraic polynomial Q such that |Q| h±1 ∈ L∞ (ρ). Theorem 3.3 ([L98a]). Let ρ be a measure on γ such that ρ > 0 a.e. on γ. Let h be a function on γ verifying (c5 ) and (c6 ). Then lim

n→∞

ϕn (hdρ; z) = Sγ (h; z), ϕn (dρ; z)

o function associated uniformly on compact subsets of C\γ, where Sγ (h; z) is the Szeg˝ to the arc γ and the function h. For an explicit expression of Sγ (h; z), see formula (59) in [L98a]. The proof of Theorem 3.3 follows the same trail as Theorem 3.2 except for the fact that now one additional step is needed. Firstly, the opposite change of variable to (3.5), z+i (3.7) ζ(z) = , z−i is performed in order to work with varying orthogonal polynomials on a compact interval of R. The varying part of the measure is now (1 + x2 )n ; clearly its zeros are far away from R. Secondly, the problem is connected to the unit circle by projecting up the varying measures in the usual way. The zeros of the polynomials Wn that appear in the denominator of these varying measures on T have their zeros uniformly bounded away from T. Hence (cf. Theorem 3.1), a Carleman-type condition is not necessary for obtaining a result on relative asymptotics on T, from which Theorem 3.3 follows. An analog of Rakhmanov’s Theorem for an arc γ of the unit circle is also given in [L98a]. The main ingredients of the proof are the change of variable (3.7), Theorem 2.8, and a formula on relative asymptotics for varying orthogonal polynomials on a compact interval of R. Set ϕn (dρ; z) = αn z n + . . . , αn > 0. Theorem 3.4 ([L98a]). Let ρ be a measure on γ such that θ2 − θ1 < 2π. Assume that ρ > 0 a.e. on γ. Then ϕn+1 (dρ; z) = Gγ (z), lim n→∞ ϕn (dρ; z) Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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A WALK THROUGH APPROXIMATION THEORY

uniformly on compact subsets of C \ γ, where Gγ is the conformal mapping of C \ γ onto E ∪ {∞} such that Gγ (∞) = ∞ and Gγ (∞) > 0. Moreover,

−1 αn+1 θ 2 − θ1 1 = sin = . lim n→∞ αn cap(γ) 4 The bottom line of these works is that we can change the support of the measure in order to work with compact supports on the real line paying the price of working with varying orthogonal polynomials. 3.2. Three-term recurrence relation and linear operator theory. Some other results by L´ opez on orthogonal polynomials and Pad´e approximants were not proved by using varying measures but linear operator theory. They are similar to the previous ones in the sense that they constitute extensions and analogs of very well-known theorems in the ﬁeld. They are related to the Nevai-Blumenthal class of measures, which, in turn, is closely connected to sequences of polynomials satisfying a three-term recurrence relation. Let {Pn }n∈N be a sequence of monic polynomials such that ⎧ ⎨ Pn+1 (z) = (z − bn ) Pn (z) − a2n Pn−1 (z), an = 0, n ∈ Z+ , (3.8) ⎩ P0 (z) ≡ 1, P−1 (z) ≡ 0, where Z+ = {0, 1, . . .} and the coeﬃcients an , bn , n ∈ Z+ , are in general complex numbers. We will associate to the recurrence relation (3.8) two other objects. Consider ﬁrst the Chebyshev continued fraction (J-fraction) (3.9)

a20

J(z) =

,

a21

z − b0 − z − b1 −

an = 0,

n ∈ Z+ .

a22 z − b2 − . ..

It is well known (see [W], p. 197) that there is a one-to-one correspondence between J-fractions (3.9) and formal power series of the type (2.1) verifying c0 c1 c2 ··· cn c1 c2 c3 · · · cn+1 c2 c c · · · cn+2 = 0, 3 4 n ∈ Z+ . hn = .. .. .. .. .. . . . . . cn cn+1 cn+2 · · · c2n The denominators Pn of the n-th partial fraction a20

πn (z) =

a21

z − b0 − z − b1 − . .

.−

a2n−1 z − bn−1

satisfy the recurrence relation (3.8) with the indicated initial conditions. The numerators Tn of πn satisfy the same recurrence relation with initial conditions T0 ≡ 0, T1 ≡ a20 . Furthermore, the rational function πn coincides with the n-th Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

44 20

B. DE LA CALLE YSERN

diagonal Pad´e approximant of f (the formal expansion (2.1) uniquely associated to the J-fraction) developed at z = ∞. Whence the notation used for the partial fraction. Therefore, J-fractions represents a wider class of formal power series (2.1) than the class of Cauchy transforms of positive measures with support on R considered in Section 2. Actually, it was in this setting of continued fractions (with real coefﬁcients an , bn , n ∈ Z+ ) where the Markov and Stieltjes Theorems were originally stated and proved. The convergence of the Pad´e approximants strongly depends on the behavior of their poles, that is, the knowledge of the zero location of a sequence of polynomials verifying the recurrence relation (3.8) would allow us to prove results about the convergence of the Pad´e approximants to formal powers series (2.1) represented by J-fractions. For that, consider the second object associated to (3.8), the tridiagonal Jacobi matrix ⎛ ⎞ b0 a1 0 · · · ⎜ a1 b1 a2 · · · ⎟ ⎜ ⎟ (3.10) G = ⎜ 0 a2 b2 · · · ⎟ . ⎝ ⎠ .. .. .. . . . . . . The principal minors of G are ⎛

b0 a1 .. .

⎜ ⎜ ⎜ Gn = ⎜ ⎜ ⎝ 0 0

a1 b1 .. .

··· ··· .. .

0 0 .. .

0 0 .. .

0 0

··· ···

bn−2 an−1

an−1 bn−1

⎞ ⎟ ⎟ ⎟ ⎟. ⎟ ⎠

Obviously, Pn (z) = det(z In −Gn ), where In is the n×n identity matrix. Therefore, the spectrum of Gn , made up only by eigenvalues, coincides with the set of zeros of Pn . It is clear that the set of limit points of the zeros of {Pn }n∈N , which we will denote by P , must be related in some way to the spectrum of G. Hence, the need to study the properties of the operator G and its spectrum arises and it is here where the linear operator theory becomes an essential tool. Suppose, for the time being, that the coeﬃcients an , bn , n ∈ Z+ , in (3.8) are real numbers. Favard’s Theorem [F] assures the existence of a positive Borel measure µ (not necessarily unique) supported on Σ ⊂ R such that the sequence {Pn }n∈N in (3.8) is the sequence of monic orthogonal polynomials with respect to µ. If the coeﬃcients an , bn , n ∈ Z+ , are uniformly bounded then the measure µ is unique. When orthogonality is involved, and to maintain the notation coherent with the rest of the sections, we will denote the sequence of monic polynomials by {Qn }n∈N . We say that µ belongs to the Nevai-Blumenthal class with parameters a ≥ 0 and b ∈ R and denote it by µ ∈ M(a, b) if (3.11)

lim an = a/2,

n→∞

lim bn = b.

n→∞

In this case it is known that Σ = [b−a, b+a]∪E, where E is at most a denumerable set whose only possible accumulation points are b ± a. Also, we have (z − b) + (z − b)2 − a2 Qn+1 (z) = , (3.12) lim n→∞ Qn (z) 2 Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

A WALK THROUGH APPROXIMATION THEORY

45 21

uniformly on compact subsets of C\Σ, where the square root is taken to be positive when the radicand is positive. The reciprocal statement is also true. That is, if a sequence {Qn }n∈N of polynomials satisfying a recurrence relation of the type (3.8) with real coeﬃcients veriﬁes (3.12), then (3.11) holds and the measure with respect to which they are orthogonal belongs to the class M(a, b). If µ ∈ M(a, b) with a > 0 so that [b − a, b + a] does not reduce to a point, in addition to (3.12), we have that for every bounded Borel-measurable function f on Σ, continuous on [b − a, b + a], it holds 1 b+a dx (3.13) lim f (x) qn2 (x) dµ(x) = f (x) , n→∞ Σ π b−a a2 − (x − b)2 where {qn }n∈N is the sequence of polynomials orthonormal with respect to µ. For more details on this class of measures and its properties see the book [N] by Nevai. As mentioned, a well-known suﬃcient condition for µ ∈ M(a, b) due to Rakhmanov [R77a, R83] is that Σ = [b − a, b + a] and µ > 0 a.e. on [b − a, b + a]. Later, Denisov [De] extended this result proving that if Σ = [b − a, b + a] ∪ E, where E is of the type described above, and µ > 0 a.e. on [b − a, b + a], then (3.12) holds. The purpose of the series of papers [L95a, L95c, L98b] was to extend some of the results mentioned above to the case when the coeﬃcients an , bn , n ∈ Z+ , are complex numbers. Let us begin with [L95a]. Suppose that the coeﬃcients an , bn , n ∈ Z+ , are uniformly bounded and that their imaginary parts tend to zero, i.e., (3.14)

lim an = lim bn = 0.

n→∞

n→∞

Under those conditions, the linear operator G may be expressed as (3.15)

G = G + i G = H + iC,

where H is a Hermitian operator and C is a compact one. Denote by σ(L), σess (L), and σp (L), the spectrum, the essential spectrum, and the point spectrum, respectively, of the operator L. Then, σess (G) = σess (H) is a compact set of R and σ(G) \ σess (H) is at most a countable set of isolated points in C \ σess (H). The key step is to make clear the connection between the set P of limit points of the zeros of the polynomials {Pn }n∈N and σ(G). It turns out that there exists a bounded real interval [A, B] such that σess (G) ⊂ [A, B] and P ⊂ (σp (G) ∪ [A, B]). The constants A and B are deﬁned in terms of the bounds for the coeﬃcients an , bn , n ∈ Z+ . Now, the authors of [L95a] make use of a result by Gonchar [G83] that basically says that sequences of diagonal Pad´e approximants converge uniformly on compact subsets of those regions where the approximants do not have poles. The result is valid for a wide class of domains. We will say that a domain Ω belongs to the class G∞ if ∞ ∈ Ω = D \ L, where D is a domain, and L is a closed subset of D such that • cap L = 0 = ∂D, where D stands for the complement of the convex hull of ∂D. • ∂D Theorem 3.5 ([G83]). Suppose that Ω ∈ G∞ . Let {πn }n∈N be a sequence of diagonal Pad´e approximants deﬁned by (2.2) such that for any compact set K ⊂ Ω, there exists nK ∈ N such that πn does not have poles on K for n ≥ nK . Then, the sequence {πn }n∈N converges uniformly on compact subsets of Ω. Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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B. DE LA CALLE YSERN

As the domain C\([A, B] ∪ σp (G)) belongs to the class G∞ , the following result is straightforwardly obtained from the above considerations and Theorem 3.5. It may be seen as an extension of Markov’s Theorem. Theorem 3.6 ([L95a]). Suppose that supn∈Z+ {|an |, |bn |} < +∞, and (3.14) holds. Then, the Pad´e approximants πn of the J-fraction (3.9) (or, equivalently, of the formal power series (2.1) associated to it) converge on each compact subset of C \ ([A, B] ∪ E), where E is at most a countable set of isolated points in C \ [A, B], and [A, B] ⊂ R is a bounded interval. We will say that the sequence of monic polynomials {Pn }n∈N ∈ MC (a, b) if the polynomials satisfy a three-term recurrence relation of type (3.8) and lim an = a/2,

n→∞

lim bn = b,

n→∞

a, b ∈ C,

where all the coeﬃcients an are not necessarily diﬀerent from zero. Theorem 3.7 below shows that the class MC (a, b) is an analog of the Nevai-Blumenthal class M(a, b) although the lack of a Favard type result for complex coeﬃcients makes it necessary to take into consideration sequences of polynomials instead of measures. Theorem 3.7 ([L95a]). Suppose that the sequence of monic polynomials {Pn } satisﬁes (3.8), then the following statements are equivalent: (i) {Pn } ∈ MC (a, b). (ii) It holds (z − b) + (z − b)2 − a2 Pn+1 (z) lim = , n→∞ Pn (z) 2 uniformly on compact subsets of C \ Σ, where Σ = [b − a, b + a] ∪ E and E is at most a countable set of isolated points in C \ [b − a, b + a]. It may be proved that Theorem 3.6 remains valid when {Pn }n∈N ∈ MC (a, b). In that case [A, B] = [b − a, b + a] is a segment in the complex plane. Once the location of the zeros of the polynomials {Pn } is established, the main tool for proving Theorem 3.7 is a classical result by Poincar´e [Po]. Theorem 3.8 (Poincar´e’s Theorem). Let (3.16)

un+k + c1 (n) un+k−1 + c2 (n) un+k−2 + · · · + ck (n) un = 0

be a linear diﬀerence equation such that lim ci (n) = ci ,

n→∞

i = 1, . . . , k.

Suppose that the roots t1 , t2 , . . . , tk , of the characteristic equation tk + c1 tk−1 + · · · + ck = 0 have distinct moduli. Then, any non-trivial solution of (3.16) fulﬁlls un+1 lim = ti n→∞ un for some i ∈ {1, . . . , k}. The characteristic roots of the recurrence (3.8) are (z − b) ± (z − b)2 − a2 . 2 Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

A WALK THROUGH APPROXIMATION THEORY

47 23

Hence, the point-wise ratio convergence follows from Poincar´e’s Theorem and the location of the zeros of the polynomials {Pn }n∈N whereas the normality of the sequence of ratios is proved by induction, which yields uniform convergence. Theorem 3.7 was generalized to the periodic limit case in [L95c]. We will say that the sequence of polynomials {Pn }n∈N has asymptotically periodic recurrence coeﬃcients if it satisﬁes (3.8) and there exists a number k ∈ N such that (3.17)

lim

n=m modk

an = a(m) ,

lim

n=m modk

bn = b(m) ,

m = 1, . . . , k.

We set k to be the smallest natural number for which the above limits hold. Theorem 3.9 ([L95c]). Suppose that the sequence of monic polynomials {Pn } veriﬁes (3.14) and has asymptotically periodic recurrence coeﬃcients. Then, there exist analytic functions Ψ1 , Ψ2 in C \ σess (G) such that lim

n→∞

Pn+k (z) = Ψ1 (z), Pn (z)

uniformly on compact subsets of C \ (C [σess (G)] ∪ σp (G)), where C[A] stands for the convex hull of the set A, and lim

n→∞

Pn+k (z) = Ψ2 (z), Pn (z)

z ∈ σp (G) \ C [σess (G)] .

As for the convergence of the Pad´e approximants, a result analogous to Theorem 3.6 is given under the conditions (3.14) and (3.17), see Theorem 1 in [L95c] for details. The improvement with respect to Theorem 3.6 consists in a more precise description of the singularities of the J-fraction. It is proved that each point in σp (G) \ C [σess (G)] attracts a speciﬁc number of zeros of Pn and this number is exactly the order of the pole of the J-fraction at this point. The study of the convergence of the J-fraction (3.9) was taken up again in [L98b] without requiring the coeﬃcients an , bn , n ∈ Z+ , to be uniformly bounded. The main result is an extension of Stieltjes’ Theorem for complex coeﬃcients. As in that classical theorem, there is need for a condition which ensures the uniqueness of the limit function. For that, let us normalize the polynomials Pn and Tn , the denominator and numerator of the partial fraction πn , respectively, according to the formulas pn (z) =

Pn (z) , a0 . . . an

tn (z) =

Tn (z) , a0 . . . an

n ∈ Z+ .

We will say that G is determinate or indeterminate when at least one of the following two series diverge or both series converge, respectively, ∞ n=0

|pn (ζ)|2 ,

∞

|tn (ζ)|2 ,

n=0

at a given ﬁxed point ζ ∈ C. The deﬁnition is consistent since if, at a given point, both these series converge then both converge at all points of the complex plane. When the coeﬃcients an , bn , n ∈ Z+ , are real, G is determinate if and only if the moment problem for the measure µ is determinate or, equivalently, the measure µ is determined by the coeﬃcients an , bn , n ∈ Z+ . Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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B. DE LA CALLE YSERN

There also exist some Carleman-type conditions which are suﬃcient for G to be determinate. Namely, if either ∞

1

|an+1 | n=0

=∞

or

∞

|bn+1 | = ∞, |a n+1 an+2 | n=0

then G is determinate. Although the arguments employed in [L98b] are rather general, it is possible to reason as follows. If G = H + C is determinate and C is bounded, then H is determinate too [Ca]. Let us use again representation (3.15). Under condition (3.14), C is compact and, as H is determinate, the properties of the spectrum of G and H are similar to the previous cases. That is, σess (G) = σess (H) ⊂ R and σp (G) is at most a countable set of isolated points in C\σess (G). Besides P ⊂ (R ∪ σp (G)). Now, in order to prove convergence of the Pad´e approximants, instead of Theorem 3.5, a result on convergence of positive deﬁnite continued fractions (see Theorem 25.4 in [W]) must be used. Theorem 3.10 ([L98b]). Suppose that G is determinate and (3.14) takes place. Then, the Pad´e approximants πn of the J-fraction (3.9) (or, equivalently, of the formal power series (2.1) associated to it) converge on each compact subset of the set C \ (R ∪ σp (G)). It may be additionally proved that each point of σp (G) \ R is a pole of the J-fraction which attracts exactly as many zeros of Pn as its order. When the coeﬃcients an , bn , n ∈ Z+ , are real and the determinate case holds, we have σ(G) ⊂ R since G is self-adjoint. Therefore, σp (G) \ R = ∅ and, by the use of Favard’s Theorem, we regain Stieltjes’ Theorem for measures supported on R. Similar ideas to those described above are used in [L99a], where an analog of the Nevai-Blumenthal class for measures supported on the unit circle T is given. Let ρ be a probability measure on the unit circle T with support Σ. Denote by Φn the n-th monic orthogonal polynomial with respect to ρ and by ϕn (z) = αn z n + . . . , αn > 0, the corresponding orthonormal polynomial. We will say that the measure ρ belongs to the class MT (a; b), with a ∈ (0, 1], if (3.18)

lim |Φn (0)| = a,

n→∞

lim

n→∞

Φn+1 (0) = b. Φn (0)

As |Φn (0)| < 1 for any sequence of monic orthogonal polynomials, if the ﬁrst limit exists, necessarily a ∈ [0, 1]. When a = 0, the possible values for b are all those with |b| ≤ 1 and it is easy to see that lim

n→∞

Φn+1 (z) = z, Φn (z)

& 2 uniformly on compact subsets of C \ D. When a ∈ (0, 1], we have ∞ n=0 |Φn (0)| = ∞ and the measure ρ does not belong to the Szeg˝ o class. The initial idea for establishing Theorem 3.11 below was the fact that the theses of Theorem 3.4 imply (3.18) while limn→∞ Φn (0) may not exist. On the other hand, if ρ ∈ MT (a; b), it may be proved, following a reasoning by Geronimus, that the support Σ has similar features to the supports of measures in the Nevai-Blumenthal class. So, the class MT (a; b) seemed to be a good candidate for characterizing measures verifying ratio asymptotics. Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

A WALK THROUGH APPROXIMATION THEORY

49 25

As for the linear operator techniques used in [L99a], the relevant operator is the unitary multiplication operator U : L2ρ −→ L2ρ given by (U f ) (z) = zf (z),

f ∈ L2ρ ,

where L2ρ is the space of square integrable functions with respect to the measure ρ. If log ρ ∈ L1 [0, 2π], as is our case whenever ρ ∈ MT (a; b), the operator U is represented by the inﬁnite Hessenberg matrix ⎛ ⎞ α0 α0 −Φ1 (0) Φ0 (0) − Φ2 (0) Φ1 (0) − Φ3 (0) Φ0 (0) · · · α α 1 2 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ α0 α1 ⎜ −Φ2 (0) Φ1 (0) − Φ3 (0) Φ1 (0) · · · ⎟ ⎜ ⎟ α1 α2 ⎜ ⎟ ⎜ ⎟, Uρ = ⎜ ⎟ α1 ⎜ ⎟ −Φ3 (0) Φ2 (0) ··· ⎟ 0 ⎜ ⎜ ⎟ α2 ⎜ ⎟ ⎝ ⎠ .. .. .. .. . . . . provided that the basis {ϕn } is used in L2ρ . It may be proved that the support Σ of ρ coincides with the spectrum of U and the set of zeros of the polynomial Φn (z) coincides with the spectrum (eigenvalues) of the ﬁnite matrix Uρ,n given by the ﬁrst n columns and the ﬁrst n rows of Uρ , that is, Φn (z) = det(z In − Uρ,n ). The key step in the proof of Theorem 3.11 is the construction, for each non-empty class MT (a; b), of a representative χa,b ∈ MT (a; b) such that the operator Uρ − Uχa,b is compact, whence it follows that the spectra of both operators coincide except for a countable set of isolated points. Hence, the derived set of Σ coincides with the derived set of the support of χa,b , which turns out to be the arc γa,b ⊂ T given by γa,b = z = eiθ , α ≤ θ − β ≤ 2π − α , where the numbers α and β are deﬁned by the relations b = eiβ and a = sin(α/2). Other main ingredients of the proof are the three-term recurrence relation satisﬁed by the polynomials {Φn } and Theorem 3.8. The degenerate case a = 1 (when the arc γa,b reduces to the point −b) needs a slightly diﬀerent proof. Theorem 3.11 ([L99a]). Let a ∈ (0, 1]. The following statements are equivalent: (i) ρ ∈ MT (a; b). (ii) it holds αn lim = 1 − a2 n→∞ αn+1 and (z + b) + (z − b)2 + 4zba2 Φn+1 (z) = , lim n→∞ Φn (z) 2 uniformly on compact subsets of C \ Σ, where the square root is taken so √ that 1 = 1. From any one of the conditions (i) and (ii) above, it follows that (z + b) − (z − b)2 + 4zba2 Φn+1 (z) = , z ∈ Σ \ γa,b , lim n→∞ Φn (z) 2 Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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and if z0 is an isolated mass point in Σ, then, for all suﬃciently small ε > 0, there exists n0 ∈ N such that for n ≥ n0 , the polynomial Φn has exactly one zero in {z ∈ C : |z − z0 | ≤ ε}. Actually, with the aid of some results from [PSte], Theorem 3.11 was given for the following wider class of measures. We will say that the measure ρ belongs to the class MT (a1 , . . . , ak ; b1 , . . . , bk ), with am ∈ (0, 1), m = 1, . . . , k, if lim

n=m modk

|Φn (0)| = am ,

lim

n=m modk

Φn+1 (0) = bm , Φn (0)

m = 1, . . . , k.

For details on the results for this class of measures, see Theorems 2-4 in [L99a]. If am = 0 for some m ∈ {1, . . . , k} then am = 0 for all m = 1, . . . , k. Then, we have Φn+1 (0) (3.19) lim |Φn (0)| = 0, = bm , lim m = 1, . . . , k. n→∞ n=m modk Φn (0) This case was studied in [L01c], obtaining ratio asymptotics of the orthogonal polynomials {Φn } inside the unit circle and describing those points that may be of accumulation of the zeros of {Φn }. On this subject, see [Si05b]. The work [L01c] also provides an interesting result relating the behavior (3.19) with the analytic properties of the numeric inverse of the Szeg˝o function (3.2). Assume that (3.19) holds, then the largest disk centered at z = 0 inside of which S −1 (ρ ; z) can be extended analytically is {z ∈ C : |z| < |b1 . . . bk |−1/k }. Denote by D1 the largest disk centered at z = 0 in which S −1 (ρ ; z) can be extended to a meromorphic function having at most 1 pole. Theorem 3.12 ([L01c]). Suppose that S −1 (ρ ; z) can be analytically extended inside the disk {z ∈ C : |z| < R} with R > 1. Then, the following assertions are equivalent: (i) The function S −1 (ρ ; z) has exactly one pole in D1 . (ii) There exists b ∈ C, with 0 < |b| < 1, such that 1/n Φn+1 (0) lim sup = δ < 1. − b Φn (0) n→∞ Notice that if statement (ii) holds & the measure ρ not only belongs to the Szeg˝ o ∞ class but it is in the Baxter class, i.e., n=0 |Φn (0)| < ∞, being absolutely continuous with respect to the Lebesgue measure, see Theorem 8.5 in [Ger]. The proof of Theorem 3.12 does not use linear operator theory but it relies on the convergence of row sequences of Fourier-Pad´e approximants. 3.3. Sobolev orthogonal polynomials. From the results commented on so far, it is clear the need of dealing with non-standard orthogonal polynomials in order to solve several kinds of approximation problems. Formulas (2.8) and (2.23) are examples of this situation and we will see others connected to multiple orthogonality in Section 4. Now, we are concerned with the Sobolev orthogonal polynomials, possibly, the furthest ones from being standard regarding their formal deﬁnition and basic properties. Such polynomials present many intriguing and distinctive features which makes them play a fundamental role in the extension of the general theory of orthogonal polynomials. Nevertheless, we will see that, in many cases, their asymptotic behavior resembles that of the standard orthogonal polynomials. Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

A WALK THROUGH APPROXIMATION THEORY

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Let {µ0 , µ1 , . . . , µm } be a set of m + 1 ﬁnite Borel measures in the complex plane. For each k = 0, 1, . . . , m, let Σk be the support of µk with Σ0 consisting of inﬁnitely many points. Set Σ = Σ0 ∪ Σ1 ∪ · · · ∪ Σm . Assume that all polynomials are integrable. On the vector space P of all polynomials deﬁne the inner product m m (k) (k) (3.20) p, qS = p (ζ) q (ζ) dµk (ζ) = p(k) , q (k) L2 (µk ) , p, q ∈ P, k=0

Σk

k=0

(k)

where f denotes the k-th derivative of the function f . The inner product (3.20) is called a Sobolev inner product. For each n ∈ N, we deﬁne the n-th Sobolev n of monic orthogonal polynomial associated to (3.20) as the monic polynomial Q least degree, not identically equal to zero, such that n S = 0, z ν , Q

ν = 0, 1, . . . , n − 1.

If all the measures µk , k = 0, 1, . . . , m, are positive, which will be assumed in the sequel unless explicitly stated, the inner product (3.20) is positive deﬁnite and the n has degree n. Information on the Sobolev orthogonal polynomials polynomial Q may be found in the surveys [AlMaRe, Mar98]. The study of this kind of orthogonal polynomial was motivated by [Le], in connection with the polynomial least squares approximation of a function and its derivatives at the same time. In this respect, see also [I90]. From the outset [Alt], these polynomials showed they did not share with the standard orthogonal polynomials some of the most important properties; for instance, their zeros may not lie in the convex hull of Σ. It is not even known whether the compactness of Σ n } are uniformly bounded. These facts mark the whole implies that the zeros of {Q theory and make the zero location a crucial issue. Another important characteristic is the lack, in general, of a three-term recur n }. The reason lies in the fact that rence relation fulﬁlled by the polynomials {Q the multiplication operator U (p)(z) = zp(z),

p ∈ P,

is not symmetric with respect to the Sobolev inner product, that is, zp, qS = p, zqS ,

(3.21)

p, q ∈ P,

due to the terms involving derivatives in (3.20). As a matter of fact, it was proved in [E] that, if Σ ⊂ R, the class of inner products such that Σ\Σ0 is a ﬁnite set of points n } satisfy a recurrence is essentially the class for which the monic polynomials {Q relation with a number of terms independent of n. This kind of Sobolev inner products are called discrete and are precisely those for which asymptotic results were proved in the ﬁrst place for general measures µ0 . For instance, consider the discrete Sobolev inner product (3.22) p, qS = p(x) q(x) dµ0 (x) + M p (c) q (c), M > 0, c ∈ R, Σ0

where Σ0 ⊂ R. The common strategy in such a situation is to compare the behavn } with that of the orthogonal polynomials {Qn } with ior of the polynomials {Q respect to the measure µ0 . The latter are required to have some kind of asymptotic behavior. Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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n } be Theorem 3.13 ([MaV]). Suppose that µ0 ∈ M(1, 0). Let {Qn } and {Q the sequences of monic orthogonal polynomials with respect to µ0 and the Sobolev inner product (3.22) respectively. Then, ⎧ ⎪1, c ∈ Σ0 , Qn (z) ⎨ 2 = (Ψ(z) − Ψ(c)) lim n→∞ Qn (z) ⎪ , c ∈ Σ0 , ⎩ 2Ψ(z) (z − c) √ uniformly on compact subsets of C \ Σ0 , where Ψ(z) = z + z 2 − 1. n } does not depend on the weight M Notice that the relative asymptotics of {Q in (3.22) and is similar to that of the polynomials orthogonal with respect to the standard inner product p, q = p(x) q(x) dµ0 (x) + N p(c) q(c), N > 0, c ∈ R, Σ0

n is attracted by the point when µ0 ∈ M(1, 0), see [N]. In particular, one zero of Q c. Thus, Theorem 3.13 suggests that, for the discrete case, the number of zeros n outside the convex hull of Σ0 may not depend on the order of the higher of Q derivative appearing in the inner product. Actually, it basically depends on the number of terms in the discrete part as was proved in [L96b]. In fact, let ∆0 be the convex hull of the set Σ0 ⊂ R with ∆0 = R and ﬁx c ∈ R such that it does not belong to the interior of ∆0 . Deﬁne m (3.23) p, qS = p(x) q(x) dµ0 (x) + Mk p(k) (c) q (k) (c), m > 0, Mm > 0, Σ0

k=0

where, for each k = 0, 1, . . . , m, Mk ∈ R. Notice that the inner product (3.23) is not n , the n-th Sobolev monic orthogonal positive deﬁnite. It is very easy to see that Q polynomial with respect to (3.23), is quasi-orthogonal of order m + 1 with respect to the measure (x − c)m+1 dµ0 (x). That is, n (x) (x − c)m+1 dµ0 (x) = 0, ν = 0, 1, . . . , n − m − 2, (3.24) xν Q Σ0

n has at least n − (m + 1) changes from which it straightforwardly follows that Q of sign in ∆0 . Something more can be proved. For each n ∈ Z+ = 0, 1, . . . , denote by s(n) the number of non-zero terms in the discrete part of (3.23) whose order of derivative is less than n. Obviously, for suﬃciently large n, s(n) is the number of non-zero terms in the discrete part of (3.23). n } be the sequence of Sobolev monic orthogonal Theorem 3.14 ([L96b]). Let {Q polynomials with respect to the inner product (3.23). Then, for each n ∈ Z+ , the n has at least n − s(n) changes of sign in the interior of ∆0 . polynomial Q This work also contains results about the interlacing property of the zeros n } (see Theorem 3.3 in [L96b]) which rely on the quasiof the polynomials {Q orthogonality relations (3.24). The paper [L95d] deals with a very general kind of discrete Sobolev inner product and has interesting connections with other papers by L´opez. We will Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

A WALK THROUGH APPROXIMATION THEORY

53 29

present a simpliﬁed form of the main result. Let µ0 ∈ M(1, 0) and deﬁne (3.25)

p, qS =

p(x) q(x) dµ0 (x) + Σ0

mi N

Mi,k p(k) (ci ) q (k) (ci ),

i=1 k=0

where, for i = 1, . . . , N , ci ∈ C \ Σ0 , mi > 0, Mi,mi = 0, and Mi,k ∈ C, k = 0, . . . , mi . Set N (z − ci )mi +1 , d = deg R. R(z) = i=1

n associated to the inner product As in the previous case, the Sobolev polynomial Q (3.25) veriﬁes the quasi-orthogonality relations n (x) R(x) dµ0 (x) = 0, ν = 0, 1, . . . , n − d − 1. (3.26) xν Q Σ0

For each n ∈ N, the authors consider the monic polynomial Pn of least degree, not identically equal to zero, such that xν Pn (x) R(x) dµ0 (x) = 0, ν = 0, 1, . . . , n − 1. Σ0

It is clear that Pn (x) R(x) =

d

cn,j Qn+d−j (x),

j=0

where {Qn } is the sequence of monic orthogonal polynomials with respect to µ0 . As the polynomials {Qn } satisfy ratio asymptotics, the situation reminds formula (2.9) which was dealt with in [L89b] which, in turn, follows [G75a]. The key is to prove that, for j = 0, 1, . . . , d, the coeﬃcients cn,j have a limit cj as n tends to ∞. Then, it may be proved that, for suﬃciently large n, Pn has degree n and fulﬁlls the relative asymptotic formula

mi +1 N Pn (z) 1 Ψ(z) − Ψ(ci ) = d lim , n→∞ Qn (z) 2 i=1 z − ci uniformly on compact subsets of C\Σ0 . Moreover, the sequence {Pn } belongs to the class MC (1, 0) deﬁned in [L95a] and, due to Theorem 3.7, veriﬁes ratio asymptotics. n /Pn . Relations (3.26) imply So, it is enough to study the relative asymptotics Q that, for suﬃciently large n, it holds n (x) = Q

d

an,j Pn−j (x),

j=0

which is managed in a analogous way as before, whence Theorem 3.15 follows. n } be Theorem 3.15 ([L95d]). Suppose that µ0 ∈ M(1, 0). Let {Qn } and {Q the sequences of monic orthogonal polynomials with respect to µ0 and the Sobolev inner product (3.25), respectively. For each i = 1, . . . , N , let m∗i stand for the number of coeﬃcients Mi,k , k = 0, 1, . . . , mi , diﬀerent from zero in (3.25). Then, n is n and each point ci attracts exactly for all suﬃciently large n, the degree of Q Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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n , i = 1, . . . , N , while the rest of the zeros concentrate on Σ0 . Also, m∗i zeros of Q we have )m∗i ( N n (z) Q (Ψ(z) − Ψ(ci ))2 lim = n→∞ Qn (z) 2Ψ(z) (z − ci ) i=1 uniformly on compact subsets of C \ Σ0 . The case when the Sobolev inner product is not discrete, usually referred to as continuous, is much more delicate to handle. Let us mention one of the ﬁrst works where such situation is dealt with to appreciate the diﬃculties that may arise. Recall that gΩ (z, ∞) stands for the Green function of the region Ω. To properly state the result, we will also need the deﬁnitions of regular measures (2.26) and normalized zero counting measure of a polynomial (2.25). Take m = 1 in (3.20). For each r > 0, denote by Vr the union of those components of the set {z ∈ C : gC\Σ (z, ∞) < r} having empty intersection with Σ0 . Put * Vr and K = ∂V ∪ (Σ \ V ) . V = r>0

For instance, if Σ0 and Σ1 are disjoint intervals of R, V is a bounded open neighborhood of Σ1 not touching Σ0 and K = ∂V ∪ Σ0 . Theorem 3.16 ([GaK]). Let m = 1 in (3.20). Suppose that Σ0 and Σ1 are compact regular sets contained in R and µ0 , µ1 ∈ Reg. Let the measure α be a weak star limit of a subsequence of {ΘQn }. Then (a) The support of α is contained in the set V ∪ Σ. (b) The balayage (sweeping out) of α onto K is equal to the balayage of λΣ onto K, where λΣ denotes the equilibrium measure of the set Σ. Under the same conditions, the authors of [GaK] prove that the sequence {ΘQ } does converge to λΣ , thus obtaining more information about the location of n the zeros of the derivatives than about the Sobolev orthogonal polynomials themselves. In the particular case when Σ1 ⊂ Σ0 , we have V = ∅ and K = Σ. Therefore, as a consequence of Theorem 3.16, the whole sequence {ΘQn } converges, in the weak star sense, to λΣ . In view of the diﬃculties, the usual way to try to prove more precise asymptotics of Sobolev orthogonal polynomials has been that of requiring that the measures appearing in (3.20) fulﬁll some type of relationship or correlation between them. For instance, a pair of positive measures µ0 and µ1 are said to be coherent if the orthogonal polynomials associated to the latter can be expressed as a linear combination (of a ﬁxed number of terms independent of the degree of the polynomials) of the derivatives of the orthogonal polynomials corresponding to the former. The concept of coherent measures was introduced in [I91] and has been used in numerous papers to prove a number of properties as well as asymptotics of the Sobolev orthogonal polynomials for particular choices of pairs of measures. For further details, see [Mar98, Me96] and the references therein. The main drawback of this concept is that there are not many coherent pairs, see [Me97]. Another kind of correlation between measures was introduced in [L99b] for measures compactly supported on R and extended later in [L01d] to measures on the complex plane C. In the following, as long as results of [L01d] are concerned, Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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A WALK THROUGH APPROXIMATION THEORY

we will assume that Σ is a compact set of C. We will say that the Sobolev inner product (3.20) is sequentially dominated if (3.27)

dµk = fk−1 dµk−1

with fk−1 ∈ L∞ (µk−1 ),

k = 1, . . . , m.

Relations (3.27) imply in particular that Σk ⊂ Σk−1 , k = 1, . . . , m. This class of Sobolev inner products enjoys nice properties linked to the multiplication operator U deﬁned by (3.21). The operator U is related to the zeros of the Sobolev orthogonal polynomials zeros in a similar way as described in Section 3.2 for the standard orthogonal polynomials. To see that, notice that the inner product (3.20) induces the norm (m )1/2 2 (k) (3.28) pS = , p ∈ P. p (ζ)| dµk (ζ) k=0

Σk

Consider the Banach space HS obtained completing the normed space (P, · S ). As usual, this is done by taking equivalence classes of Cauchy sequences. The space HS may be endowed with the structure of a Hilbert space and the resulting inner product is denoted by ·, ·S too. The operator U is extended to HS by continuity. n /Q n S . For each n ∈ N, we have Set qn = Q z qn−1 (z) =

n

cj,n−1 qj (z),

where

cj,n−1 = z qn−1 , qj S ,

j = 0, . . . , n.

j=0

The sequence { qn } of Sobolev orthonormal polynomials forms a complete basis in HS so, if the multiplication operator U is bounded, the inﬁnite Hessenberg matrix ⎛ ⎞ c0,0 c0,1 c0,2 · · · ⎜ c1,0 c1,1 c1,2 · · · ⎟ ⎜ ⎟ U =⎜ 0 c2,1 c2,2 · · · ⎟ ⎝ ⎠ .. .. .. .. . . . . is the matrix representation of U for the basis { qn }. Notice that, when m = 0 in (3.20) and Σ0 ⊂ R, the Hessenberg matrix U is the Jacobi matrix G given by (3.10). It turns out that the eigenvalues of the principal minors of U are precisely the n }. This fact was used in [L99b] to prove that if the zeros of the polynomials {Q n } are uniformly bounded. Such a operator U is bounded, then the zeros of {Q result was improved in [L01d] by giving a sharper uniform bound of the zeros. Theorem 3.17 ([L01d]). Suppose that, for the norm given by (3.28), there exists a positive constant C such that U (p)S ≤ C pS ,

p ∈ P.

n } are contained in the Then all the zeros of the Sobolev orthogonal polynomials {Q disk {z ∈ C : |z| ≤ C}. The proof of Theorem 3.17 given by the authors is extremely simple. Let z0 n . As deg Q n = n, it is clear that there exists a denote one of the zeros of Q n (z). Now, we have polynomial p of degree n − 1 such that zp(z) = z0 p(z) + Q |z0 | p(z)S = z0 p(z)S ≤ zp(z)S ≤ Cp(z)S , Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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n , pS = 0 in the ﬁrst inequality and the hypothesis in the where we have used Q second one. As p(z)S = 0, the result follows. Now, the remarkable fact is that sequentially dominated inner products give rise to bounded multiplication operators. Theorem 3.18 ([L01d]). Suppose that the Sobolev inner product (3.20) is sequentially dominated. Then, for each p ∈ P, we have U (p)S ≤ C pS , where + C = 2C12 + 2C2 (m + 1)2 ,

with

C1 = max |z|, z∈Σ0

C2 =

max

k=0,...,m−1

fk L∞ (µk ) .

Consequently, if the Sobolev inner product is sequentially dominated all the zeros of the corresponding Sobolev orthogonal polynomials are uniformly bounded. We can extend a bit further the applicability of Theorem 3.18. We say that a Sobolev inner product is essentially sequentially dominated if it deﬁnes a norm on P which is equivalent to that deﬁned by a sequentially dominated Sobolev inner product. It is clear that, for such a kind of inner products, all the zeros of the Sobolev orthogonal polynomials are also uniformly bounded. On the other hand, for Sobolev inner products with Σ ⊂ R a compact set, it has been proved in [Ro] that if the multiplication operator is bounded then the corresponding Sobolev inner product is essentially sequentially dominated. Thus, sequential domination is a suﬃcient condition for the multiplication operator to be bounded which is not far from being necessary. The above considerations do not close the zero location issue. Unlike the standard inner products (m = 0) on the real line, the fact that the zeros of the Sobolev orthogonal polynomials are uniformly bounded does not imply that the multiplication operator is bounded. Actually, the authors of [L01d] show that if m = 1 and Σ0 , Σ1 ⊂ R are disjoint compact sets, then the zeros of the Sobolev orthogonal polynomials are uniformly bounded, see Theorem 1.3 therein. The second parts of [L99b, L01d] are dedicated to asymptotics and are inspired by [GaK]. We will say that the inner product (3.20) is l-regular if there exists l ∈ {0, 1, . . . , m} such that l * Σk = Σ k=0

and, for each k = 0, 1, . . . , l, Σk is regular and µk ∈ Reg. Obviously, a sequentially dominated inner product is 0-regular provided that Σ0 is regular and µ0 ∈ Reg. The next theorem basically says that, if the Sobolev inner product is l-regular, n(j) }, j ≥ l, have regular asymptotic behavior. Recall that cap(A) the sequences {Q denotes the logarithmic capacity of the set A. Theorem 3.19 ([L01d]). Let the Sobolev inner product (3.20) be l-regular. The following assertions hold: (a) For j ≥ l, we have n(j) lim Q Σ

1/n

n→∞

= cap(Σ).

If, additionally, the interior of Σ is empty and its complement connected, then ∗ ΘQ(j) −→ λΣ , n → ∞, j ≥ l. n

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A WALK THROUGH APPROXIMATION THEORY

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(b) If the multiplication operator U is bounded with constant C, then (j) 1/n lim Q = cap(Σ) egΩ (z,∞) , j ≥ l, n (z) n→∞

uniformly on compact subsets of {z ∈ C : |z| > C} ∩ Ω, where Ω is the unbounded component of C \ Σ. Under the assumptions of Theorem 3.19 or some other particular cases, sevn(j) } actually eral authors have conjectured that the zeros of the polynomials {Q accumulate on C \ Ω. The paper [L08b] deals with strong (Szeg˝ o-type) asymptotics of Sobolev orthogonal polynomials when the measures are supported on an arc or Jordan curve of the complex plane. It may be seen as an extension of the results in [Mar00] and [MarPij] as well as of those in [BrFoMa]. Let Γ be an arc or a closed rectiﬁable Jordan curve in the complex plane. By Ω we denote the unbounded component of C \ Γ. For technical reasons, we will require that Γ fulﬁlls the regularity condition Γ ∈ C2+ . That is, if the complex valued function s(x) deﬁned on an interval [a, b] ⊂ R is the parametrization of Γ with respect to the arc length, then the second derivative of s(x) satisﬁes the Lipschitz condition |s (x) − s(y) | ≤ M |x − y|α , for some constants M, α > 0 and for all x, y ∈ [a, b]. Let µ be a ﬁnite positive Borel measure supported on Γ and µ = h(z)|dz| + µs its Lebesgue decomposition on Γ. We say that µ belongs to the Szeg˝o class SΓ if (log h(z)) |ΨΓ (z)| |dz| > −∞, Γ

where ΨΓ is the conformal mapping of Ω onto the exterior of the unit circle such that ΨΓ (z) 1 = . ΨΓ (∞) = ∞ and ΨΓ (∞) = lim z→∞ z cap(Γ) The authors of [L08b] consider a kind of inner product more general than (3.20), which basically consists in the sum of a Sobolev continuous inner product and a discrete non-diagonal one. In view of the results on weak asymptotics, all the measures in the continuous part are supported on the same set: the arc or Jordan curve Γ. Let µ0 , . . . , µm , be m + 1 ﬁnite positive Borel measures supported on Γ such that dµm (z) = hm (z) |dz|. Let {z1 , . . . , zr } ⊂ Ω be a ﬁnite set of points. Consider m−1 (3.29) p, qS = p(k) , q (k) L2 (µk ) + p(m) , q (m) m , p, q ∈ P, k=0

where ·, ·m is a discrete non-diagonal Sobolev inner product given by (3.30) p, qm = p(ζ) q(ζ) hm (ζ) |dζ| + p(Z) M q(Z)∗ , Γ

where

p(Z) = p(z1 ), . . . , p(d1 ) (z1 ), p(z2 ), . . . , p(d2 ) (z2 ), . . . , p(zr ), . . . , p(dr ) (zr ) ,

v ∗ stands for the transposed conjugate vector of v, and M is a hermitian positive deﬁnite matrix of order M = r + d1 + · · · + dr . Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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The main idea is to compare the behavior of the Sobolev monic orthogonal n } (and their derivatives) corresponding to the inner product (3.29) polynomials {Q with that of the Sobolev monic orthogonal polynomials {Pn } associated to the inner product (3.30), which is the term in (3.29) having the highest order derivative. The reason behind that way of proceeding relies on the fact that the derivatives of the polynomials are no longer monic but they are multiplied by a factor of order nk , k = 1, . . . , m, therefore playing a more important role than the Sobolev polynomials themselves. Then, it seems sensible to begin with obtaining the asymptotic behavior n(m) } as compared to {nm Pn−m }. of the dominant term {Q Notice that, when M = 0, the orthogonal polynomials {Pn } are standard. That was the case dealt with in [Mar00] for m = 1 and, for m > 1, in [MarPij]. When M = 0, the strong asymptotics of the polynomials {Pn } was obtained in [BrFoMa]. There, it was compared to the asymptotics of the polynomials {Qn } orthogonal with respect to the standard inner product given by the measure µm . Theorem 3.20 ([BrFoMa]). Let {Pn } and {Qn } be the sequences of monic orthogonal polynomials with respect to the Sobolev inner product (3.30) and the inner product given by hm (z) |dz|, respectively. Suppose that hm (z) |dz| ∈ SΓ . Then )di +1 ( r (j) Pn (z) ΨΓ (zi ) (ΨΓ (z) − ΨΓ (zi )) , j = 0, 1, . . . , lim = n→∞ Q(j) (z) ΨΓ (z)ΨΓ (zi ) − 1 n i=1 uniformly on compact subsets of Ω. In turn, the strong asymptotic behavior of the polynomials {Qn } was given in [Wi]. Under the assumption that hm (z) |dz| ∈ SΓ , it holds lim

n→∞

Qn (z) SΓ (hm ; z) , = SΓ (hm ; ∞) [cap(Γ) ΨΓ (z)]n

uniformly on compact subsets of Ω, where SΓ ( · ; z) is the Szeg˝o function associated n /Pn } to Γ, see [Wi] for details. Thus, it is clear that the relative asymptotics of {Q n }. A key step is to establish the asymptotic gives the strong asymptotics of {Q behavior of the Sobolev norms. For each n ∈ N, set (3.31)

n , Q n S κn = Q

and

τn = Pn , Pn m .

n } and {Pn } be the sequences of Sobolev monic Theorem 3.21 ([L08b]). Let {Q orthogonal polynomials with respect to the Sobolev inner products (3.29) and (3.30), respectively. Suppose that the measure µm ∈ SΓ is absolutely continuous with respect to |dz| on Γ. Then κn lim 2m = 1, n→∞ n τn−m where κn and τn are given by (3.31), and n(m) (z) Q = 1, n→∞ nm P n−m (z) lim

uniformly on compact subsets of Ω. Once the dominant term in the strong asymptotics is obtained, the rest of the terms can be recovered as well. Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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Theorem 3.22 ([L08b]). In addition to the conditions of Theorem 3.21, suppose that µk ∈ SΓ , k = 0, 1 . . . , m − 1. Then n(j) (z) Q 1 , = n→∞ nj P n−j (z) [cap(Γ) ΨΓ (z)]m−j lim

j = 0, 1, . . . ,

uniformly on compact subsets of Ω. As a consequence of Theorem 3.22, it may be proved that, for suﬃciently n . The rest large n, each point zi , i = 1, . . . , r, attracts exactly di + 1 zeros of Q accumulate on Γ ∪ V , where V = C \ Ω. Thus, the strong asymptotics picture of the Sobolev orthogonal polynomials seems to be wholly drawn; except perhaps for the fact that some conditions could be weakened like, for instance, the additional assumptions of Theorem 3.22. Such conditions are clearly not necessary since the asymptotic behavior of the polyno n } only depends on the measure µm and the matrix M. This may be mials {Q explained by the above-mentioned argument on the size of the derivatives. As in the weak asymptotics case, the consideration of measures supported on disjoint sets poses a much more diﬃcult problem. Very few results have been proved in the case when the measures involved in the inner product (3.20) have unbounded supports. Setting aside some works dealing with particular measures (see the introduction in [L06c] and the references therein), the ﬁrst work considering a general class is [GeLuMa]. The authors of that paper study the inner product (3.32) p, qS = p(x) q(x) ψ 2 (x) w2 (x) dx + λ p (x) q (x) w2 (x) dx, p, q ∈ P, R

R

where λ > 0, ψ ∈ L∞ (R) is a non-negative function which is positive on a set of positive measure, and w belongs to a class of exponential weights which includes the weights wα (x) = exp(−|x|α ), α > 1. Provided all monomials are integrable, let { qn } be the sequence of Sobolev orthonormal polynomials with respect to the inner product (3.32), that is, qn , qn S = 1, n ∈ N, and let {qn } be the sequence of standard orthonormal polynomials with respect to the measure w2 (x) dx. Among other results, the authors of [GeLuMa] prove that

1 qn − √ qn−1 w lim = 0. n→∞ λ L∞ (R) Therefore, the re-scaled (see below) strong asymptotics of { qn } may be obtained from the re-scaled strong asymptotics of {qn }, which is known. Notice the dependance on the parameter λ in contrast to the bounded case. The following major achievement was the paper [L06c] which is concerned with the weak asymptotics of Sobolev orthogonal polynomials for Freud-type weights. Let us describe a simpliﬁed version of the main result. Let {w0 , w1 , . . . , wm } be a family of positive continuous functions on R and consider the inner product (3.20) where dµk (x) = wk2 (x) dx, k = 0, 1, . . . , m. We will say that a positive continuous function w on R belongs to the class W (α, c), α > 0, c > 0, if − log w(x) lim = 1. c |x|α |x|→∞ Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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For example, exp(−c|x|α ) ∈ W (α, c). In the following, we will assume that the weights {wk } are asymptotically of Freud type, that is, wk ∈ W (αk , τk ), k = 0, 1, . . . , m. It turns out that, roughly speaking, the re-scaled Sobolev orthogonal polynomials behave as the re-scaled standard polynomials orthogonal with respect to the measure that dominates the inner product (3.20). Although it is possible to reason with more generality, we will suppose that wk (3.33) ∈ L∞ (R), k = 1, . . . , m, wk−1 which is equivalent to say that the Sobolev inner product (3.20) is sequentially dominated. Thus, condition (3.33) implies that w0 ∈ W (α0 , c0 ) is the dominant weight. Generally speaking, the zeros of the polynomials orthogonal with respect to measures supported on R are not uniformly bounded. Moreover, the number of zeros of each polynomial in any ﬁxed ﬁnite interval is o(n), that is, the zeros tend to inﬁnity. Therefore, it is necessary to re-scale the polynomials. Set α 1

Γ Γ 2 2

, γα = α > 0, 1+α 2Γ 2 where Γ(·) stands for the Gamma function. The numbers γα play an important role when re-scaling the polynomials orthogonal with respect to a Freud weight since, for each n ∈ N, it holds exp(−|x|α )pn (x)L∞ (R) = exp(−|x|α )pn (x)L∞ ([−an ,an ]) for all polynomials pn ∈ Pn , where {an } are the Mhaskar-Rakhmanov-Saﬀ numbers. It is well known that, for this type of weight, an = (nγα )1/α , n ∈ N. The ﬁrst and most important step is to prove that the Sobolev orthogonal n } are asymptotically extremal for the weighted uniform norm on R polynomials {Q given by the dominant weight w0 . For that, the main idea is to compare the Sobolev n } to the corresponding weighted uniform norm norm (3.28) of the polynomials {Q on R. This is done by making use of Markov and Nikolskii type inequalities and the fact that n S = min {P S : P (x) = xn + . . . } . Q Theorem 3.23 ([L06c]). Suppose that wk ∈ W (αk , ck ), k = 0, 1 . . . , m, and n } be the sequence of Sobolev monic orthogonal poly(3.33) takes place. Let {Q nomials with respect to the inner product (3.20) where dµk (x) = wk2 (x) dx, k = 0, 1, . . . , m. Then

1/α0 1/n 1 γα0 −1/α0 (j) α0 lim n = , j = 0, 1, . . . . Qn (x) exp (−c0 |x| ) n→∞ 2 c0 e L∞ (R) Straightforward consequences of Theorem 3.23 are the asymptotic re-scaled (or n(j) } and the weak contracted) limit distribution of the zeros of the polynomials {Q asymptotics of the re-scaled Sobolev polynomials. To properly state such results, deﬁne the Ullmann distribution uα , α > 0, as duα (t) = Uα (t) dt, where α 1 xα−1 √ dx, t ∈ [−1, 1]. Uα (t) = π |t| x2 − t2 Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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The Ullman distribution uα turns out to be the equilibrium measure of the interval [−1, 1] in the presence of the external ﬁeld given by the Freud weight exp(−γα |x|α ). The logarithmic potential of the distribution uα is 1 z xα−1 1 √ V uα (z) = − log z + z 2 − 1 − dx + + log 2, z ∈ [−1, 1]. α z 2 − x2 0 Let Pn be a monic polynomial of degree n. In order to re-scale the polynomial Pn , let us adopt the following notation ,

1/α nγ n/α c α Pn (α, c; x) = Pn x , α > 0, c > 0. c nγα The limit contracted zero distribution of the polynomials orthogonal with respect to the Freud weights w2 (x), with w(x) = exp(−c|x|α ), is known to be the Ullman distribution. The authors of [L06c] obtain, under suitable conditions, the same result for the Sobolev orthogonal polynomials. Notice that their asymptotic behavior only depends on the dominant weight w0 . Theorem 3.24 ([L06c]). Suppose that wk ∈ W (αk , τk ), k = 0, 1 . . . , m, and n } be as in Theorem 3.23. For each j ≥ 0 and n ∈ N, let (3.33) takes place. Let {Q Sn,j be such that n! (j) (x). Sn,j (α0 , c0 ; x) = Q (n + j)! n+j Then ∗ ΘSn,j −→ uα0 , n → ∞, and lim |Sn,j (z)|

1/n

n→∞

= exp {−V uα0 (z)} ,

uniformly on compact subsets of C \ R. 4. Hermite-Pad´ e approximation Let F = (f1 , . . . , fm ) be a system of functions such that each function in F has an expansion in powers of 1/z, that is, ∞ am,k fk (z) = , m+1 z m=0

k = 1, . . . , m.

We will mostly deal with analytic functions on a neighborhood of z = ∞. Like in Section 2, the choice of the point z = ∞ is not essential and it is mainly motivated by the structure of the functions considered below. We are concerned with the simultaneous approximation of the functions in F. Fix a multi-index n = (n1 , . . . , nm ) ∈ Zm + , where Z+ = {0, 1, . . .}. Set |n| = n1 + · · · + nm . A Hermite-Pad´e approximant corresponding to F and n is a vector of rational functions Rn = (Rn,1 , . . . , Rn,m ) = (Pn,1 /Qn , . . . , Pn,m /Qn ), such that (4.1)

deg Pn,k ≤ |n| − 1,

k = 1, . . . , m,

Qn ≤ |n|

Qn ≡ 0,

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where the polynomial Qn is monic, and

1 , (4.2) Qn (z)fk (z) − Pn,k (z) = O z nk +1

z → ∞,

k = 1, . . . , m.

We have three ingredients in the deﬁnition: a vector of functions F to be approximated, a vector of rational approximants Rn , and a vector of non-negative integers n which controls the degree of approximation at ∞ in each component. The main feature is that all the approximants {Rn,k }m k=1 share the same polynomial Qn in the denominator, from which the name simultaneous approximation follows. Information about the Hermite-Pad´e approximation may be found in the surveys [ApSt, Nu] and the book [NiSo]. What is diﬀerent from classical Pad´e approximation? When m is equal to 1, we regain the Pad´e approximants, so there is no diﬀerence at all. When m > 1, the Hermite-Pad´e approximant, although it exists, is not unique in general. For instance, if there exist i, k ∈ {1, . . . , m}, i = k, such that fi = fk and ni , nk > 0. Uniqueness is an essential property in order to obtain results and the lack of it is what makes it much harder to deal with simultaneous approximation than Pad´e approximation. Given a system of functions F, we will say that a multi-index n is normal if any non trivial solution Qn of (4.1) and (4.2) satisﬁes deg Qn = |n|. Normality implies uniqueness and if any multi-index n for the system F is normal we will say that the system F is perfect. Be aware that diﬀerent names appear in the literature for these concepts. We are interested in ﬁnding perfect systems or, at least, normal multiindices for a given system. The ﬁrst example given of a perfect system is the one formed by the powers of the exponential function E = {ez , e2z , . . . , emz }. Hermite worked out the Hermite-Pad´e approximants of E (with interpolation carried out at z = 0) to prove the transcendence of the number e and since then, Hermite-Pad´e approximants have proved to be useful in several problems of number theory (cf. [Ni80, BaRi]). In the sequel we will deal with systems formed by Cauchy transforms. Consider F = ( s1 , . . . , sm ), where each sk is given by dsk (x) , k = 1, . . . , m, sk (z) = z−x and S = (s1 , . . . , sm ) is a system of ﬁnite Borel measures with constant sign. The support Σk ⊂ R of each measure sk , k = 1, . . . , m, is a set, not necessarily bounded, consisting of inﬁnitely many points. The smallest interval containing Σk will be denoted by ∆k , k = 1, . . . , m. For each multi-index n, the common denominator Qn turns out to be a multiorthogonal polynomial with respect to the system of measures S since it satisﬁes the orthogonality relations (4.3) xν Qn (x) dsk (x) = 0, ν = 0, 1, . . . , nk − 1, k = 1, . . . , m. Thus, Hermite-Pad´e approximation is closely related to multiple orthogonal polynomials [Ap98, V], a subject of importance by its own with recent applications to a number of topics like random matrices [BK04a, BK04b] and Brownian motion [DK]. One of the basic facts in the proof of the classical Markov and Stieljes Theorems is that the orthogonal polynomials have all their zeros simple and lying in Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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the support of the measure. Those properties are also important in simultaneous approximation. Given a system F, we will say that the multi-index n is strongly normal if Qn has exactly |n| simple zeros and they all lie on ∆, ∆ being the smallest interval containing ∪m k=1 ∆k . If all the multi-indices for the system F are strongly normal, F is said to be strongly perfect. Obviously, being strongly perfect implies being perfect. Normality of multi-indices plays a fundamental role in the study of the convergence of the Hermite-Pad´e approximants. For that reason, research on asymptotics has been focused on particular systems of Cauchy transforms for which normality can be proved for a wide choice of multi-indices: the so-called Angelesco and Nikishin systems. A system of Cauchy transforms F is called an Angelesco system if ∆i ∩ ∆j = ∅, i = j (see [An]). From the multi-orthogonality relations (4.3) readily follows that the Angelesco systems are strongly perfect. Maybe the most important result on Angelesco systems was obtained by Gonchar and Rakhmanov [GR83] assuming that sk > 0 a. e. on ∆k , k = 1, . . . , m. As regards multi-indices, they require that the orthogonality relations are proportionally distributed among the components of n, that is, (4.4)

lim

|n|→∞

nk = ck > 0, |n|

k = 1, . . . , m.

They use a vector potential equilibrium problem to describe the |n|-th root asymptotics of Qn and the limit distributions of their zeros on the intervals {∆k }m k=1 . The distinctive feature is that the supports of the vector equilibrium measure are intervals ∆∗k ⊂ ∆k , k = 1, . . . , m, not coinciding in general with the system of intervals {∆k }m k=1 . As a consequence, domains of divergence of Rn may appear in C \ ∪m ∆ depending on the geometry of the intervals ∆k and the numbers k k=1 ck , k = 1, . . . , m, but not on the system of measures S. Convergence always holds on a neighborhood of z = ∞. For other results on these systems, see [Kal] and [Ap89]. In conclusion, the Angelesco systems, although are deﬁned in a natural way and are strongly perfect, do not have, in general, good properties regarding convergence. The Nikishin systems, introduced by Nikishin in [Ni82], are, in a sense, opposite to the Angelesco systems since all the measures of F are now supported on the same interval. In fact, Gonchar, Rakhmanov, and Sorokin by using tree graphs introduced in [GRSo] systems of Cauchy transforms, called generalized Nikishin systems, which contain Angelesco and Nikishin systems as particular and extreme cases. We will adopt their notation. Given a system of intervals ∆1 , . . . , ∆m satisfying ∆k−1 ∩ ∆k = ∅, k = 2, . . . , m, and a system of measures S1 = (σ1 , . . . , σm ), where the support of σk is contained in ∆k , k = 1, . . . , m, we will deﬁne a second system S2 = (s1 , . . . , sm ), whose Cauchy transforms will form the Nikishin system. Take the measures σ1 and σ2 . Deﬁne the following kind of product of measures dσ1 , σ2 (x) =

dσ2 (t) dσ1 (x) = σ 2 (x)dσ1 (x), x−t

so that σ1 , σ2 is a measure with support in ∆1 . Deﬁne inductively σ1 , σ2 , . . . , σj = σ1 , σ2 , . . . , σj ,

j = 1, . . . , m.

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Set sj = σ1 , σ2 , . . . , σj , j = 1, . . . , m. Observe that all such measures sj are supported on the interval ∆1 . The system of Cauchy transforms F = ( s1 , . . . , sm ) corresponding to the system of measures S2 = (s1 , . . . , sm ) is the Nikishin system generated by the measures S1 = (σ1 , . . . , σm ). Unlike the Angelesco systems, it is not known whether the Nikishin systems are strongly perfect or not. Nevertheless, the class of multi-indices for which strong normality holds has been widening over the years due to the eﬀorts of several people. The ﬁrst who found strongly normal indices for such systems was Nikishin himself. Theorem 4.1 ([Ni82]). The multi-indices n of the form n + 1, i ≤ q; (4.5) n = (n1 , . . . , nm ) where ni = 0 ≤ q ≤ m, n, i > q, are strongly normal. Also, in the case of systems formed by two functions, he obtained uniform convergence of the approximants without requiring any condition on the measures involved. Theorem 4.2 ([Ni82]). Suppose that n = (n, n) and ∆1 is bounded, then lim

n→+∞

Pn,k (z) = sk (z), Qn (z)

k = 1, 2,

uniformly on compact subsets of C \ ∆1 . A major breakthrough on the study of the Nikishin systems was achieved by L´ opez and Bustamante in [L94] extending Theorem 4.2 to an arbitrary number m of functions. Following an idea of Gonchar, they observed that, for each k = 1, . . . , m, the approximant Rn,k is close to being a multi-point Pad´e approximant of sk , thus reducing the proof to the study of multi-point Pad´e approximation and polynomials orthogonal with respect to varying measures. According to (4.2), |n| + nk + 1 interpolation conditions of Rn,k are assigned at inﬁnity by deﬁnition. The key ingredient is that a bonus of nearly |n| − nk interpolation points appears on ∆2 , adding up to nearly 2|n| + 1 interpolating conditions. The next step was to prove convergence in capacity of the approximants. For that, they require that the components of the multi-indices increase uniformly, that is, there exists a positive constant C such that |n| (4.6) nk ≥ − C, k = 1, . . . , m. m Finally, the following condition is intended to prove convergence of the approximants without the requirement that the intervals involved be bounded. ∞ 1 = +∞, c = |x|m ds1 (x), m ∈ N. (4.7) √ m 2m c m ∆1 m=1 Theorem 4.3 ([L94]). Suppose that either ∆2 is bounded or the Carleman condition (4.7) is fulﬁlled. Then, for multi-indices n of the form (4.5), it holds Pn,k (z) = sk (z), n→+∞ Qn (z) lim

k = 1, . . . , m,

uniformly on compact subsets of C \ ∆1 . Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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Typically, convergence in capacity of the Hermite-Pad´e approximants makes use of some kind of uniformity in the increasing of the components of the multiindices whereas uniform convergence needs strongly normal multi-indices since, due to Gonchar’s Lemma, uniform convergence follows from convergence in capacity and control over the zeros of Qn . As a by-product of the method used in the proof of Theorem 4.3, the Nikishin systems made up by two functions may be proved to be strongly perfect. A detailed proof of this was published by Driver and Stahl in [DrSt] where they also extended a result that Nikishin stated without proof in [Ni82]. Nikishin assured that the decreasing indices are strongly normal and Driver and Stahl obtained such property for all indices satisfying 1 ≤ j < k ≤ m ⇒ nk ≤ nj + 1. That set has been denoted by Zm + (). Concerning asymptotics, as in the Angelesco case, Gonchar, Rakhmanov, and Sorokin [GRSo], found the |n|-th root asymptotic behavior and the zero limit distribution of the polynomials {Qn }. The techniques used in the proof require that σk > 0 a.e. on the bounded interval ∆k , k = 1, . . . , m, and that the orthogonality relations be proportionally distributed among the components of n in the sense of (4.4). Additionally, the multi-indices n are taken to belong to the set Zm + (). The limit behavior is described in terms of a vector potential equilibrium problem as in the Angelesco case but the interaction between the components of the vector valued problem is diﬀerent. Now, the supports of the vector equilibrium measure are the intervals ∆k , k = 1, . . . , m. A key role in the proof is played by the functions of the second kind Ψn,k , k = 1, . . . , m, deﬁned inductively by Ψn,k−1 (x) dσk (x), k = 1, . . . , m. Ψn,0 (z) = Qn (z), Ψn,k (z) = z−x Denote by Qn,k , k = 1, . . . , m, the monic polynomial whose zeros are the zeros of Ψn,k−1 on ∆k . The polynomials Qn,k satisfy the following properties, • Each Qn,k is orthogonal with respect to a varying measure ωn,k dσk on ∆k with deg Qn,k = nk + · · · + nm . • The zero limit distributions of Qn,k , k = 1, . . . , m, are the components of the vector equilibrium measure. • Qn,1 = Qn . In particular, Qn is orthogonal with respect to the varying measure dσ1 /|Qn,2 |. As a consequence, taking ck = 1/m, k = 1, . . . , m, in (4.4), the exact rate of convergence of Rn,k to sk , k = 1, . . . , m, for compact subsets of C \ (∆1 ∪ ∆2 ), is obtained. It is not possible to describe the exact rate of convergence on ∆2 due to the extra interpolation points which appear on this set. All the results of [GRSo] are given in the more general setting of generalized Nikishin systems for which the above assertions must be adequately modiﬁed. For instance, in such a general situation domains of divergence for the simultaneous approximants do appear. Recall that one of the main contributions of [L94] was the observation that Qn was close to be a polynomial orthogonal with respect to a varying measure. Here, we ﬁnd a more elaborate structure with Qn satisfying full orthogonality relations, which in general provides more information. The varying measure ωn,k dσk involves the function Ψn,k−1 and some of the polynomials Qn,k , k = 1, . . . , m, in its deﬁnition. Therefore, when trying to prove some result on the polynomials {Qn }, there is usually need of dealing with the whole family {Qn,k }m k=1 . Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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In [L99c] the authors proved an extension to the polynomials {Qn,k }m k=1 of the well-known theorem (3.13) on weak convergence of orthogonal polynomials in the Nevai class. For each multi-index n and k = 1, . . . , m, let qn,k be the polynomial Qn,k normalized so that it is orthonormal with respect to ωn,k (x) dσk (x). Set ∆k = [ak , bk ], k = 1, . . . , m. Again, we will need a uniformity condition on the multi-indices n. We will suppose that if n = (n1 , n2 , . . . , nm ), it holds n1 − nm ≤ C,

(4.8)

where C is an absolute constant independent of n. Theorem 4.4 ([L99c]). Suppose that σk > 0 a. e. on ∆k , k = 1, . . . , m. Let Λ ⊂ Zm + () be a sequence of multi-indices such that, for all n ∈ Λ, property (4.8) takes place and n1 → ∞ as n varies over Λ. Then, for each k = 1, . . . , m, and every bounded Borel measurable function f on ∆k , we have bk 1 bk dx 2 f (x) qn,k (x) ωn,k (x) dσk (x) = f (x) . (4.9) lim n∈Λ a π ak (bk − x)(x − ak ) k The proof of Theorem 4.4 relies on a formula similar to (4.9) for orthogonal polynomials with respect to varying measures of the form dµn /A2n , where the measures µn converge in some sense to be speciﬁed below and A2n are polynomials of the type (2.11). This is one of the most general classes of varying measures L´ opez has dealt with. To be more precise, let {µn }n∈N be a sequence of ﬁnite positive Borel measures on ∆ = [−1, 1] whose supports contain inﬁnitely many points. Let {A2n }n∈N be a sequence of polynomials deﬁned by (2.11), modiﬁed in the same way as (2.14), and verifying (2.15) and (2.16); where we must replace dµ with dµn whenever it appears in those conditions. If, additionally, there exists a ﬁnite positive Borel measure µ on ∆ such that µ > 0 a.e. on ∆ and ∗ (a) µn −→ µ, n → ∞, 1 |µn (x) − µ (x)| dx = 0, (b) lim n→∞

−1

we will say that ({µn }, {A2n }, k) is strongly admissible on ∆. Let ln,m (x) = κn,m xm + . . . , κn,m > 0, be the m-th orthonormal polynomial with respect to the measure dµn (x)/A2n (x). In this setting we have a theorem analogous to Theorem 2.10. Theorem 4.5 ([L99c]). Let ({µn }, {A2n }, 2k) be strongly admissible on the interval [−1, 1] for each k ∈ N, then ln,n+j+1 (z) lim = z + z 2 − 1, j ∈ Z, n→∞ ln,n+j (z) uniformly on each compact subset of C \ [−1, 1]. After that, it is easy to obtain a result on weak convergence of the varying orthonormal polynomials ln,m . Theorem 4.6 ([L99c]). Let ({µn }, {A2n }, 2k) be strongly admissible on the interval [−1, 1] for each k ∈ N. Then, for each j ∈ Z and every bounded Borelmeasurable function f on [−1, 1], we have 1 2 ln,n+j (x) 1 1 dx lim dµn (x) = f (x) f (x) √ . n→∞ −1 A2n (x) π −1 1 − x2

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This is enough to prove Theorem 4.4, since, under the conditions required, the varying weight functions ωn,k (x) are of the form ωn,k (x) =

hn,k (x) , |Qn,k−1 (x) Qn,k+1 (x)|

k = 1, . . . , m,

with Qn,0 ≡ 1, Qn,m+1 ≡ 1, hn,1 ≡ 1, and 1 lim hn,k (x) = , |(x − bk−1 )(x − ak−1 )|

n∈Λ

k = 2, . . . , m,

uniformly on ∆k = [ak , bk ]. Therefore, the varying numerators hn,k (x) verify (a) and (b), from which Theorem 4.4 follows. For possible applications to Hermite-Pad´e approximation, it would be interesting to obtain asymptotic results for orthogonal polynomials with respect to varying measures more general than those of Theorem 4.5 weakening conditions (a) and (b). Concerning other kinds of asymptotics of {Qn } for Nikishin systems, strong convergence of {Qn } was proved by Aptekarev in [Ap99] for multi-indices of the form n = (n, . . . , n) and measures {σk } in the Szeg˝o class. Among other arguments, Theorem 4.4 is used. As for ratio asymptotics, L´opez and collaborators have proved the following result. Set n = (n1 , . . . , nm ) and deﬁne nj = (n1 , . . . , nj−1 , nj + 1, nj+1 , . . . , nm ),

j = 1, . . . , m.

Theorem 4.7 ([L05a]). Suppose that σk > 0 a. e. on ∆k , k = 1, . . . , m. Fix j ∈ {1, . . . , m}. Let Λ ⊂ Zm + () be a sequence of multi-indices such that, for all n ∈ Λ, nj ∈ Zm + () and (4.8) takes place. Then, there exists a function Fm,j analytic on C \ ∆1 such that (4.10)

lim

n∈Λ

Qnj (z) = Fm,j (z), Qn (z)

uniformly on compact subsets of C \ ∆1 . In particular, m Qn+1 (z) = Fm,j (z), n∈I Qn (z) j=1

lim

n + 1 = n + (1, . . . , 1),

uniformly on compact subsets of C \ ∆1 , where I ⊂ Zm + () is the family of multiindices given by I = {(0, . . . , 0), (1, . . . , 0), (1, 1, . . . , 0), . . . , (1, 1, . . . , 1), (2, 1, . . . , 1), . . . }. A key step in proving (4.10) was to obtain interlacing properties of the zeros of consecutive polynomials Qn from which follows that the ratios Qnj /Qn form a normal family. That the limit functions do not depend on the subsequence chosen is the next step, which is done showing that these functions satisfy a uniquely solvable system of boundary-value problem. To this end, Theorem 4.4 is used as well as ratio (Theorem 4.5) and relative asymptotics of varying orthogonal polynomials of the same kind employed in proving that theorem. Of course, the proof of Theorem 4.7 needs dealing with the whole family of polynomials {Qn,k }m k=1 and similar formulas to (4.10) are obtained for them. The limit functions Fm,j , j = 1, . . . , m, are given in terms of certain algebraic functions of order m + 1. However, explicit expressions of them are not available Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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for m > 1. When m = 1 the formula (4.10) reduces √ to the Rakhmanov theorem for orthogonal polynomials on R and F1,1 (z) = (z + z 2 − 1)/2, see (3.12). Before Theorem 4.7 had been proved, some advances concerning the algebraic aspects of the theory had been achieved. By ﬁnding an extra set of orthogonality relations for the polynomials Qn (see formula (5) in [L02b]), the problem of the perfectness of Nikishin systems was completely solved for m = 3. Theorem 4.8 ([L02b]). The Nikishin systems made up by three functions are strongly perfect. The following theorem constitutes, up to now, the best result on normality of multi-indices for Nikishin systems of an arbitrary number of functions. It says that the the multi-indices which do not have three increasing components are strongly normal. Theorem 4.9 ([L03a]). Given any Nikishin system, the multi-indices n verifying that there do not exist 1 ≤ j < k < l ≤ m such that nj < nk < nl are strongly normal. The set of multi-indices which Theorem 4.9 refers to will be denoted by Zm + (∗). This set is considerably larger than Zm () and the use of such general multi-indices + aﬀects the deﬁnition by induction of the functions of the second kind {Ψn,k }m k=1 and, in turn, the deﬁnition of the family of polynomials {Qn,k }m . That construction k=1 depends on the relative value of the components of the multi-indices. We will manage that by considering all of the possible cases. Suppose that n = (n1 , . . . , nm ). Let pn denote the permutation of {1, . . . , m} given by nj > nk , for k < j for all k ∈ {1, . . . , m}\{τ (1), . . . , τ (i−1)}. pn (i) = j if nj ≥ nk , for k > j That is, pn (1) is the subindex of the ﬁrst component of n (from left to right) which is greater than or equal to the rest, pn (2) is the subindex of the ﬁrst component of n which is second largest, and so on. For example, if n1 ≥ · · · ≥ nm , then pn is the identity. Let τ denote a permutation of {1, . . . , m}. Set m Zm + (∗, τ ) = n ∈ Z+ (∗) : pn = τ . For a ﬁxed τ , the construction of the functions of the second kind {Ψn,k }m k=1 turns out to be the same for all multi-indices n in the set Zm (∗, τ ). + The rest of the results in this section are characterized by their great generality partly due to the consideration of multi-indices in Zm + (∗). Thus, [L06a] deals with the exact rate of convergence of multi-point Hermite-Pad´e approximants for Nikishin systems constructed with multi-indices in the set Zm + (∗) and with some of the poles ﬁxed beforehand. Despite its complexity, we will describe the result with some detail because it constitutes one of the highlights of the theory. Let Λ ⊂ Zm + (∗) be a sequence of multi-indices n = (n1 , . . . , nm ) such that |n| tends to ∞ and let rn be a sequence of non-negative integers. Consider monic polynomials An and Bn with real coeﬃcients such that deg Bn = rn and deg An ≤ |n| + 2 rn + min{n1 , . . . , nm }. The zeros of An belong to a compact set F of C \ ∆1 and form the table A of interpolation points whereas the zeros of Bn2 lie on ∆1 and make up the table B of ﬁxed poles. A generalized Hermite-Pad´e approximant of the Nikishin system of functions F = ( s1 , . . . , sm ) with preassigned poles at the zeros of the polynomial Bn2 , which Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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interpolates the system F at the zeros of the polynomial An is a vector of rational functions

P Pn,m n,1 A,B A,B RnA,B = Rn,1 = , . . . , Rn,m , . . . , Qn Bn2 Qn Bn2 such that Qn and Pn,k , k = 1, . . . , m, satisfy the following conditions: (i) deg Qn ≤ |n|, Qn ≡ 0 and deg Pn,k ≤ |n| + 2 rn − 1, k = 1, . . . , m. (ii)

Qn Bn2 sk − Pn,k is an analytic function on C \ ∆1 , k = 1, . . . , m. An

(iii)

Qn (z) Bn2 (z) sk (z) − Pn,k (z) = O(1/z nk +1 ), An (z)

z → ∞,

k = 1, . . . , m.

The interpolation points and the ﬁxed poles are assumed to have limit distributions, that is, (4.11)

deg An ∗ ΘAn −→ α, |n|

n ∈ Λ,

deg Bn2 ∗ ΘBn2 −→ β, |n|

n ∈ Λ,

where the weak star limits are taken as n varies over Λ. Recall that ΘT stands for the normalized zero counting measure of the polynomial T , see deﬁnition (2.25). Following similar ideas as those of [GRSo], the |n|-th root asymptotic behavior of the polynomials Qn,k , k = 1, . . . , m, is obtained in terms of the solution of a vector valued equilibrium problem, now in the presence of an external vector ﬁeld f = (f1 , . . . , fm ). Let us denote such a solution by µ = (µ1 , . . . , µm ). For each k = 1, . . . , m, the support ∆∗k of the measure µk is contained in ∆k . Set wkµ = V µk (x) + f (x),

x ∈ ∆∗k ,

k = 1, . . . , m,

where V µ denotes the logarithmic potential of the measure µ. The external ﬁeld f represents the inﬂuence of the established tables of interpolation points and ﬁxed poles. Actually, the components of the external ﬁeld are given by (4.12)

f1 (x) = V β (x) − V α (x),

x ∈ ∆1 ,

fk (x) ≡ 0,

k = 2, . . . , m.

For further details on the potential theoretic arguments involved and the precise description of the vector equilibrium problem, see Theorem 4 in [L06a]. With the deﬁnitions and notations above, we can state the result on the |n|-th root asymptotics. Theorem 4.10 ([L06a]). Suppose that σ1 > 0 a.e. on ∆1 and σk ∈ Reg for k = 2, . . . , m. Assume that (4.4) and (4.11) take place. Then (4.13)

lim |Qn,k (z)|1/|n| = e−V

µk (z)

n∈Λ

,

uniformly on compact subsets of C \ ∆k , k = 1, . . . , m. Moreover, for each k = 1, . . . , m, we have (4.14)

lim |Ψn,k (z)|

n∈Λ

1/|n|

µ

= eUk (z) ,

uniformly on compact subsets of the set C \ (∆k ∪ ∆k+1 ), ∆m+1 ≡ ∅, where Ukµ = V µk − V µk+1 − (w1µ + . . . wkµ ),

V µm+1 ≡ 0.

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A,B For each k = 1, . . . , m, the rate of convergence of the approximants Rn,k is obtained through Theorem 4.10 and the integral formula for the remainder Bn2 (x) Qn (x) dsk (x) An (z) A,B , z ∈ C \ ∆1 . (4.15) sk (z) − Rn,k (z) = 2 Bn (z) Qn (z) ∆1 An (x) z−x

From (4.11), (4.12), and (4.13), it follows that 1/|n| µ1 An (z) = eV (z)+f1 (z) , (4.16) lim 2 n∈Λ Bn (z) Qn (z) uniformly on compact subsets of C \ (∆1 ∪ F ). Recall that F is the compact set of C \ ∆1 where the interpolation points lie. If we take upper limit instead of limit, formula (4.16) holds uniformly on compact subsets of C \ ∆1 . On the other hand, the asymptotic behavior of the integral in (4.15) is related to that of the functions Ψn,k , k = 1, . . . , m, in (4.14) by introducing an auxiliary Nikishin system, which is technically quite involved. In particular, the remainders may satisfy diﬀerent asymptotic formulas depending on the relative limiting values of the components of the multi-indices. Fix k ∈ {1, . . . , m}. For i = 1, . . . , k, deﬁne the regions Dik = z ∈ C \ ∆1 : Uiµ (z) > Ujµ (z), j = 1, . . . , k . Denote

ξk = max Ujµ , j = 1, . . . , k ,

k = 1, . . . , m.

Theorem 4.11 ([L06a]). Assume that the conditions of Theorem 4.10 hold. Let τ be a ﬁxed permutation of {1, . . . , m} such that Λ ⊂ Zm + (∗, τ ). Then, for each k = 1, . . . , m, we have 1/|n| µ1 A,B (4.17) lim sτ (k) (z) − Rn,τ (z) = eV (z)+f1 (z)+ξk (z) , (k) n∈Λ

. uniformly on compact subsets of ki=1 Dik \ F , except on at most a discrete set of points. If we take upper limit instead of limit, formula (4.17) becomes an inequality and holds uniformly on compact subsets of C \ ∆1 . Fix k ∈ {1, . . . , m}, in principle we have diﬀerent asymptotic formulas in each one of the regions Dik , i = 1, . . . , k, since ξk ≡ Uiµ on Dik . However, some of such regions may be empty. The number of diﬀerent non-empty regions is at most one more than the number of strict inequalities in the sequence cτ (1) ≥ · · · ≥ cτ (k) , where the constants ci , i = 1, . . . , k, are given by (4.4). Theorem 4.11 describes the regions of convergence and divergence at geometric rate of the approximants Rn,k . They are characterized by the sign of V µ1 + f1 + ξk , k = 1, . . . , m. Obviously, an arbitrary choice of the ﬁxed poles may impair convergence. The largest possible region of convergence is C \ ∆1 which is attained provided that the poles are adequately ﬁxed. Roughly speaking, they must be placed in such a way that the support of µ1 is the whole interval ∆1 . This is done by means of a condition of type (2.27) which is trivially fulﬁlled if β = 0. For details, see Corollary 2 in [L06a]. In [L07a] the problem of convergence of multi-point Pad´e approximants for Nikishin systems was broached from a diﬀerent point of view. It was sought to prove geometric rate of convergence under as weak assumptions as possible. Now, Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

A WALK THROUGH APPROXIMATION THEORY

71 47

ﬁxed poles are not considered and, again, the table A of interpolation points is contained in a compact subset F of C \ ∆1 . The rest of the assumptions only aﬀect the multi-indices. The main step is to prove convergence in capacity of the approximants which is obtained under the rather weak assumption on the multiindices n = (n1 , . . . , nm ), (4.18)

nk ≥ |n|/m − C|n|κ ,

C > 0,

κ < 1,

k = 1, . . . , m.

Fix ζ ∈ C \ ∆1 . Let Ψζ be the conformal mapping from C \ ∆1 onto the unit disk D such that Ψζ (ζ) = 0 and Ψζ (ζ) > 0. Theorem 4.12 ([L07a]). Let Λ ⊂ Zm + be a sequence of distinct multi-indices such that (4.18) takes place and, if m > 3, Λ ⊂ Zm + (∗). Then, on each compact subset K of C \ ∆1 , we have 1/2|n|

A lim sup Rn,k − sk K

≤ δK < 1,

k = 1 . . . , m,

n∈Λ

where δK = max {|Ψζ (z)| : z ∈ K, ζ ∈ F ∪ ∆2 ∪ {∞}}. The method of the proof borrows some ideas from Theorems 2.6 and 2.15. As compared to Theorem 4.11, Theorem 4.12 is less precise in the expression of the rate of convergence but more general in the class of measures and table of interpolation points considered. Also, notice that condition (4.18) is weaker than (4.6) and the set of multi-indices considered is much larger than that of Theorem 4.3. Finally, Theorem 4.7 on ratio asymptotics of the polynomials Qn was generalized in [L08a] in the direction followed by Denisov [De] (cf. Section 3.2). The asymptotic relation (4.10) was proved for measures σk , k = 1, . . . , m, being as in Denisov’s Theorem and multi-indices n ∈ Zm + (∗). We will say that a system of measures (σ1 , . . . , σm ) is of Denisov type if, for each k = 1, . . . , m, it holds • The support of σk is the compact set Σk = [ak , bk ] ∪ Ek ⊂ R, where Ek is a set without accumulation points in R \ [ak , bk ]. • ∆k ∩ ∆k+1 = ∅, where ∆k is the convex hull of Σk and ∆m+1 = ∅. • σk > 0 a.e. on [ak , bk ]. The usual condition on the uniform increasing of the components of the multiindices now takes the following form: let Λ be an inﬁnite sequence of distinct multi-indices with the property (4.19)

max

max {mnk − |n|} < +∞.

n∈Λ k=1,...,m

Theorem 4.13 ([L08a]). Let (σ1 , . . . , σm ) be a system of measures of Denisov type. Let Λ ⊂ Zm + (∗) be a sequence of distinct multi-indices such that (4.19) takes place. Fix j ∈ {1, . . . , m} and let τ be a ﬁxed permutation of {1, . . . , m} such that, for all n ∈ Λ, n, nj ∈ Zm + (∗, τ ). Then lim

n∈Λ

Qnj (z) = Fm,τ −1 (j) (z), Qn (z)

uniformly on compact subsets of C \ ∆1 , where the functions Fm,j , j = i, . . . , m, have the same meaning as in (4.10). Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

72 48

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As a corollary, we have m Qn+1 (z) = Fm,j (z), n∈Λ Qn (z) j=1

lim

n + 1 = n + (1, . . . , 1),

uniformly on compact subsets of C \ ∆1 , provided that Λ ⊂ Zm + (∗) veriﬁes (4.19). The proof of Theorem 4.13 runs along the same lines as that of Theorem 4.7 although it is technically more complicated. In this more general setting, there is need for using certain analogs of Theorems 4.5 and 4.6. They were proved in [L06b] and are necessary, together with condition (4.19), to obtain a weak star formula like (4.9). 5. Quadrature rules Let µ be a ﬁnite positive Borel measure whose support Σ consists of an inﬁnite set of points contained in R. If Σ is not a bounded set, we will assume that the moments of the measure µ are all ﬁnite numbers. The smallest interval containing Σ will be denoted by ∆. Let us consider the integral I(f ) = f (x) dµ(x). Σ

A quadrature rule for the integral I(f ) is any linear functional of the form In (f ) =

n

λn,i f (xn,i ),

λn,i ∈ C,

xn,i ∈ ∆,

i = 1, . . . , n.

i=1

The rule In is called positive if all the quadrature weights {λn,i }ni=1 are positive. The quantity I(f ) − In (f ) represents the error of the rule and is denoted by En (f ). We will say that the rule In is exact in a space of functions F if En (f ) = 0 for all f ∈ F. Let Pn denote the space of polynomials of degree not greater than n. If the nodes {xn,i }ni=1 are ﬁxed beforehand, the weights {λn,i }ni=1 may be chosen so that the rule In is exact at least in Pn−1 . Such a rule is called interpolatory because In (f ) = I(Ln (f )), where Ln (f ) is the Lagrange interpolating polynomial of f at the nodes {xn,i }ni=1 . The maximum degree of polynomial exactness is attained when the nodes are chosen as the zeros of the n-th orthogonal polynomial with respect to the measure µ. In that case, the rule, which is called Gaussian and denoted by InG , is exact in P2n−1 . As mentioned at the beginning of Section 2, there is a close connection between Pad´e approximation of Cauchy transforms and Gaussian quadratures, since

1 (5.1) µ (z) − πn (z) = EnG , z − (·) as deﬁned by (2.2). Furwhere πn is the n-th diagonal Pad´e approximant of µ thermore, suppose that ∆ is a bounded interval and f an analytic function on a neighborhood of ∆. A standard application to (5.1) of Cauchy’s integral formula and Fubini’s theorem gives the integral representation 1 (5.2) EnG (f ) = f (ζ) [ µ(ζ) − πn (ζ)] dζ, 2πi γ where γ is any positively oriented simple curve surrounding the interval ∆ and contained in the region of analyticity of f . Thus, by means of formula (5.2), estimates of the speed of convergence of Gaussian quadrature rules for analytic functions may Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

A WALK THROUGH APPROXIMATION THEORY

73 49

be obtained from results on the rate of convergence of the Pad´e approximants to Markov functions. Conversely, the fact that the Christoﬀel numbers {λn,i }ni=1 are positive in (2.4) is essential in order to prove the Markov theorem. 5.1. Rational quadrature rules. The existence of formulas based on n nodes and exact in a 2n-dimensional vector space is not limited to the case mentioned of P2n−1 . Krein [Kr] proved that given any Chebyshev system of 2n continuous functions on [−1, 1], there exists a unique couple of sets of n nodes contained in (−1, 1) and n strictly positive quadrature weights such that the corresponding quadrature rule is exact in the system and that gives the maximum degree of exactness. One of the main examples of this more general situation is when the space where the quadrature rule is exact is formed by rational functions. Rational quadrature rules implicitly appear in the very ﬁrst papers by L´ opez [L78, L79], see (2.13). However, his ﬁrst paper dealing exclusively with rational rules was [L82]. Ten years later, rational formulas attracted the attention of Gautschi [Ga93] and Van Assche [VVa]. Numerical experiments showed that they give good results if the poles of the rational functions used to build the rule are chosen so as to simulate the singularities of the function to be integrated. Another important property is that, according to the results of [GR86], the nodes of the rational Gaussian rules tend to the equilibrium measure in the presence of the external ﬁeld generated by the limit distribution of the poles of the rational functions. These facts have stimulated the interest in rational rules until today: computer routines implementing rational Gaussian rules are designed in [Ga99], extensions to Gauss-type rules may be found in [GaGor], and in [C] an error bound for analytic integrands is given. Let us brieﬂy describe them. For each m ∈ N, let Am be a polynomial with real coeﬃcients of degree 1 ≤ dm ≤ m such that Am (0) = 1. Set Am (x) =

Cm

(1 −

i=1

x am,i

)Mm,i .

We will also assume that am,i ∈ ∆, i = 1, . . . , Cm . Sometimes, it will be more convenient to consider the polynomial Am as though it were made up by m factors Am (x) =

m

(1 −

i=1

x ), αm,i

where numbers αm,i are allowed to appear several times and we have taken αm,i = ∞ for i = dm + 1, . . . , m whenever dm < m. The zeros of the polynomials Am , m ∈ N, form the table A = {αm,i , i = 1, . . . , m, m ∈ N}. We will say that the table A is Newtonian if, for all m ∈ N, αm,i = αi . To each polynomial Am we associate a complex vector space Rm of dimension m which is the direct sum of a space of rational functions and a space of polynomials: Rm := Qdm ⊕ Pm−dm −1 ,

dm < m,

where Qdm := span{g : g(x) =

1 , k = 1, . . . , Mm,i , i = 1, . . . , Cm } (x − am,i )k

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and Rm := Qm if dm = m. It is clear that dim Qdm = dm . It is easy to prove that the quadrature rule In is exact in the space Rm if and only if

p =0 En Am for all polynomial p of degree not greater than m − 1. As in the polynomial case, if the nodes {xn,i }ni=1 are given, the weights {An,i }ni=1 may be chosen so that the rule In is exact, at least, in Rn ; such a rule is called rational of interpolatory type and denoted by IA,n . The maximum degree of exactness is attained when both nodes and weights are chosen conveniently, in that case IA,n is exact in R2n , is G . Its existence easily follows called a rational Gaussian rule, and denoted by IA,n from the existence of the usual Gaussian quadrature rule for the positive measure dµ(x)/A2n (x). In fact, the nodal polynomial wn (x) =

n

(x − xn,i )

i=1

is orthogonal with respect to the positive measure dµ(x)/A2n (x) and the coeﬃcients n n {An,i }i=1 verify that An,i = A2n (xn,i ) λn,i , where {λn,i }i=1 are the Christoﬀel numbers corresponding to the Gauss quadrature formula for the measure dµ(x)/A2n (x) (see [VVa]). Rational Gaussian rules share with the classical Gaussian one the most basic and important properties, namely, the zeros of An (x) are simple and lie n in ∆ and the weights {An,i }i=1 are positive numbers. The ﬁrst result by L´opez on rational rules is an extension of the Steklov theorem on the convergence of quadrature formulas for Riemann integrable functions. Take ∆ = [0, +∞). The table A ⊂ C \ [0, +∞) is required to be Newtonian and the conditions √ αm = +∞, αn = αm if m = n, (5.3) 1 + |αm | √ αm >0

assure the family

1,

1 1 ,..., ,... 1 − x/α2 1 − x/αm to span a dense set in the space of functions that are continuous on [0, +∞) with ﬁnite limit at +∞.

Theorem 5.1 ([L82]). Assume that the measure µ is of the type dg, where g is an increasing function on [0, +∞). Suppose that the table A is Newtonian and veriﬁes (5.3) with α1 = ∞. Then +∞ G lim IA,n (f ) = f (x) dg(x), n→∞

0

for all functions f Riemann integrable on [0, +∞) with ﬁnite limit at +∞. The following result relates the convergence of the multi-point Pad´e approximants of a Stieltjes function to the convergence of the corresponding rational Gaussian rules. The proof relies on the formula (cf. (2.13))

1 A G µ (z) − πn (z) = EA,n . z − (·)

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A WALK THROUGH APPROXIMATION THEORY

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Theorem 5.2 ([L82]). Suppose that the table A is Newtonian and veriﬁes that α1 = ∞, then the following statements are equivalent: (i) For all functions f continuous on [0, +∞) with ﬁnite limit at +∞, it holds +∞ G f (x) dµ(x). lim IA,n (f ) = n→∞

0

(ii) It holds (z), lim πnA (z) = µ

n→∞

uniformly on compact subsets of C \ [0, +∞). Both theorems were extended to positive rational rules of interpolatory type, see [L87a] for details. An estimate of the rate of convergence of rational Gaussian rules for analytic integrands was obtained in [L84]. Theorem 5.3 ([L84]). Assume that the measure µ is supported on a compact set Σ ⊂ R. Let f be an analytic function on a neighborhood Ω of ∆ and suppose that A is contained in a compact set F of C \ Ω. Then G (f )|1/2n ≤ e−τ , τ = inf gC\Σ (z, ζ), z ∈ ∂Ω, ζ ∈ F . lim sup |EA,n n→∞

The main ingredients of the proof are the analogous representation of (5.2) for multi-point Pad´e approximants and Theorem 2.7. The rate of convergence may be improved if the table A is taken to be extremal in a certain sense with respect to the sets Σ and C \ Ω. Following the same arguments, each result on Pad´e approximation of Markov functions has application to quadrature rules for analytic functions. Let us take Theorem 2.14 as example. Consider the partial fraction decomposition of the approximant πnA,B πnA,B (z)

(5.4)

=

Nn M n,i i=1 j=0

j! λn,i,j , (z − xn,i )j+1

where the points {xn,i } are the zeros of qn Bn2 . Deﬁne the quadrature rule (5.5)

G (f ) = IA,B,n

Nn M n,i

λn,i,j f (j) (xn,i ),

G G EA,B,n (f ) = I(f ) − IA,B,n (f ).

i=1 j=0

Although the zeros of qn are simple, they may coincide with zeros of Bn . Therefore, unless k(n) = 0, the quadrature rule so constructed will have multiple nodes. It has a total of n + k(n) nodes with 2k(n) nodes ﬁxed beforehand at the zeros of Bn2 . Under those circumstances, it turns out that the maximum degree of exactness is 2n in all cases. In order to attain it, the free nodes must be precisely the zeros of the polynomial qn orthogonal with respect to the varying measure Bn2 dµ/A2n . Then, we have

p(x) p G IA,B,n dµ(x), = A2n A 2n (x) Σ for all polynomials p ∈ P2n−1 , which, together with Theorem 2.14, gives Theorem 5.4. Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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Theorem 5.4 ([L01a]). Assume that the conditions of Theorem 2.14 hold. Let f be an analytic function on a neighborhood Ω of ∆. Then G lim sup |EA,B,n (f )|1/2n ≤ e−τ , τ = inf gC\Σ (z, ζ) dα(ζ), z ∈ ∂Ω . n→∞

F

5.2. Kronrod-type rules. The works [L02a] and [L04a] deal with the socalled Gauss-Kronrod quadrature rule, [L04a] being the rational counterpart of [L02a]. Such a rule arose from the need to estimate simultaneously an approximate value of the integral and the error as well (cf. [Kro]). It takes the form GK I2n+1 (f ) =

n

An,i f (xn,i ) +

i=1

n+1

Bn,j f (yn,j ),

GK GK E2n+1 (f ) = I(f ) − I2n+1 (f ),

j=1

{xn,i }ni=1

are the Gaussian nodes while the rest of the nodes {yn,j }n+1 where j=1 and n+1 n the quadrature weights {An,i }i=1 , {Bn,j }j=1 , are chosen so that the rule has the highest possible degree of polynomial exactness, which is 3n+1. This requirement is equivalent to the fact that the nodal polynomial Sn+1 (x) = n+1 j=1 (x − yn,j ) satisﬁes the orthogonality relations xk Sn+1 (x) qn (x) dµ(x) = 0, k = 0, 1, . . . , n, (5.6) Σ n

where qn (x) = κn x + . . . , κn > 0, is the n-th orthonormal polynomial with respect GK G to µ. So, we could keep using the same notation as in (5.5) denoting I2n+1 by IA,B,n , where the table B of ﬁxed nodes is made up by the zeros of the polynomials qn and the table A is given by a2n,i = ∞, i = 1, . . . , 2n, n ∈ N. However, we will reserve GK for those cases when the rule has 2n + 1 distinct nodes on ∆ and the symbol I2n+1 G IA,B,n will stand for the generalized quadrature rule to be deﬁned below. The polynomials Sn+1 were introduced by Stieltjes [HSti] in 1894 for the case of the Legendre measure dµ(x) = dx and are named after him. They are closely connected to the second type functions qn (x) (5.7) hn (z) = dµ(x). Σ z−x Note that, by (5.6), hn (z) = O(1/z n+1 ), z → ∞. It turns out that 1 (5.8) = κn Sn+1 (z) + O (1/z) , z → ∞, hn (z) which constitutes an alternative formula for the deﬁnition of the Stieltjes polynomials. For convenience, we will denote κn Sn+1 by sn+1 . Despite the fact that the polynomials {Sn+1 } are orthogonal with respect to a non-positive measure, their zeros occasionally behave as those of the classical orthogonal polynomials, which implies the existence of the corresponding GaussKronrod rule. The most studied case has been that of the ultraspherical measures dµ(x) = (1 − x2 )λ−1/2 dx,

λ > −1/2.

Depending on λ, the Stieltjes polynomials may have zeros which are real, simple, and interlace the zeros of the corresponding orthogonal polynomial [Sz35], they may have real zeros outside ∆ [CP], or most of the zeros of Sn+1 may be complex [PPe]. When they exist, Gauss-Kronrod rules can be computed eﬃciently [La] Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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and are used in packages for automatic integration [PiDo]. For further details on Stieltjes polynomials and Gauss-Kronrod quadrature formulas, see the surveys [Ga88, Mon]. In [L02a], the asymptotic properties of the Stieltjes polynomials outside Σ are studied. For that, a new integral expression connecting the Sieltjes polynomials and the functions of the second type is derived. Namely, dζ hn (z) (5.9) sn+1 (z) hn (z) = 1 + , 2πi γ hn (ζ) (ζ − z) where γ is any positively oriented close smooth curve which surrounds ∆ such that z is contained in the unbounded component of C\γ. Formula (5.9) permits reducing the study of the asymptotic behavior of Sn+1 to that of hn , which is known. The main result is that, when Σ is an interval, the behavior of the polynomials {Sn+1 }n∈N outside Σ is similar to that of standard orthogonal polynomials. Such a result is presented here in a simpliﬁed form for the case of Σ = [−1, 1]. Recall that the class of measures Reg is deﬁned by (2.26). Theorem 5.5 ([L02a]). The following assertions hold: (a) If µ ∈ Reg, the set of accumulation points of the zeros of {Sn+1 }n∈N is contained in [−1, 1] and lim |sn+1 (z)|1/n = |z + z 2 − 1|. n→∞

(b) If µ > 0 a.e. on [−1, 1], then sn+1 (z) = z + z 2 − 1. n→∞ sn (z) lim

The limits hold true uniformly on compact subsets of C \ [−1, 1]. As a consequence, for regular measures, the zeros of the Stieltjes polynomials, although they may be complex, tend to the interval [−1, 1] and their limit distribution is the equilibrium measure of [−1, 1]. This implies that a Gauss-Kronrod type quadrature rule with possibly multiple nodes always exists for analytic integrands. The construction of that generalized rule is carried out through the Pad´e-type approximant of µ p2n πnA,B = , Sn+1 qn with preassigned poles at the zeros of the polynomial qn , which interpolates the Markov function at the inﬁnity. That is, here qn plays the role of Bn2 and Sn+1 that G of qn in (2.22), with A2n ≡ 1. The construction of the corresponding rule IA,B,n is analogous to that of (5.5) from (5.4). The rule so deﬁned has 3n + 1 as degree of GK polynomial exactness and coincides with the Gauss-Kronrod rule I2n+1 when the zeros of Sn+1 qn are simple and belong to Σ. The following result may be proved in the usual way by means of Theorem 5.5 as well as the formulas analogous to (2.24) and (5.2). Theorem 5.6 ([L02a]). Suppose that the measure µ ∈ Reg is supported on a bounded interval Σ ⊂ R. Let f be an analytic function on a neighborhood Ω of Σ. Then, we have G lim sup |EA,B,n (f )|1/3n ≤ e−τ , τ = inf gC\Σ (z, ∞), z ∈ ∂Ω . n→∞

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If the support of µ is not an interval a distinct phenomenon can be observed. In that case, the support Σ is a proper subset of its convex hull ∆ and the existence of zeros of the functions of the second type in ∆ \ Σ may provoke divergence of the polynomials Sn+1 in a neighborhood of ∆, see (5.10) below. Corresponding results for this more general situation may also be found in [L02a]. The main diﬃculty in extending the above results to rules exact in a vector space of rational functions of dimension 3n + 2 was to deﬁne the corresponding Stieltjes polynomials in a proper way. The following deﬁnition was given in [L04a]. For each n ∈ N, let A2n+1 be a polynomial as in (2.11). Let Qn,k be the k-th monic orthogonal polynomial with respect to the varying measure dµ/A2n+1 . In addition, let {Vn+1 } be another family of polynomials with the same features as {A2n+1 } and deg Vn+1 ≤ n + 1, n ∈ N. Then, the varying Stieltjes polynomial Sn,k+1 is the monic polynomial of least degree verifying dµ(x) xν Sn,k+1 (x) Qn,k (x) = 0, ν = 0, 1, . . . , k. A2n+1 (x) Vn+1 (x) It may be proved that deg Sn,k+1 = k + 1 if k + 1 ≥ deg Vn+1 . For these varying Stieltjes polynomials analogous results to those of [L02a] are given in [L04a]. In particular, a multi-point Pad´e-type approximant p2n πnA,B = Sn,n+1 Qn,n is constructed where the table A is made up by the zeros of the polynomials {A2n+1 Vn+1 } and B by the zeros of {Qn,n }. In contrast with (2.22) the ﬁxed G veriﬁes poles need not to be doubled. The corresponding rational rule IA,B,n

p G = 0, p ∈ P3n+1 . EA,B,n A2n+1 Vn+1 The conditions required are similar to those of Theorem 2.14 so we will refer to them using the same notation: (1) Table A is contained in a compact set F of C \ ∆. ∗ ∗ (2) ΘA2n+1 −→ α, n → ∞, ΘVn+1 −→ α, n → ∞. (3) The support Σ is a compact regular set. (4) µ ∈ Reg. Let us state the result in its full generality (when Σ is not an interval). For that, set τ (z) = gC\Σ (z, ζ) dα(ζ) ⊃ ∆ given by and deﬁne ∆ = ∆

(5.10)

F

z ∈ C : τ (z) ≤ max τ (w) . w∈∆

From some examples in [L02a] it follows that it is not possible to obtain convergence for the whole class of regular measures. When Σ is an interval inside the set ∆ = ∆ = Σ. ∆ Theorem 5.7 ([L04a]). Suppose that conditions (1)–(4) are fulﬁlled. Then, we have on each compact subset K of C \ ∆, 1/3n lim µ − πnA,B ≤ exp { − τ K } exp {τ ∆ } < 1. n→∞

K

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then Moreover, let f be an analytic function on a neighborhood Ω of ∆, G (f )|1/3n ≤ exp { − τ ∂Ω } exp {τ ∆ } < 1. lim sup |EA,B,n n→∞

The work [L09a] is concerned with the extension of the Gauss-Kronrod scheme to quadrature rules on the unit circle T. Now, the problem is to approximate the integral T (f ) = f (ζ) dρ(ζ), T

where ρ is a ﬁnite positive Borel measure whose support Σ is contained on T and has inﬁnitely many points. This is the suitable setting when integrating periodic functions with known periodicity. A quadrature rule for the integral T (f ) is any expression of the form Tn (f ) =

n

λn,i f (zn,i ),

zn,i ∈ T,

i = 1, . . . , n.

i=1

The error T (f ) − Tn (f ) will be denoted by Rn (f ). Let Λp,q , p, q ≥ 0, denote the space of all rational functions of the form h/z p , h ∈ Pp+q . Because of the Weierstrass theorem on approximation of continuous functions on T, it is natural to require that Tn be exact in a space Λp,q with p + q ≥ n − 1. Given a set of nodes {zn,i }ni=1 ⊂ C \ {0} such a rule always exists and is called interpolatory. Which is the largest possible space of exactness Λp,q achieved by a rule Tn ? Let ϕn (z) = αn z n + . . . , αn > 0, be the orthonormal polynomial of degree n with respect to ρ. The polynomial ϕn has all its zeros in the unit disk D so it cannot be used to construct on T an analog of the Gauss-Jacobi rule. Instead, polynomials called para-orthogonal are considered. A polynomial Wn of degree n is said to be para-orthogonal with respect to the measure ρ if it is of the form Wn = cn (ϕn + τn ϕ∗n ),

τn ∈ T,

cn ∈ C \ {0},

where ϕ∗n (z) = z n ϕn (1/z). Set wn = ϕn + τn ϕ∗n and write Wn for monic paraorthogonal polynomials. It is obvious that para-orthogonal polynomials depend on the parameter τn , yet omitting that dependance in the notation will not be misleading. It is well known that the zeros of Wn are simple and lie on the unit circle T. If the nodes {zn,i }ni=1 are chosen to be the zeros of Wn , there exist positive numbers {λn,i }ni=1 such that Rn (h) = 0, h ∈ Λn−1,n−1 . Such quadrature rules are called Szeg˝ o rules and denoted by TnS . It may be proved that the space Λn−1,n−1 is the largest possible space of exactness of type Λp,q for any rule Tn , provided that the n nodes belong to the unit circle. For information on para-orthogonal polynomials and Szeg˝ o quadrature rules, see [J]. The next step is to construct a rule retaining the n Szeg˝ o nodes and adding m new ones. In order to increase the degree of exactness of TnS it is necessary to add at least n new nodes and, as in the case of the Stieltjes polynomials, the polynomial whose zeros are the optimal additional nodes must satisfy certain orthogonality relations. Fix n ∈ N and τn ∈ T. In contrast with the extension of Gauss-Jacobi rules, we ﬁnd here that there are diﬀerent spaces Λp,q of the same dimension verifying Λp,q ⊃ Λn−1,n−1 , which explains the need of an extra parameter Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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m ∈ {0, 1, . . . , n − 1}. Let Sn,m be the monic polynomial of least degree such that Sn,m (ζ) ζ ν−m Wn (ζ) dρ(ζ) = 0, ν = 0, 1 . . . , n − 1. T

We will say that Sn,m is the (n, m) monic Stieltjes-type polynomial with respect to ρ. The polynomial Sn,m veriﬁes deg Sn,m = n and Sn,m (0) = 0. It may be proved that the interpolatory quadrature rule with nodal polynomial Wn Sn,m is exact in the space Λn+m,2n−m−1 . Such a quadrature rule is called a Szeg˝o-Kronrod rule and SK . The polynomial Sn,m does not have in general its zeros on T, so denoted by T2n SK T2n has its (possibly multiple) nodes in C \ {0}. Stieltjes-type polynomials are strongly connected to functions analogous to the second type functions (5.7). Deﬁne ζ wn (ζ) Hn (z) = dρ(ζ), n ∈ N. T z−ζ Set sn,m = αn Sn,m and sn,m = sn,m,1 +sn,m,2 , where sn,m,1 contains the ﬁrst m+1 terms of sn,m . Then ⎧ ⎨ sn,m,2 (z) + O(z m ), z → ∞, 1 = (5.11) Hn (z) ⎩ sn,m,1 (z) + O(z m+1 ), z → 0. Formula (5.11) is the analog of (5.8). This special relationship with the function Hn permits obtaining asymptotics for the Stieltjes-type polynomials under suitable conditions. Basically, the orthonormal polynomials {ϕn } must satisfy n-th root asymptotics. So, the measure ρ will be required to belong to the class Reg T which means that lim |ϕn (z)|1/n = |z|, n→∞

uniformly on compact subsets of C \ D. We will also assume that m depends on n, i.e., m = mn . Denote mn mn = ξ and lim sup = ξ. lim inf n→∞ n n n→∞ Obviously, 0 ≤ ξ ≤ ξ ≤ 1. Theorem 5.8 ([L09a]). Assume that ρ ∈ Reg T and 0 < ξ ≤ ξ < 1. Then ⎧ ⎨ |z|, z ∈ C \ D, lim |sn,mn (z)|1/n = n→∞ ⎩ 1, z ∈ D, uniformly on compact subsets of the indicated regions. Once Theorem 5.8 is proved, the rest follows along the lines of [L02a]. The zeros of the Stieltjes-type polynomials tend to the unit circle and their limit distribution SK is the equilibrium measure of T. Therefore, the quadrature rules T2n exist for analytic functions deﬁned on a neighborhood of T. The estimate of the rate of convergence given by Theorem 5.9 below is proved by using a sequence of rational interpolants of the Caratheodory functions ζ +z Fρ (z) = dρ(ζ) T ζ −z and Theorem 5.8. Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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Theorem 5.9 ([L09a]). Assume that ρ ∈ Reg T and 0 < ξ ≤ ξ < 1. Let f be an analytic function on the annulus {z : r1 < |z| < r2 , r1 < 1 < r2 }. Then SK 1/n 1+ξ lim sup R2n < 1. (f ) ≤ max (1/r2 )2−ξ , r1 n→∞

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[L81b] G. L´ opez Lagomasino, On the convergence of the Pad´ e approximants for meromorphic functions of Stieltjes type, Math. USSR Sb. 39 (1981), 281–288. [L82] G. L´ opez Lagomasino, J. Ill´ an Gonz´ alez, F´ ormulas de cuadratura para intervalos no acotados, Cienc. Mat. (Havana) 3, no. 3 (1982), 29–47. [L83] G. L´ opez Lagomasino, On the moment problem and the convergence of Pad´ e approximants for meromorphic functions of Stieltjes type, Constructive Function Theory 1981, Bulgar. Acad. Sci., Soﬁa, 1983, pp. 419–424. [L84] G. L´ opez Lagomasino, J. Ill´ an Gonz´ alez, A note on generalized quadrature formulas of Gauss-Jacobi type, Constructive Function Theory 1983, Bulgar. Acad. Sci., Soﬁa, 1984, pp. 513–518. [L87a] G. L´ opez Lagomasino, J. Ill´ an Gonz´ alez, Sobre los m´ etodos interpolatorios de integraci´ on num´ erica y su conexi´ on con la aproximaci´ on racional, Cienc. Mat. (Havana) 8, no. 2 (1987), 31–44. [L87b] G. L´ opez Lagomasino, On the asymptotics of the ratio of orthogonal polynomials and convergence of multipoint Pad´ e approximants, Math. USSR Sb. 56 (1987), 207–219. [L89a] G. L´ opez Lagomasino, Asymptotics of polynomials orthogonal with respect to varying measures, Constr. Approx. 5 (1989), 199–219. [L89b] G. L´ opez Lagomasino, Convergence of Pad´ e approximants of Stieltjes type meromorphic functions and comparative asymptotics for orthogonal polynomials, Math. USSR Sb. 64 (1989), 207–227. [L90] G. L´ opez Lagomasino, Relative asymptotics for orthogonal polynomials on the real axis, Math. USSR Sb. 65 (1990), 505–529. [L94] J. Bustamante, G. L´ opez Lagomasino, Hermite-Pad´ e approximation for Nikishin systems of analytic functions, Russian Acad. Sci. Sb. Math. 77 (1994), 367–384. [L95a] D. Barrios, L´ opez Lagomasino, E. Torrano, Location of zeros and asymptotics of polynomials satisfying three-term recurrence relations with complex coeﬃcients, Russian Acad. Sci. Sb. Math. 80 (1995), 309–333. e [L95b] G. L´ opez Lagomasino, A. Mart´ınez Finkelshtein, Rate of convergence of two-point Pad´ approximants and logarithmic asymptotics of Laurent-type orthogonal polynomials, Constr. Approx. 11 (1995), 255–286. [L95c] D. Barrios, G. L´ opez Lagomasino, E. Torrano, Polynomials generated by a three-term recurrence relation with asymptotically periodic complex coeﬃcients, Sb. Math. 186 (1995), 629–659. [L95d] G. L´ opez Lagomasino, F. Marcell´ an, W. Van Assche, Relative asymptotics for polynomials orthogonal with respect to a discrete Sobolev inner product, Constr. Approx. 11 (1995), 107– 137. [L95e] F. Cala Rodr´ıguez, G. L´ opez Lagomasino, From Pad´ e to Pad´ e-type approximants. Exact rate of convergence, Approx. Optim., vol. 8, Lang, Frankfurt am Main, 1995, pp. 155–163. [L95f] P. Gonz´ alez-Vera, G. L´ opez Lagomasino, R. Orive, J. C. Santos, On the convergence of quadrature formulas for complex weight functions, J. Math. Anal. Appl. 189 (1995), 514–532. [L96a] P. Gonz´ alez-Vera, M. Jim´enez Paid, G. L´ opez Lagomasino, R. Orive, On the convergence of quadrature formulas connected with multipoint Pad´ e-type approximation, J. Math. Anal. Appl. 202 (1996), 747–775. [L96b] M. Alfaro, G. L´ opez Lagomasino, M. L. Rezola, Some properties of zeros of Sobolev-type orthogonal polynomials, J. Comput. Appl. Math. 69 (1996), 171–179. [L98a] M. Bello Hern´ andez, G. L´ opez Lagomasino, Ratio and relative asymptotics of polynomials orthogonal on an arc of the unit circle, J. Approx. Theory 92 (1998). [L98b] D. Barrios Rolan´ıa, G. L´ opez Lagomasino, A. Mart´ınez Finkelshtein, E. Torrano Gim´enez, On the domain of convergence and poles of complex J-fractions, J. Approx. Theory 93 (1998), 177–200. [L98c] F. Cala Rodr´ıguez, G. L´ opez Lagomasino, Multipoint Pad´ e-type approximants. Exact rate of convergence, Constr. Approx. 14 (1998), 259–272. [L99a] D. Barrios Rolan´ıa, G. L´ opez Lagomasino, Ratio asymptotics for polynomials orthogonal on arcs of the unit circle, Constr. Approx. 15 (1999), 1–31. [L99b] G. L´ opez Lagomasino, H. Pijeira Cabrera, Zero location and nth root asymptotics of Sobolev orthogonal polynomials, J. Approx. Theory 99 (1999), 30–43. [L99c] B. de la Calle Ysern, G. L´ opez Lagomasino, Weak convergence of varying measures and Hermite-Pad´ e orthogonal polynomials, Constr. Approx. 15 (1999), 553–575.

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[L99d] A. Fern´ andez Infante, G. L´ opez Lagomasino, Overconvergence of subsequences of rows of Pad´ e approximants with gaps, J. Comput. Appl. Math. 105 (1999), 265–273. [L01a] F. Cala Rodr´ıguez, G. L´ opez Lagomasino, Multipoint rational approximants with preassigned poles, J. Math. Anal. Appl. 256 (2001), 142–161. [L01b] B. de la Calle Ysern, G. L´ opez Lagomasino, Convergence of multipoint Pad´ e-type approximants, J. Approx. Theory 109 (2001), 257–278. [L01c] D. Barrios Rolan´ıa, G. L´ opez Lagomasino, E. B. Saﬀ, Asymptotics of orthogonal polynomials inside the unit circle and Szeg¨ o-Pad´ e approximants, J. Comput. Appl. Math. 133 (2001), 171–181. [L01d] G. L´ opez Lagomasino, I. P´erez Izquierdo, H. Pijeira Cabrera, Sobolev orthogonal polynomials in the complex plane, J. Comput. Appl. Math. 127 (2001), 219–230. [L02a] M. Bello Hern´ andez, B. de la Calle Ysern, J. J. Guadalupe Hern´ andez, G. L´ opez Lagomasino, Asymptotics for Stieltjes polynomials, Pad´ e-type approximants, and Gauss-Kronrod quadrature, J. Anal. Math. 86 (2002), 1–23. [L02b] U. Fidalgo Prieto, G. L´ opez Lagomasino, On perfect Nikishin systems, Comput. Methods Funct. Theory 2 (2002), 415–426. [L03a] A. Branquinho, J. Bustamante, A. Foulqui´ e Moreno, G. L´ opez Lagomasino, Normal indices in Nikishin systems, J. Approx. Theory 124 (2003), 254–262. [L03b] D. Barrios Rolan´ıa, G. L´ opez Lagomasino, E. B. Saﬀ, Determining radii of meromorphy via orthogonal polynomials on the unit circle, J. Approx. Theory 124 (2003), 263–281. [L04a] M. Bello Hern´ andez, B. de la Calle Ysern, G. L´ opez Lagomasino, Generalized Stieltjes polynomials and rational Gauss-Kronrod quadrature, Constr. Approx. 20 (2004), 249–265. [L04b] U. Fidalgo Prieto, J. Ill´ an, G. L´ opez Lagomasino, Hermite-Pad´ e approximation and simultaneous quadrature formulas, J. Approx. Theory 126 (2004), 171–197. [L05a] A. I. Aptekarev, G. L´ opez Lagomasino, I. A. Rocha, Ratio asymptotics of Hermite–Pad´ e polynomials for Nikishin systems, Sb. Math. 196 (2005), 1089–1107. [L06a] U. Fidalgo Prieto, G. L´ opez Lagomasino, Rate of convergence of generalized Hermite-Pad´ e approximants of Nikishin systems, Const. Approx. 23 (2006), 165–196. [L06b] D. Barrios Rolan´ıa, B. de la Calle Ysern, G. L´ opez Lagomasino, Ratio and relative asymptotics of polynomials orthogonal with respect to varying Denisov-type measures, J. Approx. Theory 139 (2006), 223–256. [L06c] G. L´ opez Lagomasino, F. Marcell´ an Espa˜ nol, H. Pijeira Cabrera, Logarithmic asymptotics of contracted Sobolev extremal polynomials on the real line, J. Approx. Theory 143 (2006), 62–73. [L07a] U. Fidalgo Prieto, G. L´ opez Lagomasino, General results on the convergence of multipoint Hermite-Pad´ e approximants of Nikishin systems, Constr. Approx. 25 (2007), 89–107. [L07b] M. Bello Hern´ andez, G. L´ opez Lagomasino, J. M´ınguez Ceniceros, Fourier-Pad´ e approximants for Angelesco systems, Constr. Approx. 26 (2007), 339–359. [L07c] U. Fidalgo Prieto, J. Ill´ an Gonz´ alez, G. L´ opez Lagomasino, Convergence and computation of simultaneous rational quadrature formulas, Numer. Math. 106 (2007), 99–128. [L08a] A. L´ opez Garc´ıa, G. L´ opez Lagomasino, Ratio asymptotic of Hermite-Pad´ e orthogonal polynomials for Nikishin systems. II, Adv. Math. 218 (2008), 1081–1106. [L08b] G. L´ opez Lagomasino, A. Mart´ınez-Finkelshtein, I. P´erez Izquierdo, H. Pijeira Cabrera, Strong asymptotics for Sobolev orthogonal polynomials in the complex plane, J. Math. Anal. Appl. 340 (2008), 521–535. [L09a] B. de la Calle Ysern, G. L´ opez Lagomasino, L. Reichel, Stieltjes-type polynomials on the unit circle, Math. Comp. 78 (2009), 969–997. [L09b] G. L´ opez Lagomasino, J. M´ınguez Ceniceros, Fourier-Pad´ e approximants for Nikishin systems, Constr. Approx. 30 (2009), 53–69. [MaV] F. Marcell´ an, W. Van Assche, Relative asymptotics for orthogonal polynomials with a Sobolev inner product, J. Approx. Theory 72 (1993), 193–209. [Mark] A. A. Markov, Deux demostrations de la convergence de certains fractions continues, Acta Math. 19 (1895), 93–104. [Mar98] A. Mart´ınez Finkelshtein, Asymptotic properties of Sobolev orthogonal polynomials, J. Comput. Appl. Math. 99 (1998), 491–510. [Mar00] A. Mart´ınez Finkelshtein, Bernstein-Szeg˝ o theorem for Sobolev orthogonal polynomials, Constr. Approx. 16 (2000), 73–84.

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[Mar06] A. Mart´ınez Finkelshtein, Equilibrium Problems of Potential Theory in the Complex Plane, Lecture Notes in Math., vol. 1883, Springer-Verlag, Berlin, 2006, pp. 79–117. [MarPij] A. Mart´ınez Finkelshtein, H. Pijeira Cabrera, Strong asymptotics for Sobolev orthogonal polynomials, J. Anal. Math. 78 (1999), 143–156. [MNT84] A. M´ at´e, P. Nevai, V. Totik, What is beyond Szeg˝ o’s theory of orthogonal polynomials?, Lecture Notes in Math., vol. 1105, Springer-Verlag, Berlin, 1984, pp. 502–510. [MNT85] A. M´ at´e, P. Nevai, V. Totik, Asymptotics for the ratio of leading coeﬃcients of orthogonal polynomials, Constr. Approx. 1 (1985), 63–69. [MNT87] A. M´ at´ e, P. Nevai, V. Totik, Extensions of Szeg˝ o’s theory of orthogonal polynomials, II, III, Constr. Approx. 3 (1987), 51–72; 73–96. [Me96] H. G. Meijer, A short history of orthogonal polynomials in a Sobolev space I. The nondiscrete case, Nieuw Arch. Wiskd. 14 (1996), 93–113. [Me97] H. G. Meijer, Determination of all coherent pairs, J. Approx. Theory 89 (1997), 321–343. [MhS] H. N. Mhaskar, E. B. Saﬀ Extremal problems for polynomials with exponential weights, Trans. Amer. Math. Soc. 285 (1984), 203–234. [Mon] G. Monegato, Stieltjes polynomials and related quadrature rules, SIAM Rev. 24 (1982), 137–158. [N] P. Nevai, Orthogonal Polynomials, Memoirs Amer. Math. Soc. vol. 213, Amer. Math. Soc., Providence, Rhode Island, 1979. [Ni80] E. M. Nikishin, On the irrationality of the values of the functions F (x, s), Math. USSR Sb. 37 (1980), 318–338. [Ni82] E. M. Nikishin, On simultaneous Pad´ e approximants, Math. USSR Sb. 41 (1982), 409–425. [NiSo] E. M. Nikishin, V. N. Sorokin, Rational Approximation and Orthogonality, Transl. Math. Monogr., vol. 92, Amer. Math. Soc., Providence, Rhode Island, 1991. [Nu] J. Nuttall, Asymptotics of diagonal Hermite-Pad´ e polynomials, J. Approx. Theory 42 (1984), 299–386. [PPe] F. Peherstorfer, K. Petras, Ultraspherical Gauss-Kronrod quadrature is not possible for λ > 3, SIAM J. Numer. Anal. 37 (2000), 927–948. [PSte] F. Peherstorfer, R. Steinbauer, Asymptotic behavior for orthogonal polynomials on the unit circle with asymptotically periodic reﬂection coeﬃcients, J. Approx. Theory 88 (1997), 316–353. ¨ D. K. Kahaner, QUADPACK: A [PiDo] R. Piessens, E. de Doncker-Kapenga, C. W. Uberhuber, subroutine package for automatic integration, Springer Ser. Comput. Math., vol 1., Springer, Berlin, 1983. [Po] H. Poincar´ e, Sur les equations lin´ eaires aux diﬀ´ erentielles ordinaires et aux diﬀ´ erences ﬁnies, Amer. J. Math. 7 (1885), 203–258. [Pom] Ch. Pommerenke, Pad´ e approximants and convergence in capacity, J. Math. Anal. Appl. 41 (1973), 775–780. [R77a] E. A. Rakhmanov, On the asymptotics of the ratio of orthogonal polynomials, Math. USSR Sb. 32 (1977), 199–213. [R77b] E. A. Rakhmanov, Convergence of diagonal Pad´ e approximants, Math. USSR Sb. 33 (1977), 243–260. [R83] E. A. Rakhmanov, On the asymptotics of the ratio of orthogonal polynomials, II, Math. USSR Sb. 46 (1983), 105–117. [R84] E. A. Rakhmanov, On asymptotic properties of polynomials orthogonal on the real axis, Math. USSR Sb. 47 (1984), 155-193. [R87] E. A. Rakhmanov, On asymptotic properties of polynomials orthogonal on the circle with weights not satisfying Szeg˝ o’s condition, Math. USSR Sb. 58 (1987), 149-167. [Ra] T. Ransford, Potential Theory in the Complex Plane, London Mathematical Society, Student Texts 28, Cambridge University Press, Cambridge, 1995. [Ro] J. M. Rodr´ıguez, The multiplication operator in Sobolev spaces with respect to measures, J. Approx. Theory 109 (2001), 157-197. [ST] E. B. Saﬀ, V. Totik, Logarithmic Potentials with External Fields, A Series of Comprehensive Studies in Mathematics, vol. 316, Springer-Verlag, Berlin, 1997. [Si05a] B. Simon, Orthogonal Polynomials on the Unit Circle, Amer. Math. Soc. Colloq. Publ., vol. 54, Amer. Math. Soc., Providence, Rhode Island, 2005. [Si05b] B. Simon, Fine structure of the zeros of orthogonal polynomials, II. OPUC with competing exponential decay, J. Approx. Theory 135 (2005), 125–139.

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[St] H. Stahl, The convergence of Pad´ e approximants to functions with branch points, J. Approx. Theory 91 (1997), 139–204. [StT] H. Stahl, V. Totik, General orthogonal polynomials, Encyclopedia of Mathematics and its Applications, vol. 43, Cambridge University Press, Cambridge, 1992. [Sti] T. J. Stieltjes, Recherches sur les fractions continues, Ann. Fac. Sci. Toulouse Math. 8 (1894), J1–J122; 9 (1895), A1–A47. Reprinted in Oeuvres completes, vol. 2, Noordhoﬀ, Groningen, 1918, pp. 402–566. ¨ [Sz35] G. Szeg˝ o, Uber gewisse orthogonale Polynome, die zu einer oszillierenden Belegungsfunktion geh¨ oren, Math. Ann. 110 (1935), 501–513. [Sz75] G. Szeg˝ o, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, Rhode Island, 1975. [V] W. Van Assche, Multiple orthogonal polynomials, irrationality, and transcendence, Contemp. Math., vol. 236, Amer. Math. Soc., Providence, Rhode Island, 1999, pp. 325–342. [VVa] W. Van Assche, I. Vanherwegen, Quadrature formulas based on rational interpolation, Math. Comp. 61 (1993), 765–783. [W] H. S. Wall, Analytic Theory of Continued Fractions, Chelsea, New York, 1973. [Wi] H. Widom, Extremal polynomials associated with a system of curves in the complex plane, Adv. Math. 3 (1969), 127–232. ´tica Aplicada, ETS Ingenieros Industriales, Universidad Departamento de Matema Polit´ ecnica de Madrid, Jos´ e Guti´ errez Abascal 2, 28006 Madrid, Spain E-mail address: [email protected]

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http://dx.doi.org/10.1090/conm/507/09955

Contemporary Mathematics Volume 507, 2010

Asymptotic Uniqueness of Best Rational Approximants to Complex Cauchy Transforms in L2 of the Circle Laurent Baratchart and Maxim Yattselev Dedicated to Guillermo Lop` ez Lagomasino on the occasion of his 60-th birthday.

Abstract. For all n large enough, we show uniqueness of a critical point in best rational approximation of degree n, in the L2 -sense on the unit circle, to functions of the form dµ(t) + r(z), dµ = µ˙ dω[a,b] , f(z) = z−t with r a rational function and µ˙ a complex-valued Dini-continuous function on a real segment [a, b] ⊂ (−1, 1) which does not vanish, and whose argument is of bounded variation. Here ω[a,b] stands the normalized arcsine distribution on [a, b].

1. Introduction Best rational approximation of given degree to a holomorphic function, in the least squares sense on the boundary of a disk included in the domain of analyticity, is a classical issue for which early references are [22, 30, 26, 20, 17]. The interplay between complex and Fourier analysis induced by the circular symmetry confers to such an approximation a natural character, and the corresponding approximants provide one with nice examples of locally convergent sequences of diagonal multipoint Pad´e interpolants. The problem can be recast as best rational approximation of given degree in the Hardy space H 2 of the unit disk and also, upon reﬂecting the functions involved across the unit circle, in the Hardy space of the complement of the disk which is the framework we shall really work with. Because of the natural isometry between Hardy spaces of the disk and the half-plane, that preserves rationality and the degree [25, Ch. 8], the question can equivalently be stated as best rational L2 -approximation of given degree on the line to a function holomorphic in a half-plane. Further motivation for this type of approximation stems from Control Theory and Signal Processing. Indeed, the transfer-function of a stable linear control system belongs to the Hardy space of the half-plane or of the complement of the disk, depending whether the setting is in continuous or discrete time, and it is rational if the system is ﬁnite-dimensional. Moreover, by Parseval’s theorem, the L2 norm 1991 Mathematics Subject Classiﬁcation. Primary 41A52, 41A20, 30E10. Key words and phrases. Uniqueness of best approximation, rational approximation. 1

c 2010 American Mathematical Society

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on the line or the circle of this transfer function coincides with the norm of the underlying convolution operator from L2 [0, ∞) to L∞ [0, ∞) in the time domain [19]. Further, in a stochastic context, it coincides with the variance of the output when the input is white noise [24]. Therefore approximating the transfer function by a rational function of degree n, in L2 of the line or the circle, is tantamount to identify the best system of order n to model the initial system with respect to the criteria just mentioned. Also, since any stationary regular stochastic process is the output of a linear control system fed with white noise [29, 18], this approximation yields the best ARMA-process to model the initial process while minimizing the variance of the error. A thorough discussion of such connections with System Theory, as well as additional references, can be found in [2]. From the constructive viewpoint no algorithm is known to constructively solve the question we raised, and from a computational perspective this is a typical nonconvex minimization problem whose numerical solution is often hindered by the occurence of local minima. It is therefore of major interest in practice to establish conditions on the function to be approximated that ensure uniqueness of a local minimum. This turns out to be diﬃcult, like most uniqueness issues in nonlinear approximation. New ground for the subject was broken in [1], where a diﬀerential-topological method was introduced to approach the uniqueness issue for critical points, i.e. stationary points of the approximation error. Uniqueness of a critical point implies uniqueness of a local minimum, but is a stronger property which is better suited to analysis. In fact, the above-mentioned method rests on the so-called index theorem [5] that provides us with a relation between the Morse indices of the critical points, thereby reducing the proof of uniqueness, which is a global property, to checking that each critical point has Morse index 0, which is a local issue. The latter is in turn equivalent to each critical point being a non-degenerate local minimum. This approach was taken in [12] to handle the case where the approximated function is of Markov type, that is, the Cauchy transform of a positive measure on a real segment, when that measure is supported within some absolute bounds. Subsequently, in [8], the property of being a local minimum was connected to classical interpolation theory and the technique was applied to prove asymptotic uniqueness of a critical point in best L2 rational approximation to e1/z /z on the unit circle, as well as in best L2 rational approximation to generic holomorphic functions over small circles; here, asymptotic uniqueness means uniqueness in degree n for all suﬃciently large n. The criterion derived in [8] for being a local minimum was further reﬁned in [11], where it is shown that asymptotic uniqueness of a critical point holds for Markov functions whose deﬁning measure satisﬁes the Szeg˝o condition. The result is sharp in that the Szeg˝ o condition cannot be omitted in general [10]. A general condition on the logarithmic derivative of the approximated function was derived [3] but it only ensures uniqueness in degree 1. A criterion for best approximation in degree n − 1 to a rational function of degree n can further be found in [9, Thm. 9.1], based on fast geometric decay of the error in lower degree. Altogether, these works indicate the fact, perhaps unexpected, that uniqueness of a critical point in best L2 rational approximation is linked to a regular decrease of the error. The present paper can be viewed as a sequel to [11]. Indeed, the latter reference expressed hope that the techniques set up there could be adapted to handle more

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general Cauchy integrals than Markov functions. Below, we take a step towards carrying out this program. Speciﬁcally, we consider Cauchy transforms of complex measures that are absolutely continuous with respect to the equilibrium distribution on a real segment [a, b] ⊂ (−1, 1). The density will be required to be Dini-smooth and non-vanishing. In addition, it should admit an argument function of bounded variation on [a, b]. Moreover, we handle with little extra-pain the case where a rational function is added to such a Cauchy tranform. For functions of this kind, we establish an analog to [11, Thm. 1.3], namely asymptotic uniqueness of a critical point in best rational approximation for the L2 -norm on the unit circle. This is the ﬁrst uniqueness result in degree greater than 1 for Cauchy integrals with complex densities, more generally for non-rational functions without conjugate symmetry. In contrast, say, to [12, Thm. 3], it is only fair to say that such a statement is not really constructive in that no estimate is provided for the degree beyond which uniqueness prevails. However, considering our restricted knowledge on uniqueness in non-linear complex approximation, our result sheds considerable light on the behaviour of best rational approximants to Cauchy transforms, and it is our hope that suitable reﬁnements of the technique will eventually produce eﬀective bounds. Our method of proof follows the same pattern as [11]. Namely, the index theorem is invoked to reduce the question of uniqueness to whether each critical point is a non-degenerate local minimum. Next, a criterion for being a local minimum is set up, based on a comparison between the error function generated by the critical point under examination and the error function attached to a particular multipoint Pad´e interpolant of lower degree; we call it for short the comparison criterion. Finally, the fact this criterion applies when the degree is suﬃciently large depends on strong asymptotic formulas for the error in rational interpolation to functions of the type we consider, that were recently obtained in [15, 13, 31], and on a speciﬁc design of interpolation nodes to build the particular multipoint Pad´e interpolant of lower degree that we need. With respect to [11], however, two main diﬀerences arise. The ﬁrst is that the comparison criterion was set up there for conjugate-symmetric functions only, i.e. for those functions having real Fourier coeﬃcients. Because we now adress Cauchy transforms of complex densities on a segment, we handle complex Fourier coeﬃcients as well. Although the corresponding changes are mostly mechanical, they have to be carried out thoroughly for they impinge on the computation of the Hessian quadratic form and on the nondegeneracy thereof. The second diﬀerence causes more serious diﬃculties. Indeed, the construction of the special interpolant of lower degree needed to apply the comparison criterion requires rather precise control on the poles of multipoint Pad´e interpolants to the approximated function. For Markov functions, it is known that such poles are the zeros of certain orthogonal polynomials with respect to a positive measure on the segment [a, b], therefore they lie on that segment. But for Cauchy transforms of complex measures, the poles are the zeros of some non-Hermitian orthogonal polynomial with respect to a complex measure on [a, b], and their behaviour does not lend itself to analysis so easily. We resort here to the work in [4] on the geometry of non-Hermitian orthogonal polynomials to overcome this diﬃculty, and this is where the boundedness of the variation of the density’s argument becomes important.

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Let us brieﬂy indicate some generalizations of our results that were not included here. Firstly, asymptotic uniqueness of a critical point extends to Cauchy transforms of absolutely continuous measures whose density with respect to Lebesgue (rather than equilibrium) measure satisﬁes similar assumptions, e.g. nonvanishing and H¨ older-smoothness; in fact, densities with respect to any Jacobi weight could be handled the same way under suitable regularity requirements. We did not mention them, however, because such an extension depends on asymptotics for non-Hermitian orthogonal polynomials with respect to this type of weight which are yet unpublished [14]. Secondly, ﬁnitely many zeros in the density would still be acceptable, provided they are of power type with suﬃciently small exponent. Developing the precise estimates would make the paper heavier (compare [13, Thm. 4] and [31, Thm. 5]), so we felt better omitting this stronger version. The organization of the article is as follows. In Section 2 we present the rational approximation problem under study and we state the main result of the paper, which is Theorem 2.1. Section 3 introduces the critical points in H 2 -rational approximation and develops their interpolating properties; this part is adapted to complex Fourier coeﬃcients from [8]. The index theorem and the comparison criterion are expounded in section 4, paralleling the treatment for real Fourier coeﬃcients given in [11]. Section 5 recalls the necessary material on interpolation from [15, 13, 31], which is needed to carry out the comparison between critical points and interpolants of lower degree required in the comparison criterion. Finally, elaborating on [11], we prove Theorem 2.1 in Section 6. 2. Notation and Main Results Let T be the unit circle and D the unit disk. We let L2 (resp. L∞ ) stand for the space of square-integrable (resp. essentially bounded) measurable functions on T. Denote by H 2 the familiar Hardy space of the unit disk, consisting of those L2 functions whose Fourier coeﬃcients of strictly negative index are zero. The space H 2 identiﬁes with traces on T of those holomorphic functions in D whose L2 -means on circles centered at zero with radii less than 1 are uniformly bounded above. In fact, any such function has non-tangential boundary values almost everywhere on T that deﬁne a member of L2 from which the function can be recovered by means of a Cauchy integral [21]. The space of bounded holomorphic functions on D is denoted by H ∞ and is endowed with the L∞ norm of the trace. ¯ 2 be the orthogonal complement of H 2 in L2 with respect to the standard Let H 0 scalar product |dτ | , f, g ∈ L2 . (2.1) f, g := f (τ )g(τ ) 2π T ¯ 02 , in turn, identiﬁes with traces of those holomorphic functions in C\D The space H that vanish at inﬁnity and whose L2 -means on circles centered at zero with radii greater then 1 are uniformly bounded above. In what follows, we denote by · 2 ¯ 2 induced by the scalar product (2.1). On one occasion, the norm on L2 , H 2 , and H 0 ¯ 2 (C \ Dρ ), which is deﬁned similarly except that we shall refer to the Hardy space H 0 D gets replaced by Dρ := {|z| < ρ} where ρ > 0. Set Pn for the space of algebraic polynomials of degree at most n and Mn for the subset of monic polynomials with exactly n zeros in D. Note that q ∈ Mn if and Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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¯ 02 and 1/q(z) ∼ z −n at inﬁnity. From the diﬀerential viewpoint, we only if 1/q ∈ H regard Pn as Cn+1 and Mn as an open subset of Cn , upon taking the coeﬃcients as coordinates (except for the leading coeﬃcient in Mn which is ﬁxed to unity). Deﬁne pn−1 z n−1 + pn−2 z n−2 + · · · + p0 p(z) = (2.2) Rn := : p ∈ P , q ∈ M n−1 n . q(z) z n + qn−1 z n−1 + · · · + q0 It is easy to check that Rn consists of those rational functions of degree at most n ¯ 2 , and we endow it with the corresponding topology. Coordinatizthat belong to H 0 ing Pn−1 and Mn as above, it is straightforward to see that the canonical surjection J : Pn−1 × Mn → Rn is smooth (i.e., inﬁnitely diﬀerentiable) when viewed as a ¯ 2 -valued map. Note that J is not injective, due to possible cancellation between H 0 p and q, but it is a local homeomorphism at every pair (p, q) such that p, q are coprime. We shall be concerned with the following problem. ¯ 2 and n ∈ N, ﬁnd r ∈ Rn to minimize f − r2 . Problem 1. Given f ∈ H 0 Let us point out two equivalent formulations of Problem 1 that account for early discussion made in the introduction. ¯ 2 , as is subsumed in the Firstly, it is redundant to assume that r lies in H 0 deﬁnition of Rn . Indeed, by partial fraction expansion, a rational function of degree ¯ 2 and r2 ∈ H 2 have degree at most n in L2 can be written as r1 + r2 , where r1 ∈ H 0 2 2 ¯ at most n. By orthogonality of H0 and H , we get f − r22 = f − r1 22 + r2 22 so that r1 is a better candidate approximant than r, showing that Problem 1 is in fact equivalent to best rational approximation of given degree to f in L2 . ¯ 2 onto Secondly, composing with z → 1/z, which is an L2 -isometry mapping H 0 2 zH while preserving rationality and the degree, Problem 1 transforms to best approximation in H 2 of functions vanishing at the origin by rational functions of degree at most n that vanish at the origin as well. However, by Parseval’s identity, any best rational approximant to g in H 2 has value g(0) at 0. Therefore Problem 1 is equivalent to Problem 2. Given g ∈ H 2 and n ∈ N, ﬁnd a rational r of degree at most n in H 2 to minimize f − r2 . Problem 1 is the one we shall work with, and we refer to it as the rational ¯ 2 -approximation problem to f in degree n. It is well-known (see [3, Prop. 3.1] H 0 for a proof and further bibliography on the subject) that the minimum is attained and that a minimizing r, called a best rational approximant of degree n to f , lies in Rn \ Rn−1 unless f ∈ Rn−1 . Uniqueness of such an approximant is a delicate matter. Generically, there is only one best approximant by a theorem of Stechkin on Banach space approximation from approximately compact sets [16]. However, the proof is non-constructive and does not allow us to determine which functions have a unique best approximant and which functions do not. Moreover, from the computational viewpoint, the main interest lies not so much with uniqueness of a best approximant, but rather with uniqueness of a local best approximant for such places are all what a numerical search can usually spot. By deﬁnition, a local best approximant is a function rl ∈ Rn such that f − rl 2 ≤ f − r2 for all r ∈ Rn in some neighborhood of rl . Like best approximants, local best approximants lie in Rn \ Rn−1 unless f ∈ Rn−1 [3]. Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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Still more general is the notion of a critical point, which is deﬁned as follows. ¯ 2 and put Fix f ∈ H 0 (2.3)

Φf,n : Rn r

→ [0, ∞) → f − r22 .

A pair (p, q) ∈ Pn−1 ×Mn is called critical if all partial derivatives of Φf,n ◦J vanish at (p, q). Subsequently, a rational function rc ∈ Rn is said to be a critical point of Φf,n if there is a critical pair (pc , qc ) such that rc = pc /qc . Critical points fall into two classes: they are termed irreducible if they have exact degree n, and reducible if they have degree strictly less than n. Note that rc is irreducible if and only if pc and qc are coprime in some (hence any) representation rc = pc /qc . Clearly a local best approximant is a particular instance of a critical point, and it is irreducible unless f ∈ Rn−1 . In the present work, we dwell on a diﬀerential topological approach to uniqueness of a critical point introduced in [1] and further developed in [5, 2]. In this approach, global uniqueness is deduced from local analysis of the map Φf,n . Specifically, to conclude there is only one critical point, which is therefore the unique local minimum (and a fortiori the global minimum as well), one needs to show that each critical point is irreducible, does not interpolate the approximated function on T, and is a nondegenerate local minimum; here nondegenerate means that the second derivative is a nonsingular quadratic form. This method turns out to be fruitful when studying rational approximation to Cauchy transforms of measures supported in (−1, 1), i.e., functions of the form dµ(t) (2.4) fµ (z) := , supp(µ) ⊂ (−1, 1). z−t The ﬁrst result in this direction was obtained in [12, Thm. 3] when fµ is a Markov function, meaning that µ in (2.4) is a positive measure. It goes as follows. Theorem A. Let√µ be supported on [a, b] ⊂ (−1, 1) where a a positive measure and b satisfy b − a ≤ 2 1 − max a2 , b2 . Assume further that µ has at least n points of increase, i.e., fµ ∈ / Rn−1 . Then there is a unique critical point in rational ¯ 02 -approximation of degree n to fµ . H Removing the restriction on the size of the support makes the situation more diﬃcult. The following theorem [11, Thm. 1.3] asserts that rational approximants are asymptotically unique for Markov functions whose deﬁning measure is suﬃciently smooth. Hereafter, we denote by ω[a,b] the normalized arcsine distribution on [a, b] given by dω[a,b] (t) = dt/(π (t − a)(b − t)). Theorem B. Let µ be a positive measure supported on [a, b] ⊂ (−1, 1) and let us write dµ = µ dt + dµs , where µs is singular and µ is integrable on [a, b]. If µ satisﬁes the Szeg˝ o condition: log µ dω[a,b] > −∞, then there is a unique critical ¯ 02 -approximation of degree n to fµ for all n large enough. point in rational H As an additional piece of information, the following negative result [10, Thm. 5] shows that the asymptotic nature of the previous theorem is indispensable. Theorem C. For each n0 ∈ N there exists a positive measure µ satisfying the Szeg˝ o condition such that for each odd n between 1 and n0 there exist at least two diﬀerent best rational approximants of degree n to fµ . Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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Our goal is to extend Theorem B to a class of complex measures which is made precise in the deﬁnition below. Recall that a function h with modulus of continuity ωh is said to be Dini-continuous if ωh (t)/t is integrable on [0, ] for some (hence any) > 0. Definition (Class M). A measure µ is said to belong to the class M if supp(µ) ⊂ (−1, 1) is an interval, say [a, b], and dµ = µdω ˙ [a,b] , where µ˙ is a Dinicontinuous non-vanishing function with an argument of bounded variation on [a, b]. Observe that we deal here with µ, ˙ the Radon-Nikodym derivative of µ with respect to the arcsine distribution, rather then with µ , the Radon-Nikodym derivative with respect to the Lebesgue measure. Our main result is: Theorem 2.1. Let f := fµ + r, where µ ∈ M and r ∈ Rm has no poles on ¯ 2 -approximation of supp(µ). Then there is a unique critical point in rational H 0 degree n to f for all n large enough. Before we can prove the theorem, we must study in greater detail the structure of critical points, which is the object of Sections 3 and 4 to come. 3. Critical Points The following theory was developed in [5, 6, 7, 8] when the function f to be approximated is conjugate-symmetric, i.e., f (¯ z ) = f (z), and the rational approximants are seeked to be conjugate-symmetric as well. In other words, when a function with real Fourier-Taylor expansion at inﬁnity gets approximated by a rational function with real coeﬃcients. Surprisingly enough, this is not subsumed in Problem 1 in that conjugate-symmetric functions need not have a best approximant out of Rn which is conjugate-symmetric. For Markov functions, though, it is indeed the case [4]. Below, we develop an analogous theory for Problem 1, that is, without conjugate-symmetry assumptions. This involves only technical modiﬁcations of a rather mechanical nature. Hereafter, for any f ∈ L2 , we set f σ (z) := (1/z)f (1/¯ z ). Clearly, f → f σ 2 2 ¯ is an isometric involution mapping H onto H0 and vice-versa. Further, for any z ) and deﬁne its reciprocal polynomial (in Pk ) to be p ∈ Pk , we set pˇ(z) := p(1/¯ z ). Note that p has the same modulus as p on T and its p (z) := z k pˇ(z) = z k p(1/¯ zeros are reﬂected from those of p across T. 3.1. Critical Points as Orthogonal Projections. Fix f ∈ H02 and let Φn := Φf,n be given by (2.3). It will be convenient to use complex partial derivatives with respect to pj , qk , p¯j , q¯k , where, for 0 ≤ j, k ≤ n − 1, the symbols pk and qk refer to the coeﬃcients of p ∈ Pn−1 and q ∈ Mn as in equation (2.2). By complex derivatives we mean the standard Wirtinger operators, e.g., if pj = xj + iyj is the decomposition into real and imaginary part then ∂/∂pj = (∂/∂xj − i∂/∂yj )/2 and ∂/∂ p¯j = (∂/∂xj + i∂/∂yj )/2. The standard rules for derivation are still valid, obviously ∂g(pj )/∂ p¯j = 0 if g is holomorphic, and it is straightforward ¯ p¯j for any function h. In particular, since Φn is real, that ∂h/∂pj = ∂ h/∂ ∂Φn ∂Φn ∂Φn ∂Φn (3.1) and . = = ∂ p¯j ∂pj ∂ q¯k ∂qk Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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Thus, writing Φn ◦ J(p, q) = f − p/q, f − p/q and diﬀerentiating under the integral sign, we obtain that a critical pair (pc , qc ) of Φn ◦ J is characterized by the relations

j ∂Φn z pc (3.2) = 0, j ∈ {0, . . . , n − 1}, (pc , qc ) = ,f − ∂pj qc qc

k ∂Φn z pc pc (3.3) = 0, k ∈ {0, . . . , n − 1}. (pc , qc ) = − , f − ∂qk qc2 qc Equation (3.2) means that pc /qc is the orthogonal projection of f onto Vqc , where for any q ∈ Mn we let Vq := {p/q : p ∈ Pn−1 } to be the n-dimensional linear ¯ 2 consisting of rational functions with denominator q. In what follows, subspace of H 0 we consistently denote the orthogonal projection of f onto Vq by Lq /q, where Lq ∈ Pn−1 is uniquely characterized by the fact that

Lq p , = 0 for any p ∈ Pn−1 . (3.4) f− q q Taking equation (3.3) into account, we conclude from what precedes that critical points of Φn are precisely Rn -functions of the form Lqc /qc , where qc ∈ Mn satisﬁes

k Lq c z Lq c = 0, k ∈ {0, . . . , n − 1}. , f − (3.5) qc 2 qc Now, it is appearent from (3.4) that Lq is a smooth function of q, therefore we deﬁne a smooth map Ψn on Mn by setting (3.6)

Ψn = Ψf,n : Mn q

→ [0, ∞) . → f − Lq /q22

By construction, Φn attains a local minimum at r = Lql /ql if and only if Ψn attains a local minimum at ql , and the assumed values are the same. More generally, r ∈ Rn is a critical point of Φn if and only if r = Lqc /qc , where qc ∈ Mn , is a critical point of Ψn . This is readily checked upon comparing (3.5) with the result of the following computation:

∂Ψn Lq (∂Lq /∂ q¯k ) (∂Lq /∂qk )q − z k Lq Lq + f− ,− (q) = ,f − ∂qk q2 q q q

k z Lq Lq (3.7) , k ∈ {0, . . . , n − 1}, = − ,f − q2 q where we applied (3.4) using that the derivatives of Lq lie in Pn−1 and that ∂q/∂ q¯k = 0. For simplicity, we drop from now on the subscript “c” we used so far as a mnemonic for “critical”. Altogether we proved the following result: ¯ 2 , let Φn and Ψn be deﬁned by (2.3) and (3.6), Proposition 3.1. For f ∈ H 0 respectively. Then r ∈ Rn is a critical point of Φn if and only if r = Lq /q and q ∈ Mn is a critical point of Ψn . In view of Proposition 3.1, we shall extend to Ψn the terminology introduced for Φn and say that a critical point q ∈ Mn of Ψn is irreducible if Lq and q are coprime. Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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3.2. Interpolation Properties of Critical Points. If we denote with a ¯ 2 , it is elementary to check that superscript “⊥” the orthogonal complement in H 0 q

¯2 , H ¯ 2 = Vq ⊕ V ⊥ . u: u∈H (3.8) Vq⊥ = 0 0 q q ¯ 02 such that Hence, by (3.4), there exists uq = uf,q ∈ H (3.9)

f q − Lq = q uq .

Relation (3.9) means that Lq /q interpolates f at the reﬂections of the zeros of q across T. Assume now that q ∈ Mn is a critical point of Ψn . Then, combining (3.9) and (3.7), we derive that (3.10) pLq p(τ ) (Lq uσq )(τ ) dτ pLq q uq Lq 0= = = , f − , , p ∈ Pn−1 . q2 q q2 q q (τ ) 2πi T q(τ ) q ∈ H 2 , we see by letting p range over elementary divisors of q and Since Lq uσq /

applying the residue formula that (3.10) holds if and only if each zero of q is a zero of Lq uσq of the same multiplicity or higher. That is, q is a critical point of Ψn if and only if q divides Lq uσq in H 2 . Let d ∈ Mk be the monic g.c.d. of Lq and q, with 0 ≤ k ≤ n − 1. Writing Lq = dp∗ and q = dq ∗ , where p∗ and q ∗ are coprime, we deduce that q ∗ divides uσq ¯ 2 . Besides, it follows from (3.4), in H 2 or equivalently that uq = qˇ∗ h for some h ∈ H 0 ∗ applied with p = dv and v ∈ Pn−k−1 , that p = Lq∗ . Therefore, upon dividing (3.9) by d, we get q qˇ∗ h, (3.11) f q ∗ − Lq ∗ = d implying that uq∗ = d qˇ∗ h/d. In particular, q, thus a fortiori q ∗ , divides uσq∗ in H 2 , whence q ∗ is critical for Ψn−k by what we said before. Finally, dividing (3.11) by q ∗ and taking into account the deﬁnition of the reciprocal polynomial, we ﬁnd that we established the following result. Proposition 3.2. Let q be a critical point of Ψn and d ∈ Mk be the monic g.c.d. of Lq and q with 0 ≤ k ≤ n − 1. Then q ∗ = q/d ∈ Mn−k is an irreducible 2ˇ critical point of Ψn−k , and Lq∗ /q ∗ interpolates f at the zeros of qˇ∗ d/z in Hermite’s sense on C \ D, that is, counting multiplicities including at inﬁnity. The converse is equally easy: if q ∗ is an irreducible critical point of Ψn−k , and 2ˇ Lq∗ /q ∗ interpolates f at the zeros of qˇ∗ d/z in Hermite’s sense for some d ∈ Mk , then q = q ∗ d is a critical point of ϕn and d is the monic g.c.d. of q and Lq . This we shall not need. It is immediately seen from Proposition 3.2 that a critical point of Φf,n must interpolate f with order 2 at the reﬂections of its poles across T; for best approximants, this property is classical [22, 26]. 3.3. Smooth Extension of Ψn . One of the advantages of Ψn , as compared to Φn , is that its domain of deﬁnition can be compactiﬁed, which is essential to rely on methods from diﬀerential topology. To do that, however, we need to place an additional requirement on f . ¯ 0 the subset of H ¯ 02 comprised of functions that extend Let us denote by H ¯ 0 and pick ρ = holomorphically across T. Hereafter we will suppose that f ∈ H Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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ρ(f ) < 1 such that f is holomorphic in {|z| > ρ − } for some > 0. In particular, f is holomorphic across Tρ := {|z| = ρ}. 1/ρ Denote by Mn and Mn respectively the closure of Mn and the set of monic polynomials with zeros in D1/ρ := {|z| < 1/ρ}; as usual, we regard these as subsets of Cn when coordinatized by their coeﬃcients except the leading one. This way 1/ρ Mn becomes an open neighborhood of the compact set Mn , which is easily seen to consist of polynomials with zeros in D. Also, q lies on the boundary ∂Mn = Mn \ Mn if and only if it is a monic polynomial of degree n having at least one zero of modulus 1 and no zero of modulus strictly greater then 1. For q ∈ Mn , since q/

q is unimodular on T, it follows from (3.9) that 2 Lq = uq 2 = (uq uσ )(τ ) dτ . (3.12) Ψn (q) = f − 2 q q 2 2πi T q in D, In addition, taking into account the Cauchy formula, the analyticity of Lq /

the analyticity of f across Tρ , and the deﬁnition of the σ-operation, we obtain f (τ )q(τ ) dτ 1 (3.13) uq (z) = uf,q (z) = , |z| > ρ, 2πi Tρ q (τ ) z − τ f σ (τ )

q (τ ) dτ 1 (3.14) , |z| < 1/ρ uσq (z) = 2πi T1/ρ q(τ ) τ − z f (τ ) q(z)

q (τ ) − q (z)q(τ ) dτ (3.15) Lq (z) = Lf,q (z) = , |z| > ρ. q

(τ ) z−τ 2πi Tρ 1/ρ

Now, if q ∈ Mn , then q has all its zeros of modulus greater then ρ, therefore (3.15) and (3.13) are well-deﬁned and smooth with respect to the coeﬃcients of q, ¯ 02 (C \ Dρ ) respectively. Because evaluation at τ ∈ T is with values in Pn−1 and H ¯ 2 (C \ Dρ ), Ψn in turn extends smoothly uniformly bounded with respect to τ on H 0 1/ρ to Mn in view of (3.12). Moreover, diﬀerentiating under the integral sign, we obtain ∂uσq ∂Ψn ∂uq dτ (3.16) (q) = (τ )uσq (τ ) + uq (τ ) (τ ) , ∂qj ∂qj ∂qj 2πi T with (3.17) (3.18)

∂uq (z) ∂qj ∂uσq (z) ∂qj

= =

1 2πi 1 2πi

Tρ

τ j f (τ ) dτ , q (τ ) z − τ

T1/ρ

|z| > ρ,

−τ j f σ (τ )

q (τ ) dτ , q 2 (τ ) τ −z

|z| < 1/ρ,

for j = 0, . . . , n − 1. To recap, we have proved: ¯ 0 . Then Ψn extends to a smooth function in some Proposition 3.3. Let f ∈ H ¯ 2 (C \ Dρ ) neighborhood of Mn and so do Lq and uq with values in Pn−1 and H 0 respectively. In addition, (3.16), (3.17), and (3.18) hold. We shall continue to denote the extension whose existence is asserted in Proposition 3.3 by Ψf,n , or simply Ψn if f is understood from the context. Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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3.4. Critical points on the boundary. Having characterized the critical points of Ψn on Mn in Section 3.2, we need now to describe the critical points that it may have on ∂Mn . We shall begin with the case where all the roots of the latter lie on T. ¯ 0 and assume v(z) = (z − ξ)k , ξ ∈ T, is a critical point of Ψk . It Let f ∈ H immediately follows from (3.13), the Cauchy formula, and the deﬁnition of v that (3.19)

v = eiθ v,

where

¯ k, eiθ := (−ξ)

Lv ≡ 0,

and uv = e−iθ f.

In this case, equations (3.17) and (3.18) become ∂uv τ j f (τ ) dτ e−iθ , |z| > ρ, (z) = ∂qj 2πi Tρ v(τ ) z − τ ∂uσv τ j f σ (τ ) dτ −eiθ (z) = , |z| < 1/ρ. ∂qj 2πi T1/ρ v(τ ) τ − z Plugging these expressions into (3.16), we obtain τ j (f f σ )(τ ) dτ ∂Ψk , (v) = 0= ∂qj v(τ ) 2πi ∂Aρ

j ∈ {0, . . . , k − 1},

where Aρ := {ρ < |z| < 1/ρ} with positively oriented boundary ∂Aρ , and we used the Fubini-Tonelli theorem. By taking linear combinations of the previous equations, we deduce from the Cauchy formula that (f f σ )(τ ) dτ (f f σ )(l−1) (ξ) 0= = , l ∈ {1, . . . , k}. l (l − 1)! ∂Aρ (τ − ξ) 2πi Hence v divides f f σ , when viewed as a holomorphic function in Aρ . Consequently, since ζ ∈ T is a zero of f if and only if it is a zero f σ , we get that f vanishes at ξ with multiplicity (k + 1)/2, where x is the integer part of x. Next we consider the case where q is a critical point of Ψn having exactly one root on T: q = vq ∗ with v(z) = (z − ξ)k , ξ ∈ T, and q ∗ ∈ Mn−k . Denote by Q and V 1/ρ 1/ρ some neighborhoods of q ∗ and v, in Mn−k and Mk respectively, taking them so small that each χ ∈ Q is coprime to each ν ∈ V; this is possible since q ∗ and v are coprime. Then, (χ, ν) → χν is a diﬀeomorphism from Q × V onto a neighborhood 1/ρ of q in Mn . In particular, the fact that q is a critical point of Ψn means that q ∗ is a critical point of Θ and v a critical point of Ξ, where Θ:Q χ

→ [0, ∞) → Ψn (vχ)

and

Ξ:V ν

→ [0, ∞) → Ψn (νq ∗ ).

, where eiθ is as in (3.19), it follows from (3.13) that uvχ = e−iθ uχ , Since v χ = eiθ v χ and therefore by (3.12) that Θ = Ψn−k|Q , implying that q ∗ is a critical point of Ψn−k . In another connection, shrinking V if necessary, we may assume there exists 1/

> ρ such that V ⊂ Mk . Put for simplicity w := uq∗ = uf,q∗ , which is clearly an ¯ 0 by (3.13). Computing with the latter formula yields for any ν ∈ V element of H that (wν)(τ ) dτ 1 (f q ∗ )(t) dt ν(τ ) dτ 1 1 = uw,ν (z) = 2πi T ν (τ ) z − τ 2πi T 2πi Tρ q ∗ (t) τ − t ν (τ ) z − τ ∗ (f νq )(t) dt 1 = uf,νq∗ (z), |z| > , = 2πi Tρ (νq ∗ )(t) z − t Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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where we used the Fubini-Tonelli theorem and the Cauchy integral formula. Thus, we derive from (3.12) that Ξ(ν) = Ψf,n (νq ∗ ) = uf,νq∗ 22 = uw,ν 22 = Ψw,k (ν),

ν ∈ V.

As v is a critical point of Ξ, we see that it is also critical for Ψw,k , so by the case previously considered we conclude that w = uf,q∗ vanishes at ξ with multiplicity (k + 1)/2. By (3.9), this is equivalent to the fact that Lq∗ /q ∗ = Lq /q interpolates f at the zeros of d(z) = (z − ξ)(k+1)/2 in Hermite’s sense. Finally, the case where q is arbitrarily located on ∂Mn is handled the same way upon writing q = q ∗ v1 . . . v , where vj (z) = (z − ξj )kj for some ξj ∈ T, and introducing a product neighborhood Q × V1 × . . . × V of q ∗ v1 . . . v to proceed with the above analysis on each of the corresponding maps Θ, Ξ1 , . . . , Ξ . Thus, taking into account Proposition 3.2 and the fact that vj and vˇj have the same zeros in C, we obtain: ¯ 0 and q = vq ∗ , where v = (z − ξj )kj , ξj ∈ T, Proposition 3.4. Let f ∈ H deg(v) = k, and q ∗ ∈ Mn−k . Assume that q is a critical point of Ψn = Ψf,n . Then q ∗ is a critical point of Ψn−k . Moreover, if we write q ∗ = q1 d1 where d1 is the monic g.c.d. of Lq∗ and q ∗ , then Lq∗ /q ∗ = Lq /q interpolates f at the zeros of ˇ in Hermite’s sense on C \ D, where d(z) = (z − ξj )(kj +1)/2 . qˇ1 2 dˇ1 d/z Again the converse of Proposition 3.4 is true, namely the properties of q ∗ and v asserted there imply that q = q ∗ v is critical for Ψn . This is easy to check by reversing the previous arguments, but we shall not use it. 4. A Criterion for Local Minima ¯ 0 and Ψn = Ψf,n be the extended map obtained in Proposition 3.3, Let f ∈ H based on (3.12) and (3.13). The latter is a smooth real-valued function, deﬁned on an open neighborhood of Mn identiﬁed with a subset of Cn ∼ R2n by taking as coordinates all coeﬃcients but the leading one. By deﬁnition, a critical point of Ψn is a member of Mn at which the gradient ∇Ψn vanishes. This notion is of course independent of which coordinates are used, and so is the signature of the second derivative, the so called Hessian quadratic form1. A critical point q is called nondegenerate if the Hessian form is nonsingular at q, and then the number of negative eigenvalues of this form is called the Morse index of q denoted by M (q). Observe that nondegenerate critical points are necessarily isolated. From ﬁrst principles of diﬀerential topology [23] it is known that (−1)M (q) , which is called the index of the nondegenerate critical point q, is equal to the socalled Brouwer degree of the vector ﬁeld ∇Ψn /∇Ψn e on any suﬃciently small sphere centered at q, where · e is the Euclidean norm in R2n . One can show that ∂Mn is a compact manifold2, so if Ψn has no critical points on ∂Mn and only nondegenerate critical points in Mn , then the sum of the indices of the critical points is equal to the Brouwer degree of ∇Ψn /∇Ψn e on ∂Mn . The surprising fact is that the latter is independent of f (see [1], [5, Sec. 5], and [2, Thm. 2]) and is actually equal to 1. Altogether, the following analogue of the Poincar´e-Hopf theorem holds in the present setting. 1This is not true at non-critical points. 2We skim through technical diﬃculties here, because this manifold is not smooth; the inter-

ested reader should consult the references we give.

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¯ 0 and Cf,n be the set of the critical points The Index Theorem. Let f ∈ H of Ψf,n in Mn . Assume that all members of Cf,n are nondegenerate, and that Cf,n ∩ ∂Mn = ∅. Then (−1)M (q) = 1. q∈Cf,n

To us, the value of the index theorem is that if can show every critical point is a nondegenerate local minimum and none of them lies on ∂Mn , then the critical point is unique. To see this, observe that local minima have Morse index 0 and therefore index 1. To make this criterion eﬀective, we need now to analyze the Morse index of a critical point, starting with the computation of the Hessian quadratic form. Let q be a critical point of Ψn . It is easy to check that the Hessian quadratic form of Ψn at q is given by ⎛ ⎞ n−1 2 2 n−1 ∂ Ψ ∂ Ψ n n vj vk (q) + vj v¯k (q) ⎠ , (4.1) Q(v) = 2Re ⎝ ∂qk ∂qj ∂ q¯k ∂qj j=0 k=0

n−1

where we have set v(z) = j=0 vj z j for a generic element of Pn−1 , the latter being naturally identiﬁed with the tangent space to Mn at q, and we continue to consider qj , q¯j , j ∈ {0, . . . , n − 1}, as coordinates on Mn . Clearly, q is a nondegenerate local minimum if and only if Q is positive deﬁnite, i.e., (4.2)

Q(v) > 0

for

v ∈ Pn−1 , v = 0.

Let us assume that q is irreducible, hence q ∈ Mn by Proposition 3.4. To derive conditions that ensure the validity of (4.2), we commence by reworking the expression for Q. Any polynomial in P2n−1 can be written p1 Lq +p2 q for suitable p1 , p2 ∈ Mn−1 , due to the coprimeness of Lq and q. Therefore

Lq p = 0 for any p ∈ P2n−1 ,f − (4.3) q2 q by (3.4) and (3.7). In view of (4.3), diﬀerentiating (3.4) with respect to qj and evaluating at q leads us to

p Lq ∂ , = 0, j ∈ {0, . . . , n − 1}, p ∈ Pn−1 , (4.4) ∂qj q q which means that ∂(Lq /q)/∂qj belongs to Vq⊥ . Hence, we get from (3.8) that q∂Lq /∂qj − z j Lq Lq q νj ∂ = =: 2 , νj ∈ Pn−1 , j ∈ {0, . . . , n − 1}. (4.5) ∂qj q q2 q As Lq and q are coprime, the polynomials νj are linearly independent by construction, thus we establish a one-to-one linear correspondence on Pn−1 by setting (4.6)

v(z) =

n−1 j=0

vj z j

↔

ν(z) = −

n−1

vj νj (z).

j=0

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LAURENT BARATCHART AND MAXIM YATTSELEV

Moreover, from Proposition 3.2 where d = 1 and q ∗ = q, we can write (compare (3.11)) Lq q qˇ (4.7) f− = wqσ for some wq ∈ H 2 . q q σ ¯ Note that wq ∈ H0 since f does, hence wq is holomorphic across T. Now, it follows from (3.7) and (3.1) that

Lq ∂ ∂ Lq Lq Lq ∂ 2 Ψn ∂2 ,f − + , (q) = − ∂ q¯k ∂qj ∂ q¯k ∂qj q q ∂qj q ∂qk q

νj νk q νj q νk (4.8) , = , = q2 q2 q q by (4.3), (4.5), and the fact that q /q is unimodular on T. Furthermore

∂ 2 Ψn Lq ∂Lq /∂ q¯j ∂ Lq Lq ∂2 ,f − + , (q) = − ∂qk ∂qj ∂qk ∂qj q q ∂qj q q

q qˇ Lq ∂2 , wq = − ∂qk ∂qj q q by (4.4) and (4.7). Now, a simple computation using (4.5) yields q νk q(∂ 2 Lq /∂qk ∂qj ) − z k (∂Lq /∂qj ) + z j (∂Lq /∂qk ) Lq ∂2 = − 2z j 3 , ∂qk ∂qj q q2 q and since the ﬁrst fraction on the above right-hand side belongs to P2n−1 /q, we deduce from (4.4) and what precedes that

j

j z νk σ z q νk q qˇ σ ∂ 2 Ψn w , w = 2 , (q) = 2 , (4.9) q ∂qk ∂qj q3 q q q since q /q is unimodular while qˇ = q on T. So, we get from (4.8), (4.9), and (4.6) that

n−1 n−1 ∂ 2 Ψn ν vν σ σ , wq = −2 , (vwq ) vj vk (q) = −2 ∂qk ∂qj q q j=0 k=0

and

n−1 n−1 j=0 k=0

∂ 2 Ψn vj v¯k (q) = ∂ q¯k ∂qj

ν ν , q q

2 ν = q . 2

Therefore, in view of (4.1) the quadratic form Q/2 can be rewritten as 2 2

σ ν ν 1 ν ν σ Q(v) = − 2Re , (vwq ) (4.10) = − 2Re , vwq . 2 q 2 q q 2 q To manage the above expression, we assume that Lq /q does not interpolate f on T, i.e. that wq has no zeros there, and we let Q ∈ Ml have the same zeros

where oq is as wq in D, counting multiplicities. Thus we can write wq = oq Q/Q, holomorphic and zero-free on a neighborhood of D, while |oq | = |wq | on T since

is unimodular there3. Consider now the Hankel operator Γ, with symbol Q/Q sq := Lq /(oq q

q ), i.e. ¯ 02 Γ : H2 → H u → P− (sq u) , 3The function o is none but the outer factor of w in H 2 , see [21, Thm. 2.8]. q q

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ASYMPTOTIC UNIQUENESS OF BEST RATIONAL APPROXIMANTS

101 15

¯ 02 . Observe that Γ is well where P− is the orthogonal projection from L2 onto H deﬁned because sq is bounded on T, and since the latter is meromorphic in D with poles at the zeros of q, counting multiplicities. It is elementary [27] that Γ(H 2 ) = Vq , that Ker Γ = (q/

q )H 2 , and that Γ : Vq → Vq is an isomorphism, where Vq := Pn−1 /

q is readily seen to be the orthogonal complement of Ker Γ in H 2 . Thus, there exists an operator Γ# : Vq → Vq, which is inverse to Γ|Vq . To evaluate Γ# , observe from (4.5) that n−1 j n−1 vLq z Lq q(∂Lq /∂qj ) − q νj = = vj P− vj P− Γ(voq ) = P− q

q q

q q

q j=0 j=0 =

n−1 j=0

vj P−

−

q νj q

q

= P−

ν ν = , q q

¯ 02 . Hence we may write where we used that (∂Lq /∂qj )/

q ∈ H 2 and that ν/q ∈ H ν q = voq + u, with u ∈ H 2 = Ker Γ, (4.11) Γ# q q

and since wq /oq ∈ H ∞ it follows that uwq /oq ∈ Ker Γ as well, entailing by (4.10) and (4.11) that 2

σ ν 1 ν wq # ν − 2Re Q(v) = , Γ q 2 q oq q 2 because (ν/q)σ ∈ Vq = (Ker Γ)⊥ . Altogether, we see that 2 σ ν 2 ν 1 wq # ν − 2 ≥ 1 − 2Γ# ν Q(v) ≥ , Γ q q 2 q oq q 2 2 by the Schwarz inequality and since |wq /oq | = 1 on T while the σ operation preserves the norm. The inequalities above imply that Q is positive deﬁnite as soon as Γ# < 1/2. This last inequality is equivalent to saying that 2 is strictly less than the smallest singular value of Γ|Vq , which is also the n-th singular value of Γ since Vq has dimension n and is the orthogonal complement of Ker Γ in H 2 . By the Adamjan-Arov-Krein theorem [27], the singular value in question is equal to the ∞ ∞ error in L∞ -best approximation to sq from Hn−1 , where Hn−1 stands for the set ∞ of functions of the form h/χ where h ∈ H and χ ∈ Mn−1 . Let us indicate this approximation number by σn−1 : σn−1 :=

inf

∞ g∈Hn−1

sq − gL∞ .

As sq is holomorphic on a neighborhood of D, it follows from the Adamjan-Arov∞ Krein theory that the inﬁmum is uniquely attained at some gn−1 ∈ Hn−1 which is holomorphic on a neighborhood of T, that |sq − gn−1 |(ξ) = σn−1 for all ξ ∈ T, and that wT (sq − gn−1 ) ≤ −2n + 1 as soon as σn−1 > 0, where wT stands for the usual winding number of a non-vanishing continuous function on T. We will appeal to a de la Vall´ee-Poussin principle for this type of approximation, to the eﬀect that (4.12)

σn−1 ≥ inf |sq − g| , T

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LAURENT BARATCHART AND MAXIM YATTSELEV

∞ whenever g ∈ Hn−1 is such that

wT (sq − g) ≤ 1 − 2n. This principle is easily deduced from the Rouch´e theorem, for if (4.12) did not hold then the inequality |(gn−1 − g) − (sq − g)| = |gn−1 − sq | = σn−1 < |sq − g| would imply that wT (gn−1 − g) = wT (sq − g) ≤ 1 − 2n, which is impossible unless gn−1 = g because gn−1 − g is meromorphic with at most 2n − 2 poles in D. Hence, with our assumptions, that q is an irreducible critical point and that f − Lq /q has no zero on T, we ﬁnd that Q will be positive deﬁnite if there exists Πq ∈ Rn−1 such that (4.13)

2|f − Lq /q| < |Πq − Lq /q|

on

T and

wT (f − Πq ) ≤ 1 − 2n.

Indeed, in this case, we will get Lq /q − Πq Lq Πq Πq = − = sq − (4.14) 2 0. Note that φ is conjugate-symmetric. Recall that the logarithmic energy of a positive Borel measure σ, compactly supported in C, is given by − log |z − t|dσ(z)dσ(t), which is a real number or +∞. Definition (Admissibility). An interpolation scheme E is called admissible if the sums e∈En |φ(e) − φ(¯ e)| are uniformly bounded with n, supp(E ) ⊂ Df , and Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

104 18

LAURENT BARATCHART AND MAXIM YATTSELEV

the probability counting measure of En converges weak∗ to some Borel measure σ with ﬁnite logarithmic energy5. The weak∗ convergence in the above deﬁnition is understood upon regarding complex measures on C as the dual space of continuous functions with compact support. To an admissible scheme E , we associate a sequence of functions on Df by putting φ(z) − φ(e) (5.2) Rn (z) = Rn (E ; z) := , z ∈ D. 1 − φ(z)φ(e) e∈En

Each Rn is holomorphic in D, has continuous boundary values from both sides of [a, b], and vanishes only at points of En . Note from the conjugate-symmetry of φ that φ(z) − φ(e) φ(z) − φ(e) φ(¯ e) − φ(e) = 1+ . 1 − φ(z)φ(e) 1 − φ(z)φ(e) φ(z) − φ(¯ e) Thus, Rn is a Blaschke product with zero set φ(En ) composed with φ, times an inﬁnite product which is boundedly convergent on any curve separating [a, b] from supp(E ) by the admissibility conditions. In particular, {Rn } converges to zero locally uniformly in D. To describe asymptotic behavior of multipoint Pad´e approximants, we need two more concepts. Let h be a Dini-continuous function on [a, b]. Then the geometric mean of h, given by Gh := exp

log h(t)dω[a,b] (t) ,

is independent of the actual choice of the branch of the logarithm [13, Sec. 3.3]. Moreover, the Szeg˝ o function of h, deﬁned as log h(t) w(z) 1 Sh (z) := exp dω[a,b] (t) − log h(t)dω[a,b] (t) , z ∈ D, 2 z−t 2 does not depend on the choice of the branch either (as long as the same branch is taken in both integrals) and is the unique non-vanishing holomorphic function in D that has continuous boundary values from each side of [a, b] and satisﬁes h = Gh Sh+ Sh− and Sh (∞) = 1. The following theorem was proved in [13, Thm. 4] when r = 0 and in [31] for the general case. Theorem 5.1. Let f be as in Theorem 2.1, E an admissible interpolation scheme, and {Πn } the sequence of diagonal Pad´e approximants to f associated with E . Then (5.3)

(f − Πn )w = [2Gµ˙ + o(1)](Sµ˙ Rn /R)2

locally uniformly in Df , where Rn is as in (5.2) and R(z) := (φ(z) − φ(e))/(1 − φ(z)φ(e)), the product deﬁning R being taken over the poles of r according to their multiplicity. 5Note that σ may not be compactly supported. In this case, pick z ∈ C \ supp(E ) such that 0

/ supp(σ) and set Mz0 (z) := 1/(z −z0 ). Then, all the sets Mz0 (En ) are contained in a common z0 ∈ compact set and their counting measures converge weak∗ to σ such that σ (B) := σ(Mz−1 0 (B)) for any Borel set B ⊂ C. What we require is then the ﬁniteness of the logarithmic energy of σ .

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105 19

ASYMPTOTIC UNIQUENESS OF BEST RATIONAL APPROXIMANTS

Let now {qn } be a sequence of irreducible critical points for Ψf,n . Put qn (z) = Π1≤j≤n (z − ξj,n ). It follows from Proposition 3.2 that Lqn /qn interpolates f at every 1/ξ¯j,n with order 2, hence Lqn /qn is the n-th diagonal Pad´e approximants associated with E{qn } := {{1/ξ¯j,n }nj=1 }. This interpolation scheme of course depends on qn , which accounts for the nonlinear character of the L2 -best rational approximation problem. The next theorem contains in its statement the Green equilibrium distribution of supp(µ) = [a, b], for the deﬁnition of which we refer the reader to [28]. Theorem 5.2. Let f be as in Theorem 2.1 and {qn } be a sequence of irreducible critical points for f. Then E{qn } is an admissible interpolation scheme, and moreover n (5.4) j=1 |Im(ξj,n )| ≤ const. where const. is independent of n. Also, the probability counting measures of the zeros of qn converges to the Green equilibrium distribution on supp(µ). In addition, it holds that (5.5)

(f − Lqn /qn )w = [2Gµ˙ + o(1)](Sµ˙ Rn /R)2

locally uniformly in Df , where Rn is as in (5.2) and R is as in Theorem 5.1. A few comments on Theorem 5.2 are in oder. First, the weak∗ convergence of the counting measures of the qn was obtained in [15, Thm. 2.1]. It entails that the probability counting measures of the sets E{qn } converge weak∗ to the reﬂection of the Green equilibrium measure across T, which has ﬁnite energy. The admissibility of E{qn } follows easily from this and from the bound (5.4) which was proven in [4], see [31, Lem. 8]. Then relation (5.5) is a consequence of (5.3). 6. Proof of Theorem 2.1 To prove Theorem 2.1, we follow the line of argument developed in [10, Thm. 1.3]. The main diﬀerence is that in the present case the critical points are no longer a priori irreducible and their poles no longer belong to the convex hull of the support of the measure. As we shall see, these diﬃculties can be resolved with the help of Theorem 5.2. Proof of Theorem 2.1. We claim there exists N = N (f) ∈ N such that all the critical points of Ψn = Ψf,n in Mn are irreducible for n > N . Indeed, assume to the contrary that there exists an inﬁnite subsequence of reducible critical point, say {qnj }. It follows from Propositions 3.2 and 3.4 that each qnj has a factor qn∗ j such that qn∗ j ∈ Mnj −knj is an irreducible critical point of Ψnj −knj , and the diﬀerence 2

∗ f−Lqn∗ /qn∗ j vanishes at the zeros of q nj dnj where dnj is a non-constant polynomial j of degree at least (kj + 1)/2 ≥ 1 having all its zeros in {|z| ≥ 1}. Suppose ﬁrst that (nj − knj ) → ∞ as j → ∞. Then, the asymptotic behavior of f − Lqn∗ /qn∗ j is j

2

∗ governed by (5.5), in particular it can only vanish at the zeros of q nj for all large n which contradicts the assumption that dnj is non-constant. Second, suppose that nj − knj remains bounded. Up to a subsequence, we may suppose that nj − knj = l for some integer l. As Ml is compact, we may assume that qn∗ j converges to some q ∈ Ml . Since Lv is a smooth function of v in some neighborhood of Ml (see Subsection 3.3), the polynomials Lqn∗ converge to Lq hence qn∗ j f − Lqn∗ converges j

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j

106 20

LAURENT BARATCHART AND MAXIM YATTSELEV

to qf − Lq locally uniformly in Df . In particular, if we pick 0 < ρ < 1 such that Dρ contains the zeros of q, we get that f − Lqn∗ /qn∗ j is a normal family of functions j

converging to f − Lq /q in |z| > ρ. But since the number of zeros it has in C \ D increases indeﬁnitely (because it vanishes at the zeros of dnj which are at least (n − l + 1)/2 in number), we conclude that f = Lq /q, which is impossible since f is not rational. This contradiction proves the claim. As we just showed, each critical point q of Ψn in Mn , is irreducible for all n large enough, in particular it belongs to Mn and moreover Lq /q does not interpolate f on T. Assume further that, for all such n, there exists a rational function Πq ∈ Rn−1 such that (4.15) holds with f = f. Then q is a local minimum by Theorem 4.1, and therefore it is the unique critical point of order n in view of the Index Theorem. Thus, to ﬁnish the proof, we need only construct some appropriate function Πq for each critical point q of Ψn , provided that n is large enough. Let E{qn } be the interpolation scheme induced by {qn } and Eν some admissible interpolation scheme, with supp(Eν ) ⊂ {|z| > 1}. Set {Πn } to be the sequence of diagonal Pad´e approximants to f associated with Eν . Then Theorems 5.1 and 5.2 imply that when n → ∞ 2 (f − Πn−1 )(z) Rn−1 (Eν ; z) (6.1) uniformly on T. = [1 + o(1)] (f − Lqn /qn )(z) Rn (E{qn } ; z) Moreover, for all n large enough, (f − Πn−1 ) is holomorphic outside of D by (5.3) and it has 2n − 1 zeros there, namely those of Rn−1 (Eν ; ·), counting multiplicities, plus one at inﬁnity. Consequently wT (f − Πn−1 ) = 1 − 2n for all such n, and so (4.15) will follow from (6.1) upon constructing Eν such that, for n large enough, 2 Rn−1 (Eν ; z) (6.2) > 2 on T. 1 − Rn (E{qn } ; z) For convenience, let us put I := [a, b] = supp(µ) and I −1 := {x : 1/x ∈ I}, together with Ω := C \ (I ∪ I −1 ). Set : Ω → {1 < |z| < A} to be the conformal map such that (I) = T, (I −1 ) = TA , limz→b+ (z) = 1; as is well known, the number A is here uniquely determined by the so-called condenser capacity of the pair (I, I −1 ) [28]. Note also that, by construction, is conjugate-symmetric. Deﬁne hν (z) =

1 − ( (z)A)2 ,

2 (z) − A2

z ∈ Ω,

which is a well-deﬁned holomorphic function in Ω := C \ (I ∪ I −1 ). It is not diﬃcult to show (cf. the proof of [10, Thm 1.3] after eq. (6.24)) that |1 − hν | > 2 on T. Thus, to prove our theorem, it is suﬃcient to ﬁnd Eν such that 2 Rn−1 (Eν ; z) = [1 + o(1)]hν (z) uniformly on T. (6.3) Rn (E{qn } ; z) For this, we shall make use of the fact, also proven in the course of [10, Thm 1.3], that hν can be represented as 1 − φ(z)φ(x) dν(x) , log hν (z) := exp φ(z) − φ(x) where ν is a signed measure of mass 2 supported on I −1 . Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

ASYMPTOTIC UNIQUENESS OF BEST RATIONAL APPROXIMANTS

107 21

Denote by {ξj,n }nj=1 the zeros of qn and by {xj,n }nj=1 their real parts. Observe from (5.5) that any neighborhood of the poles of r which is disjoint from I, contains exactly m zeros of qn for all n large enough. We enumerate these as ξn−m+1,n , . . . , ξn,n . The rest of the zeros of qn we order in such a manner that a < x1,n < x2,n < . . . < xdn ,n < b, while those j ∈ {dn + 1, . . . , n − m} for which xj,n either lies outside of (a, b) or else coincides with xk,n for some k ∈ {1, . . . , dn }, are numbered arbitrarily. Again from (5.5), any open neighborhood of I contains {ξj,n }n−m j=1 for all n large enough, and therefore (6.4)

δnim :=

max

j∈{1,...,dn }

|Im(ξj,n )| → 0 as

n → ∞.

In addition, as the probability counting measures of the zeros of qn converge to a measure supported on the whole interval I, namely the Green equilibrium distribution, we deduce that dn /n → 1 and that (6.5) δnre := max (x1,n − a), (b − xdn ,n ), max (xj,n − xj−1,n ) → 0 as n → ∞. j∈{2,...,dn }

Deﬁne νˇ to be the image of ν under the map t → 1/t, so that νˇ is a signed measure on I of mass 2. Let further ϕ(z) := φ(1/z) be the conformal map of C\I −1 onto D, normalized so that ϕ(0) = 0 and ϕ (0) > 0, Finally, set ϕ(z) − ϕ(t) . K(z, t) := log 1 − ϕ(z)ϕ(t) To deﬁne an appropriate interpolation scheme Eν , we consider the coeﬃcients: x1,n + x2,n , c1,n := νˇ a, 2 xj−1,n + xj,n xj,n + xj+1,n , , j ∈ {2, . . . , dn − 1}, cj,n := νˇ 2 2 xdn −1,n + xdn ,n ,b . cdn ,n := νˇ 2 Subsequently, we deﬁne two other sets of coeﬃcients ⎧ j ⎪ xj,n + xj+1,n ⎪ ⎪ ck,n = νˇ a, ⎨ bj,n := 2 k=1 ⎪ xj,n + xj+1,n ⎪ ⎪ ⎩ aj,n := 2 − bj,n = νˇ ,b 2

j ∈ {1, . . . , dn − 1},

and b0,n = adn ,n := 0. It follows in a straightforward manner from the deﬁnitions that 2 − cj,n = bj−1,n + aj,n , j ∈ {1, . . . , dn }, and therefore (6.6) dn dn d d n −1 n −1 2 K(z, ξj,n ) − cj,n K(z, ξj,n ) = bj,n K(z, ξj+1,n ) + aj,n K(z, ξj,n ). j=1

j=1

j=1

j=1

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Next, we introduce auxiliary points yj,n by setting yj,n := Observe that (6.7)

aj,n ξj,n + bj,n ξj+1,n , 2

j ∈ {1, . . . , dn − 1}.

bj,n ˇ ν (ξj+1,n − ξj,n ) ≤ |ξj+1,n − ξj,n |, |yj,n − ξj,n | = 2 2

where ˇ ν is the total variation of νˇ. Let K be compact in Ω and U ⊂ D be a neighborhood of I whose closure is n ⊂ U and disjoint from K. By (6.4), (6.5), and (6.7) we see that both {ξj,n }dj=1 dn −1 {yj,n }j=1 ⊂ U for all n large enough. Thus, for such n and z ∈ K, we can write the ﬁrst-order Taylor expansions: ∂ (6.8) K(z, ξj,n ) − K(z, yj,n ) = K(z, yj,n )(ξj,n − yj,n ) + O (ξj,n − yj,n )2 , ∂t ∂ K(z, yj,n )(ξj+1,n −yj,n )+O (ξj+1,n − yj,n )2 , ∂t and adding up (6.8) multiplied by aj,n to (6.9) multiplied by bj,n we obtain (6.10) bj,n K(z, ξj+1,n ) + aj,n K(z, ξj,n ) − 2K(z, yj,n ) = O (ξj+1,n − ξj,n )2 ,

(6.9) K(z, ξj+1,n )−K(z, yj,n ) =

where we took (6.7) into account and, of course, the three symbols big “Oh” used above indicate diﬀerent functions. By the smoothness of K on C \ I −1 × C \ I −1 and the compactness of K × U, these big “Oh” can be made uniform with respect to z ∈ K, being majorized by 2 ∂ K ζ → 2ˇ ν sup 2 (z, t) |ζ|2 . ∂t (z,t)∈K×U

In another connection, it is an immediate consequence of (6.4), (6.5), and (5.4) that d n −1

|ξj+1,n − ξj,n |2

≤

j=1

|xj+1,n − xj,n |2 + 2

j=1

≤

(6.11)

d n −1

(b −

dn

|Im(ξj,n )|2

j=1

a)δnre

+

const.δnim

= o(1).

Therefore, we derive from (6.11) upon adding equations (6.10) for j ∈ {1, . . . , dn −1} that dn −1 d d n −1 n −1 (6.12) bj,n K(z, ξj+1,n ) + aj,n K(z, ξj,n ) − 2 K(z, yj,n ) = o(1), j=1 j=1 j=1 where o(1) is uniform with respect to z ∈ K. In view of (6.6), equation (6.12) can be rewritten as dn d dn n −1 2 (6.13) K(z, ξ ) − 2 K(z, y ) − c K(z, ξ ) j,n j,n j,n j,n = o(1). j=1 j=1 j=1 Now, it follows from (6.5) and the deﬁnitions of cj,n and hν that dn (6.14) cj,n K(z, xj,n ) → K(z, t)dˇ ν (t) = − log |hν (1/z)| as j=1

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n → ∞,

ASYMPTOTIC UNIQUENESS OF BEST RATIONAL APPROXIMANTS

109 23

uniformly with respect to z ∈ K. Moreover, we deduce from (6.4) and (5.4) that dn

|cj,n (K(z, ξj,n ) − K(z, xj,n ))| ≤

dn

C

j=1

|cj,n ||Im(ξj,n )|

j=1

≤

(6.15)

Cˇ ν δnim → 0,

as n → ∞, where C = sup(z,t)∈K×U |∂K/∂t(z, t)|. Hence, combining (6.14) and (6.15) with (6.13), we get (6.16)

2

d n −1

K(z, yj,n ) − 2

j=1

dn

K(z, ξj,n ) → log |hν (1/z)|

as n → ∞,

j=1

uniformly on K. Deﬁne ⎛

d n −1

gn (z) := ⎝

j=1

⎞2 dn ϕ(z) − ϕ(yj,n ) ϕ(z) − ϕ(ξj,n ) ⎠ / , 1 − ϕ(z)ϕ(yj,n ) j=1 1 − ϕ(z)ϕ(ξj,n )

which is holomorphic in Ω. By (6.16), it holds that log |gn (z)| → log |hν (1/z)| as n → ∞ uniformly on K, and since the latter was arbitrary in Ω this convergence is in fact locally uniform there. Thus, {gn } is a normal family in Ω, and any limit point of this family is a unimodular multiple of hν (1/·). However, limz→b+ hν (z) = 1 while it follows immediately from the properties of ϕ that each gn has a well-deﬁned limit at 1/b which is also 1. So, {gn } is, in fact, a locally uniformly convergent sequence in Ω and its limit is hν (1/·). Finally, set Eν := {Eν,n }, where Eν,n = {ζj,n }, ζj,n := 1/yj,n+1 when j ∈ {1, . . . , dn+1 − 1}, and ζj,n := 1/ξ¯j+1,n+1 when j ∈ {dn+1 , . . . , n}. Then 2 Rn−1 (Eν ; z)/Rn (E{qn } ; z) = gn (1/z) and (6.3) follows from the limit just proved that {gn } → hν (1/·). Thus, it only remains to prove that Eν is admissible. To show the ﬁrst admissibility condition, put n−1 Xn := |φ(ζj,n−1 ) − φ(ζ¯j,n−1 )|. j=1

Then, since ˇ ν |Im(ξj,n )| + |Im(ξj+1,n )| , 2 by the very deﬁnition of yj,n , we get |Im(yj,n )| ≤

Xn

=

d n −1 j=1

≤

|ϕ(yj,n ) − ϕ(¯ yj,n )| + ⎛

2 sup |ϕ | ⎝ ⎛

|Im(yj,n )| +

j=1

ν < 2 sup |ϕ | ⎝2ˇ U

|ϕ(ξj+1,n ) − ϕ(ξ¯j+1,n )|

j=dn

d n −1

U

n−1

1 ≤ j ≤ dn − 1,

n−1

⎞ |Im(ξj+1,n )|⎠

j=dn dn j=1

|Im(ξj,n )| +

n

⎞ |Im(ξj,n )|⎠

j=dn +1

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110 24

LAURENT BARATCHART AND MAXIM YATTSELEV

which is uniformly bounded by (5.4). Further, since each Eν,n is contained in U −1 , we have that supp(Eν ) ⊂ {|z| > 1}. So, it only remains to show that the probability counting measures of Eν,n converges weak∗ to some Borel measure with ﬁnite logarithmic energy. Now, since dn /n → 1 as n → ∞, and by the remark made in footnote 3, it is enough to prove that this property holds for the probability counting measures of the points {yj,n }. But from (6.7) and (6.11), the latter have the same asymptotic distribution as the points {ξj,n }, namely the Green equilibrium distribution on I by Theorem 5.2. This ﬁnishes the proof of Theorem 2.1. References 1. L. Baratchart, Sur l’approximation rationelle L2 pour les syst` emes dynamiques lin´ eaires, Ph.D. thesis, Universit´ e de Nice, 1987. , Rational and meromorphic approximation in Lp of the circle: system-theoretic moti2. vations, critical points and error rates, Computational Methods and Function Theory (World Scientiﬁc Publish. Co, River Edge, N.J.) (N. Papamichael, St. Ruscheweyh, and E. B. Saﬀ, eds.), Approximations and Decompositions, vol. 11, 1999, pp. 45–78. , A remark on uniqueness of best rational approximants of degree 1 in L2 on the circle, 3. Electron. Trans. Numer. Anal. 25 (2006), 54–66. 4. L. Baratchart, R. K¨ ustner, and V. Totik, Zero distribution via orthogonality, Ann. Inst. Fourier 55 (2005), no. 5, 1455–1499. 5. L. Baratchart and M. Olivi, Index of critical points in l2 -approximation, Systems Control Lett. 10 (1988), 167–174. 6. L. Baratchart, M. Olivi, and F. Wielonsky, Asymptotic properties in rational l2 approximation, Lecture Notes in Control and Inform. Sci. 144 (1990), 477–486. , On a rational approximation problem in real Hardy space H2 , Theoret. Comput. Sci. 7. 94 (1992), 175–197. 8. L. Baratchart, E.B. Saﬀ, and F. Wielonsky, A criterion for uniqueness of a critical points in ematique 70 (1996), 225–266. H 2 rational approximation, J. Analyse Math´ 9. L. Baratchart and F. Seyfert, An Lp analog of AAK theory for p ≥ 2, J. Funct. Anal. 191 (2002), no. 1, 52–122. 10. L. Baratchart, H. Stahl, and F. Wielonsky, Non-uniqueness of rational approximants, J. Comput. Appl. Math. 105 (1999), 141–154. , Asymptotic uniqueness of best rational approximants of given degree to Markov func11. tions in L2 of the circle, Constr. Approx. 17 (2001), 103–138. 12. L. Baratchart and F. Wielonsky, Rational approximation in real Hardy space H2 and Stieltjes integrals: A uniqueness theorem, Constr. Approx. 9 (1993), 1–21. 13. L. Baratchart and M. Yattselev, Convergent interpolation to Cauchy integrals over analytic arcs, To appear in Found. Comput. Math., http://www.springerlink.com/content/a7j287827k164v2x/. , Convergent interpolation to Cauchy integrals over analytic arcs of Jacobi-type 14. weights, In preparation. , Meromorphic approximants to complex Cauchy transforms with polar singularities, 15. Mat. Sb. 200 (2009), no. 9, 3–40. 16. D. Braess, Nonlinear approximation theory, Computational Mathematics, vol. 7, SpringerVerlag, Berlin, 1986. 17. Jean Della Dora, Contribution ` a l’approximation de fonctions de la variable complexe au sens de Hermite–Pad´ e et de hardy, Th` ese d’´ etat, Univ. Scient. et Medicale de Grenoble, 1980. 18. J.L. Doob, Stochastic processes, John Wiley, 1953. 19. J.C. Doyle, B.A. Francis, and A.R. Tannenbaum, Feedback control theory, Macmillan Publishing Company, 1992. 20. M. Duc-Jacquet, Approximation des fonctionelles lin´ eaires sur les espaces Hilbertiens ` a noyaux reproduisants, Th` ese d’´ etat, Univ. Scient. et Medicale de Grenoble, 1973. 21. P. Duren, Theory of H p spaces, Dover Publications, Inc., New York, 2000. 22. V.D. Erohin, On the best approximation of analytic functions by rational functions with free poles, Dokl. Akad.Nauk SSSR 128 (1959), 29–32, in Russian. 23. V. Guillemin and A. Pollack, Diﬀerential topology, Englewood Cliﬀs, N.J., Prentice-Hall, 1974.

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ASYMPTOTIC UNIQUENESS OF BEST RATIONAL APPROXIMANTS

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24. E.J. Hannan and M. Deistler, The statistical theory of linear systems, Wiley, New York, 1988. 25. K. Hoﬀman, Banach spaces of analytic functions, Dover, 1988. 26. A.L. Levin, The distribution of poles of rational functions of best approximation and related questions, Math. USSR Sbornik 9 (1969), no. 2, 267–274. 27. V.V. Peller, Hankel operators and their applications, Springer Monographs in Mathematics, Springer-Verlag, New York, 2003. 28. E.B. Saﬀ and V. Totik, Logarithmic potentials with external ﬁelds, Grundlehren der Math. Wissenschaften, vol. 316, Springer-Verlag, Berlin, 1997. 29. A.N. Shiryaev, Probability, Springer, 1984. 30. J.L. Walsh, Interpolation and approximation by rational functions in the complex domain, A.M.S. Publications, 1962. 31. M. Yattselev, On uniform approximation of rational pertubations of Cauchy integrals, Accepted for publication in Comput. Methods Funct. Theory, http://arxiv.org/abs/0906.0793. INRIA, Project APICS, 2004 route des Lucioles — BP 93, 06902 Sophia-Antipolis, France E-mail address: [email protected] INRIA, Project APICS, 2004 route des Lucioles — BP 93, 06902 Sophia-Antipolis, France E-mail address: [email protected]

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http://dx.doi.org/10.1090/conm/507/09956

Contemporary Mathematics Volume 507, 2010

Quadrature rules on the unit circle. A survey. L. Garza and F. Marcellán Dedicated to Guillermo López Lagomasino on the occasion of his sixtieth birthday Abstract. In this paper a survey about two kinds of quadrature rules with respect to a nontrivial probability measure supported on the unit circle is presented. The connection with the standard like Gaussian quadrature rules with respect to measures supported on the real line is analyzed. Some examples related to Bernstein-Szeg˝o measures on the unit circle and their counterpart on the real line using the Szeg˝o transformation are shown.

1. Introduction. Orthonormal polynomials on the unit circle T = {z ∈ C : |z| = 1} are defined by ϕn (z)ϕm (z)dµ = δm,n , (1.1) ϕn , ϕm = T

m, n ∈ {0, 1, 2, . . .}, where µ is a nontrivial probability measure, i.e. its support is an infinite subset of T and ϕn (z) = κn zn + lower degree terms, with κn > 0. {Φn }n0 , the sequence of monic polynomials orthogonal with respect to the measure µ, is defined by Φn (z) = ϕn (z)/κn , for every n ∈ {0, 1, 2, . . .}. We will denote by Λ = span{zk }k∈Z the linear space of Laurent polynomials with complex coeﬃcients and by Λm,n = { nk=m ck zk ; m n, ck ∈ C} the linear subspace of Λ generated by {zk }nk=m . Furthermore, Pn = Λ0,n and P = ∞ n=0 Pn . The sequence of monic orthogonal polynomials {Φn }n0 satisfies the so-called backward (resp. forward) recurrence relations Φn+1 (z) = Φn+1 (z) =

(1 − |Φn+1 (0)|2 )zΦn (z) + Φn+1 (0)Φ∗n+1 (z), zΦn (z) + Φn+1 (0)Φ∗n (z), n 0,

n 0,

2010 Mathematics Subject Classification. Primary 33C47, 42C05; Secondary 41A55. Key words and phrases. Orthogonal polynomials, para-orthogonal polynomials, Carathéodory functions, Szeg˝o quadrature rules, Gaussian quadrature rules, Szeg˝o transformation. The work of the first author has been supported by a grant of Universidad Autónoma de Tamaulipas. The work of the second author has been partially supported by Dirección General de Investigación, Ministerio de Educación y Ciencia of Spain, grant MTM06-13000-C03-02. Both authors have been supported by project CCG07-UC3M/ESP-3339 with the financial support of Comunidad de Madrid/Universidad Carlos III de Madrid. 1

c 2010 American Mathematical Society

113 Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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where Φ∗n (z) = zn Φn (¯z−1 ) is the reversed polynomial associated with Φn (see [14], [17], [18], [28], and [29] among others). The complex numbers Φn (0) are called Verblunsky parameters. Notice that in this case, |Φn (0)| < 1, n 1. On the other hand, the n-th reproducing kernel, Kn (z, y), associated with µ is usually defined as n (1.2) Kn (z, y) = ϕ j (z)ϕ j (y). j=0

Notice that (1.3)

Kn (z, y), P(z) = P(y),

for every

P ∈ Pn .

From the Christoﬀel-Darboux formula (see [17], [18], [28], [29]) we get Kn (z, y) =

ϕ∗n+1 (z)ϕ∗n+1 (y) − ϕn+1 (z)ϕn+1 (y) , 1 − z¯y

or, equivalently, Kn (z, y) =

(1.4)

ϕ∗n (z)ϕ∗n (y) − z¯yϕn (z)ϕn (y) . 1 − z¯y

The parallelism that exists between orthogonal polynomials on the real line and on the unit circle has been extensively elaborated for years. Indeed, analytic results for polynomials orthogonal with respect to a measure supported on a bounded interval of the real line can be deduced using techniques of classical theory of functions for orthogonal polynomials on the unit circle. The works by G. Szeg˝o ([29]), Ya. L. Geronimus ([17], [18]), G. Freud ([14]), and the most recent monograph by B. Simon ([28]) constitute a good sample of it. Consider the multiplication operator with respect to the family of orthonormal polynomials. For measures supported on the real line, the matrix representation of such an operator is a tridiagonal symmetric matrix called Jacobi matrix (see [8], [16]). The eigenvalues of such a matrix turn out to be the zeros of the corresponding orthogonal polynomials. On the other hand, for measures supported on the unit circle we have zϕ(z) = Hϕ ϕ(z), where ϕ(z) = ϕ0 (z), ϕ1 (z), . . . , ϕn (z), . . . t and Hϕ = (hn, j )∞ n, j=0 is a lower Hessenberg matrix with entries (1.5)

⎧ κn ⎪ ⎪ ⎪ ⎪ ⎨ κn+1κ j =⎪ − κn Φn+1 (0)Φ j (0) ⎪ ⎪ ⎪ ⎩ 0

j = n + 1, j n, j > n + 1, n = 1 − |Φn+1 (0)|2 , all entries of Hϕ where κn is the leading coeﬃcient of ϕn (z). Since κκn+1 can be expressed in terms of the Verblunsky parameters. Moreover, (1.6)

hn, j

if if if

Proposition 1.1. [28] The infinite matrix Hϕ satisfies (i) Hϕ H∗ϕ = I, (ii) H∗ϕ Hϕ = I − λ∞ (0)ϕ(0)ϕ(0)∗ , where I is the identity matrix and λ∞ = limn→∞ 1/Kn (0, 0). Furthermore, the zeros of ϕn (z) are the eigenvalues of H(n) ϕ , the n × n principal leading submatrix of Hϕ . Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

QUADRATURE RULES ON THE UNIT CIRCLE. A SURVEY.

1153

However, in some instances such a similarity is not that apparent. For example, the zeros of orthogonal polynomials on the real line are all of them real, simple, and located in the interior of the convex hull of the support of the measure. On the other hand, the zeros of orthogonal polynomials with respect to measures supported on the unit circle lie inside the open unit disk D = {z ∈ C : |z| < 1}. For this reason, in the real line we can approach an integral f dν R using the interpolatory quadrature rules In ( f ) = ni=1 ωn,i f (xn,i ). Here ν is a nontrivial probability measure supported on the real line. It is very well known that if {xn,i }ni=1 are the zeros of the n-th orthogonal polynomial with respect to ν and {ωn,i }ni=1 are the Christoﬀel constants associated with these zeros, then f dν In ( f ) = R

for every f ∈ P2n−1 , the linear space of polynomials with real coeﬃcients of degree at most 2n − 1, and this is the optimal degree of exactness. Moreover, under some restrictions on ν, we get the convergence of the quadrature rule, i.e., for every continuous function on the support of the measure In ( f ) → f dν. R

From a functional theoretical point of view this means that the sequence of discrete measures dνn = ni=1 ωn,i δ(x − xn,i ) converges in the ∗-weak sense to the measure dν (see [12], [14], [16], [24]). Taking into account that the zeros of orthogonal polynomials on the unit circle can be located far away from the support of the measure, we can not introduce the analog approach to the explained above on the real line (the so-called Gaussian quadrature rules). There are at least two ways to address this issue. The first one is based on the approach to the measure by the family of discrete measures supported on the unit circle, i.e. an interpolatory quadrature rule based on zeros of some polynomials associated with {Φn }n0 . They constitute the so-called Szeg˝o quadrature rules which have been extensively studied in the literature in the recent years (see, for instance [5], [7], [9], [19], [20], [21], [23], among others). The second approach is based on the zeros of orthogonal polynomials themselves. In such a situation, we obtain a family of absolutely continuous measures that converge to the orthogonality measure. Surprisingly, little attention was paid to this case, the so-called Gaussian quadrature rules (see, for instance [9], [21] and the references therein). Notice that they are the basic tool to give a Favard’s theorem on the unit circle ([13]). The goal of our work is to present an updated survey about recent progress on quadrature rules on the unit circle. In Section 2, we will focus our attention on Szeg˝o quadrature rules. The background about para-orthogonal polynomials will be presented. Section 3 deals with Gaussian quadrature formulas and their connection with two-point Padé approximants to Carathéodory functions. Section 4 analyzes quadrature rules for Bernstein-Szeg˝o measures and the degree of its exactness in the linear space P of polynomials with complex coeﬃcients. Finally, in Section 5 we establish the connection between quadrature rules on R (mainly Gaussian and Gaussian-like rules) and Szeg˝o quadrature rules on T based on the Szeg˝o transformation. Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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2. Szeg˝o quadrature on T. 2.1. Para-orthogonal polynomials. Consider the function gn (z) =

zϕn (z) , ϕ∗n (z)

which is a Blaschke product of degree n+1, with all zeros in D. By the argument principle, gn (z) goes around the unit circle exactly n + 1 times when z wraps around the origin. Furthermore, arg gn is a strictly increasing and continuous function. Thus, for each β ∈ T, the equation gn (z) = β has exactly n + 1 diﬀerent solutions on T. Thus, for a given ω ∈ T we will consider the equation gn (z) = gn (ω). We denote its solutions by {ξn, j }nj=0 ordered in such a way that ξn,0 = ω and arg ξn, j < arg ξn, j+1 , j = 0, 1, . . . , n. Let us write zn = zn (ω) = {ξn,0 (ω), ξn,1 (ω), . . . , ξn,n (ω)}. From (1.4),

ϕ∗ (z)ϕ∗n (ω) gn (z) 1− . Kn (z, ω) = n 1 − zω ¯ gn (ω) Kn (z, ω) is a polynomial of degree n in the variable z, and its zeros are exactly {ξn, j }nj=1 . Moreover, Kn (ξn,0 , ω) = Kn (ω, ω) > 0. Taking into account Kn (z, ω) = Kn (ω, z) we get zn (ξn,p ) = zn (ξn,q ) and, as a consequence, for p q Kn (ξn,p , ξn,q ) = 0.

(2.1)

Notice that according to the reproducing property (1.3), (2.1) means that the polynomials {Ln, j }nj=0 where Ln, j (z) =

Kn (z, ξn, j ) Kn (ξn, j , ξn, j )

,

j = 0, 1, . . . , n,

constitute an orthonormal basis in Pn . Thus, if P ∈ Pn then from n P(z) = pk Ln,k (z), k=0

with

P(ξn,k ) pk = P(z), Ln,k (z) = , Kn (ξn,k , ξn,k ) we have, as a consequence, (2.2)

P, Q =

n P(ξn,k )Q(ξn,k ) , Kn (ξn,k , ξn,k ) k=0

for every P, Q ∈ Pn . Notice that the right hand side of (2.2) represents an inner product on Pn with respect to a discrete measure n dµn = µn,k δ(z − ξn,k ) k=0

where µn,k = 1/Kn (ξn,k , ξn,k ), k = 0, 1, . . . , n. The polynomials Bn+1 (z, ω) = (1 − zω)K ¯ n (z, ω) = ϕ∗n+1 (z)ϕ∗n+1 (ω) − ϕn+1 (z)ϕn+1 (ω), n 0, Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

QUADRATURE RULES ON THE UNIT CIRCLE. A SURVEY.

1175

are known in the literature as para-orthogonal polynomials associated with the measure µ. (2.2) is known as the Szeg˝o quadrature rule since n p(z)dµ = p(ξn,k )µn,k T

k=0

for every p ∈ Λ−n,n or equivalently T

q(θ)dµ =

n

q(θn,k )µn,k

k=0

for every trigonometric polynomial q of degree at most n. Here ξn,k = cos θn,k + i sin θn,k , k = 0, 1, . . . , n. Remark 2.1. Notice that Bn+1 (z, ω) = =

2 κn+1 (Φ∗n+1 (z)Φ∗n+1 (ω) − Φn+1 (z)Φn+1 (ω)), Φ∗n+1 (ω) ∗ 2 Φ (z) . −κn+1 Φn+1 (ω) Φn+1 (z) − Φn+1 (ω) n+1

Taking into account that for |ω| = 1, |Φ∗n+1 (ω)/Φn+1 (ω)| = 1, if we denote Bˆ n+1 (z, τn+1 ) = Φn+1 (z)+τn+1 Φ∗n+1 (z) with τn+1 = Φ∗n+1 (ω)/Φn+1 (ω), then we recover the standard definition of para-orthogonal polynomials { Bˆ n+1 (z; τn+1 )} (see for instance [6], [?], and [23]). Remark 2.2. Taking into account that dimΛ−n,n = 2n + 1 = 2(n + 1) − 1, where n + 1 is the number of nodes, and that cannot exist an (n + 1)-point Szeg˝o quadrature rule valid on Λ−(n+1),n , the degree of exactness of (2.2) is 2n + 1 in the sense that it is valid on Λ−n,n at most. Remark 2.3. The Szeg˝o quadrature rules are strongly related to the truncated trigonometric moment problem (see [1], [22], [23]). Given a sequence of complex numbers {cn }n∈Z with c0 = 1 and c−n = c¯ n , the trigonometric moment problem asks when it can be represented as cn =

z−n dµ,

T

n ∈ Z,

where µ is some probability measure supported on T. The truncated moment problem asks N the same question but for a finite section of moments {cn }−N . A necessary and suﬃcient condition for both relations to hold is c j−k x j x¯k > 0 j,k

for any choice of a finite vector (x1 , x2 , . . . , x M ) (0, 0, . . . , 0), (M = N in the truncated case and every M ∈ N in the general case). The trigonometric moment problem has one solution while the truncated moment problem has infinitely many ones as the choices of nodes for para-orthogonal polynomials of degree n show. 2.2. Zeros of para-orthogonal polynomials. It is very well known (see [8], [14], [16], [29]) that given a probability measure ν on the real line, if {xn,k }nk=1 are the zeros of the n-th orthogonal polynomial with respect to ν then they interlace with the zeros of the (n − 1)-th orthogonal polynomial with respect to ν, i.e. xn,k < xn−1,k < xn,k+1 ,

k = 0, 1, . . . , n − 1.

This property plays a key role in the theory of Gaussian quadrature rules. The zeros of para-orthogonal polynomials satisfy a similar property, as follows. Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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n Theorem 2.4. [19], [23] The sets A = {ξn−1, j (ω)}n−1 j=0 and B = {ξn, j (ω)} j=1 alternate. This is, between two consecutive points of A there is exactly one point of B.

There is also an analog of the separation property of the zeros on the real line case for the unit circle. Indeed Theorem 2.5. [19] Let Γ = (α, β) ⊂ T be an arc that is a gap in supp(µ), i.e. supp(µ) Γ = ∅. Then, for each n, the polynomial Bn has at most one zero in Γ = [α, β]. Recent results concerning the zeros of paraorthogonal polynomials appear in [7] and [30]. On the other hand, the capability of the Szeg˝o quadrature rule can be illustrated by the following result providing the bounds for the distance between consecutive zeros of the para-orthogonal polynomial Bn+1 . dθ Theorem 2.6. [19] Let dµ = µ 2π + dµ s be the Lebesgue decomposition of µ. (i) If µ belongs to the Szeg˝o class, i.e. ln µ ∈ L1 (T), then

|ξn,k+1 − ξn,k | C(µ)n−1/2 for k = 0, 1, . . . , n and the convention ξn,n+1 = ξn,0 . dθ , (ii) If µ is absolutely continuous on T with a.e bounded derivative, i.e. dµ = µ 2π 0 < A µ B, then 4 A 1/2 4πB |ξn,k+1 − ξn,k | < n B (n + 1)A for k = 0, 1, . . . , n and the convention ξn,n+1 = ξn,0 .

Definition 2.7. A sequence {ωn, j }nj=0 , n = 0, 1, . . ., is said to be uniformly distributed on T if 2π n 1 dθ lim f (ωn,k ) = f (eiθ ) n→∞ n + 1 2π 0 k=0 for every f ∈ C(T). Theorem 2.8. [19] The sequence {ξn, j }nj=0 , n = 0, 1, . . ., of zeros of para-orthogonal polynomials Bn+1 (·, ω), with ω a fixed point of the unit circle, is uniformly distributed on T if and only if 1 ∗ dθ Kn (eiθ , eiθ )dµ → . n+1 2π Notice that once we have some ∗-weak convergence related to the orthogonality measure we can derive certain type of distribution of zeros of para-orthogonal polynomials and viceversa. Finally, consider the n × n leading principal submatrix of Hϕ . Replacing Φn (0) by β ∈ (n) , as the result of the modification of C, with |β| = 1, we get a new Hessenberg matrix H (n) Hϕ on the last row. This new matrix turns out to be unitary, and it can be shown that their eigenvalues are the zeros of the para-orthogonal polynomial Ψn (z) = Φn (z)+τn Φ∗n (z), where τn = [β − Φn (0)]/[1 − βΦn (0)]. In other words, the zeros of para-orthogonal polynomials are the eigenvalues of certain unitary Hessenberg matrices. Moreover, given some perturbations to a nontrivial probability measure supported on the unit circle (namely the Christoﬀel, Uvarov and Geronimus transformations, among others) the eﬀect of these perturbations on the corresponding Hessenberg matrices has been Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

QUADRATURE RULES ON THE UNIT CIRCLE. A SURVEY.

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studied in [11], [15], and [25], among others. Basically, one can obtain the Hessenberg matrix associated to the perturbed measure by means of the QR factorization of the Hessenberg matrix associated to the initial measure. The eﬀect of such perturbations related to the corresponding para-orthogonal polynomials and their zeros constitutes an open problem. 3. Gaussian quadrature on T. We consider in this section a quadrature rule of the form In ( f ) =

(3.1)

s α m −1

Am, j f ( j) (ξn,m ),

m=1 j=0

s

αm = n,

m=1

which is exact in Λ−p,q , with p and q depending on n, p q and they are large enough. αm denotes the multiplicity of the nodes ξn,m . If the nodes are chosen to be the zeros of the para-orthogonal polynomial Bn (z, τ) then (3.1) becomes the Szeg˝o quadrature formula. As we have shown in Section 2, the weights An, j are all positive real numbers. It is possible to obtain a quadrature formula of the form (3.1) which is exact in Λ−(n−1),n (or similarity, in Λ−n,n−1 ), using as nodes the zeros of the sequence of monic polynomials {Φn }n0 orthogonal with respect to µ, which lie in D. For this reason, since f (and its derivatives) has to be evaluated in points in D, f is required to be analytic in D. It is well known that (see [14], [17], [18], [22], [28]) 2π 1 kn 2π P(z) (3.2) P(z)dµ(θ) = dθ, z = eiθ , 2π 0 2π 0 |Φn (z)|2 for all P ∈ Pn , where kn = Φn 2 . Thus, let us define kn 2π f (z) (3.3) In ( f ) = dθ, z = eiθ , 2π 0 |Φn (z)|2 as well as 1 Iµ ( f ) = 2π

(3.4)

2π

f (z)dµ(θ),

z = eiθ ,

f ∈Λ

f ∈ Λ.

0

∗

Notice that In → Iµ in the linear space of continuous functions on T. Consider the n-th monic polynomial Φn orthogonal with respect to µ and denote by α the multiplicity of z = 0 if it is a zero of Φn , and by αm , m = 1, . . . , s, the multiplicities of its remaining zeros. s s Thus, we can write Φn (z) = zα m=1 (z − ξn,m )αm , with m=1 αm = n − α. If α = n, then Φn (z) = zn and Iµ (z j ) = Iµ (z− j ) = 0,

1 j n.

For j = 0, we get Iµ (1) = In (1) = 1 and, as a consequence, if L ∈ Λ−n,n , namely L(z) = n j j=−n β j z , then Iµ (L) = In (L) = β0 . In the sequel, we will exclude this case, i.e, we will assume ˜ n−α (z), Φn (z) = zα Φ

(3.5)

0 α < n,

where (3.6)

˜ n−α (z) = Φ

s m=1

(z − ξm )αm ,

0 < |ξm | < 1,

s

αm = n − α.

m=1

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Proposition 3.1. Let α be a nonnegative integer such that 0 α < n and let In ( f ) =

s α m −1

ξm 0,

Am, j f ( j) (ξm ),

m = 1, . . . , s,

m=1 j=0

be an exact quadrature rule in Λ−(k−α),2n−k−1 , 0 k 2n. Then, the polynomial rn (z) = zα

s

(z − ξm )αm

m=1

satisfies the following orthogonality properties with respect to µ rn (z), z p µ = 0,

k − n + 1 p k,

0 k 2n.

Notice that, in particular, for k = n − 1, In ( f ) is exact in Λ−(n−α−1),n , and then rn (z) ≡ Φn (z). On the other hand, if k = n, then In ( f ) is exact in Λ−(n−α),n−1 and rn (z) ≡ Φ∗n (z). Now, to compute the weights Am, j of In ( f ), we will assume f is analytic in D and ¯ We have continuous in D. Φn (z)Φ∗n (z) , z = eiθ . |Φn (z)|2 = Φn (z)Φn (z) = zn Thus, applying the Residue Theorem, f (z)zn−α−1 kn kn 2π f (z)zn dθ = In ( f ) = dz, ˜ n−α (z)Φ∗n (z) 2π 0 Φn (z)Φ∗n (z) 2πi T Φ

s s 1 dαm −1 (z − ξm )αm zn−α−1 lim f (z) , Res(z = ξm ) = kn = kn ˜ n−α (z)Φ∗n (z) (αm − 1)! z→ξm dzαm −1 Φ m=1 m=1

s α m −1 dαm −1− j (z − ξm )αm zn−α−1 kn lim α −1− j f ( j) (ξm ), = ˜ n−α (z)Φ∗n (z) z→ξm dz m (α − 1 − j)! j! Φ m m=1 j=0 and, therefore,

(3.7)

Am, j =

dαm −1− j (z − ξm )αm zn−α−1 kn lim α −1− j . ˜ n−α (z)Φ∗n (z) (αm − 1 − j)! j! z→ξm dz m Φ

In the particular case when all zeros are simple and α = 0, we obtain Am, j =

(3.8)

kn ξ n−1 j Φ n (ξ j )Φ∗n (ξ j )

On the other hand, if f is analytic in C that the weights are given by (3.9)

Am, j =

,

j = 1 . . . , n.

¯ and continuous in C D

D, one can show

dαm −1− j (z − ξ¯m−1 )αm zn−α−1 kn lim α −1− j . ˜ n−α (z)Φ∗n (z) (αm − 1 − j)! j! z→ξ¯m−1 dz m Φ

Next, necessary and suﬃcient conditions for the exactness of In ( f ) are given. Proposition 3.2. In ( f ) is exact in Λ−(n−α−1),n if and only if the weights are given as in (3.9) and the nodes are the nonvanishing zeros of Φn (z), the monic polynomial orthogonal ˜ n−α (0) 0. ˜ n−α (z), Φ with respect to µ, where Φn (z) = zα Φ

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1219

Proof. ⇒ If In ( f ) is exact in Λ−(n−α−1),n , we have proved previously that the weights are given by (3.9). From Proposition (3.1), for k = n − 1, the nodes ξm , m = 1, . . . , s are the nonvanishing zeros of Φn (z). ⇐ From (3.2), for −n p n we get 2π 1 kn 2π z p p z dµ(θ) = dθ, z = eiθ . 2π 0 2π 0 |Φn (z)|2 Let 1 p n − α − 1. Then, 2π 1 kn 2π z−p −p −p z dµ(θ) = dθ, Iµ (z ) = 2π 0 2π 0 |Φn (z)|2 zn−p−α kn = dz, ˜ n−α (z)Φ∗n (z) 2πi T zΦ s Res(z = ξm ) = kn Res(z = 0) + In (z−p ) = In (z−p ). = kn Res(z = 0) + kn m=1

We obtain the same result for n − α p n. Now let q = p − n + α + 1. Then, 1 q α and

1 dq−1 1 Res(z = 0) = lim q−1 0. ˜ n−α (z)Φ∗n (z) (q − 1)! z→0 dz Φ There exists an alternative expression for the computation of the weights Am, j , as follows ⎛ ⎞ α k −1 2π ⎟⎟⎟ 1 ⎜⎜⎜⎜ 1 k iθ α −1 k ⎟⎟⎠⎟ , Ak, j = α − j ⎜⎝⎜ L (e )dµ(θ) − M A k,r r j 2π 0 Mj k r= j+1 ∀ j = 0, . . . , αk − 1, ∀k = 1, . . . , m, where q Mj

=

j (q+ j) 1 Φn (ξk ) j (−1) j−l , (q − 1)! l=0 l q+ j−l

Llp (z)

=

Φn (z) , (z − ξ p )α p −1

and the integral 1 2π

2π 0

Lkj (eiθ )dµ(θ)

l = 0, . . . , α p − 1, p = 1, . . . , m,

dα p −l−1 zn−1 kn lim . = (α p − l − 1)! z→ξ p dzα p −l−1 Φ∗n (z)

3.1. Relation with two-point Padé approximants. Next, we will show a connection between Gaussian quadrature rules and two-point Padé approximants to Caratheodory functions defined by the Herglotz-Riesz transform. We follow an analogue approach to the stated for Gaussian quadrature rules for measures supported on the real line.The results of this section and the next one appear in [9]. Since {z j }qj=−p , p + q = n − α − 1, is a Chebyshev system on every set A ⊂ C such ˜ n−α (z), the weights Am, j of In ( f ) can be uniquely that 0 A, and z = 0 is not a zero of Φ determined so In ( f ) = Iµ ( f ), for all f ∈ Λ−p,q . Besides, it can be shown that there exists a unique Laurent polynomial Ln (z) ∈ Λ−p,q such that L(nj) (ξm ) = f ( j) (ξm ),

j = 0, . . . , αm − 1, m = 1, . . . , s.

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In addition, we can write Lm (z) = L˜ (l) m, j (ξm ) = δm,k δ j,l ,

s m=1

αm −1 ˜ ( j) ˜ j=0 Lm, j (z) f (ξm ), where Lm, j ∈ Λ−p,q and

l, j = 0, . . . , αm − 1, m, k = 1, . . . , s,

and δ j,k is the well known Kronecker’s delta. It can be shown that Iµ (Ln (z)) =

s α m −1

Iµ (L˜ m, j (z)) f ( j) (ξm ) = In ( f ),

m=1 j=0

where Iµ (L˜ m, j (z)) = Am, j . For this reason, In ( f ) is said to be an interpolatory type quadrature formula in Λ−p,q . There is a connection between the interpolatory quadrature rule and the Gaussian quadrature rule, that we will show in the sequel. Proposition 3.3. [9] Let α be an nonnegative integer number such that 0 α < n. Then the quadrature rule In ( f ) =

s α m −1

Am, j f ( j) (ξm ),

ξm 0, m = 1, . . . , s,

m=1 j=0

is exact in Λ−(n−1−α),n if and only if (i) In ( f ) is of interpolatory type in Λ−p,q where p and q are arbitrary nonnegative integers such that p + q = n − α − 1. s s (ii) The nodes {ξm }m=1 with multiplicity {αm }m=1 , ishing zeros of Φn (z).

s m=1

αm = n − α, are the nonvan-

Now, consider the Herglotz-Riesz transform of µ, given by 2π ξ+z 1 dµ, ξ = eiθ , z T. F(z) = 2π 0 ξ − z It is well known that the following series expansions around 0 and ∞ hold F0 (z) =

c0 + 2

∞

c jz j,

|z| < 1,

j=1

F∞ (z) =

−c0 − 2

∞

c− j z − j ,

|z| > 1,

j=1

2π 1 e−ikθ dµ(θ), k ∈ Z and where ck is the k-th moment associated with µ defined by ck = 2π 0 we assume c0 = 1. On the other hand, if Ωn (z) is the associated polynomial of second kind for Φn (z), i.e. 2π ξ+z 1 Ωn (z) = (Φn (ξ) − Φn (z))dµ, 2π 0 ξ − z the rational function Ωn (z)/Φn (z) satisfies (see [2], [17], [26]) Φn (z)F0 (z) + Ωn (z) = Φn (z)F∞ (z) + Ωn (z) =

O(zn ), |z| < 1, O(z−1 ), |z| > 1.

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123 11

This is, −Ωn (z)/Φn (z) represents the weak (n, n) two-point Padé approximant of order (n, n + 1) for the pair (F0 , F∞ ). Moreover, if Φn (0) 0, i.e. α = 0, then Ωn (z) Φn (z) Ωn (z) F∞ (z) + Φn (z) F0 (z) +

|z| < 1,

=

O(zn ),

=

O(z−(n+1) ) |z| > 1.

In such a case, −Ωn (z)/Φn (z) represents the (n, n) two-point Padé approximant of order (n, n + 1) for the pair (F0 , F∞ ). Now, if we apply the quadrature rule In ( f ) to h(ξ, z) =

ξ+z , ξ−z

|z| > 1,

we get In (h(•, z)) =

Q˜ n (z) , ˜ n (z) Φ

where Q˜ n (z) is a polynomial of degree at most n − α. Furthermore, one can prove Proposition 3.4. [9] The rational function In (h(•, z)) =

Ωn (z) Q˜ n (z) zα Q˜ n (z) = α =− ˜ ˜ Φn (z) Φn (z) z Φn (z)

represents the weak (n, n) two-point Padé approximant of order (n, n + 1) for the pair (F0 , F∞ ). In the particular case Φn (0) 0, then such a rational function is the (n, n) two-point Padé approximant of order (n, n + 1) for the pair (F0 , F∞ ) in the strong sense. 3.2. Error estimates and convergence of the Gaussian quadrature. A second step in our analysis of the Gaussian quadrature rules deals with the analysis of the error estimates and, as a consequence, with the study of its convergence. Theorem 3.5. [9] Let f (z) be an analytic function in a bounded domain G containing ¯ and let Γ be its boundary. Then, D

Ωn (z) 1 Iµ ( f ) − In ( f ) = g(z)dz, F(z) + 2πi Γ Φn (z) where g(z) = − f (z)/2z. In other words, In ( f ) converges uniformly to Iµ ( f ), since −[Ωn (z)/Φn (z)] converges ¯ There is another result concernuniformly to F(z) for |z| > 1, and Γ is contained in C D. ing the convergence rate of the quadrature formula, as follows ¯ and let Γ Theorem 3.6. [9] Let f (z) be analytic in a bounded domain G containing D be its boundary. If p(n) + α(n) lim = s, 0 < s < 1, n→∞ n then 1 1/n < 1, lim |Iµ ( f ) − In ( f )| = r(Γ, s) = max n→∞ t∈Γ |t|1−s where α(n) denotes the multiplicity of z = 0 in Φn (z) and {p(n)} is an arbitrary sequence of nonnegative integers such that 0 p(n) n − α(n) − 1. Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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L. GARZA AND F. MARCELLÁN

The same result can be proved eliminating the condition limn→∞ p(n)+α(n) = s, 0 < n s < 1. Let Hn be the unique Hermite-Laurent interpolatory polynomial in Λ−(n−2α−1),n such that Hn( j) (ξm )) = f ( j) (ξm ),

j = 0, . . . , 2αm − 1; m = 1, . . . , s.

It can be proved that In ( f ) = Iµ (Hn ), and then ¯ and let Proposition 3.7. [9] Let f (z) be analytic in a bounded domain G containing D Γ be its boundary. Then, 1 1/n lim | f (z) − Hn (z)| max 0 there exists a Bernstein-Szeg˝o measure |dz| dµ(z) = 2π|Q(z)| 2 such that ! !! !!w(z) − 1 !!! < , for every z ∈ T. ! |Q(z)|2 ! Therefore, we can obtain quadrature rules for this class of measures based on the quadrature rule defined in Theorem 4.2. Namely, Theorem 4.5. [3] Let dσ be a measure supported on T, absolutely continuous with respect to the Lebesgue normalized measure, with weight function w, which is continuous on T. Given > 0 there exists a quadrature rule, which uses m = m(, σ) values of the function and its derivatives and m weights, of the following type: Im (P) =

s σ i −1

λi, j P( j) (zi ).

i=1 j=0

Furthermore, !! !! s α i −1 !! !! ( j) !! P(z)dσ(z) − λi, j P (zi )!! < P ∞ , for every P ∈ P. (4.5) !! !! T i=1 j=0 Proof. Applying the previous lemma, given > 0 there exists a Bernstein-Szeg˝o |dz| 1 measure dµ(z) = 2π|Q(z)| 2 such that |w(z) − |Q(z)|2 | < , ∀z ∈ T. If the degree of Q(z) is m, then we consider the zeros z1 , · · · , z s of the corresponding orthogonal polynomial of degree m according to their multiplicities α1 , · · · , α s and we construct the quadrature rule given in Theorem 4.2 Im (P) =

s α i −1

λi, j P( j) (zi ),

i=1 j=0

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QUADRATURE RULES ON THE UNIT CIRCLE. A SURVEY.

129 17

which is exact in P. Then for P ∈ P we get !

! !! !! |dz| !!! !! P(z)dα(z) − I (P)!!! = !! P(z) w(z) − 1 ! ≤ P ∞ . m ! !! ! |Q(z)|2 2π ! T T

1 w(z)|dz| with weight function w(z) = |z − 4.1. Numerical example. Let dσ(z) = 2π 1 .1| . We will approximate dσ by a Bernstein-Szeg˝o measure dµ(z) = 2π|Q(z)| 2 |dz|. According −10 to Lemma 4.4, if we set ε = 10 then it is possible to obtain Q(z) ∈ P such that !! ! !!w(z) − 1 !!! < 10−10 , for every z ∈ T. ! |Q(z)|2 ! 1 Indeed, for z ∈ T we have |z − .1|2 = |1 − .1z|2 and |1 − .1z|2 = ∞ . If we take | n=0 (.1z)n |2 10 1 − .111 z11 is not diﬃcult to show that (.1z)n = Q(z) = 1 − .1z n=0 !! ! !!w(z) − 1 !!! < 10−10 . ! |Q(z)|2 ! Therefore, in this case 1 dµ(z) = 10 n n 2 |dz| 2π| n=0 .1 z | and the sequence {Φn }n∈N of monic orthogonal polynomials with respect to dµ is given by Φn (z) = zn−10 Φ10 (z), for every n ≥ 10, with 2

Φ10 (z) = Q∗ (z) =

10

.110−n zn .

n=0

Thus, the nodes {zi }10 i=1 of the quadrature rule I10 are the zeros of Φ10 (z), and the quadare the solutions of the following system rature weights {λi }10 i=1 c−k =

10

λi zki , for k = 0, · · · , 9,

i=1

where {ck }k∈Z are the moments of the measure dµ. They are shown in the following Table. Table 3 Nodes z1 = −0.09594 + 0.02817i z2 = −0.09594 − 0.02817i z3 = −0.06548 + 0.07557i z4 = −0.06548 − 0.07557i z5 = −0.01423 + 0.09898i z6 = −0.01423 − 0.09898i z7 = 0.04154 + 0.09096i z8 = 0.04154 − 0.09096i z9 = 0.08412 + 0.05406i z10 = 0.08412 + 0.05406i

Coeﬃcients λ1 = 0.24587 − 0.02568i λ2 = 0.24587 + 0.02568i λ3 = 0.32458 − 0.11459i λ4 = 0.32458 + 0.11459i λ5 = 0.55240 − 0.12487i λ6 = 0.55240 + 0.12487i λ7 = 0.71459 − 0.09475i λ8 = 0.71459 + 0.09475i λ9 = 0.79658 − 0.00547i λ10 = 0.79658 + 0.00547i

Next, we compute integrals of the form T f (z)|z − .1|2 |dz| for some analytic functions f using our quadrature formula I10 ( f ). We compare the results with the values obtained Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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L. GARZA AND F. MARCELLÁN

using Mathematica and obtain the error. We consider the functions exp(z), sin(z), and cos(z), in order to obtain the exact value of the integrals easily. The following Table shows the obtained results. Table 4 Function exp(z) sin(z) cos(z)

Approx. I10 Approx. NIntegrate 0.84756 − 4.01478E − 13i 0.84756 −0.25487 + 3.12247E − 13i −0.25487 0.92547 + 4.16635E − 14i 0.92547

Bound Error 10−10 10−12 10−12

5. The connection between quadratures on R and T through the Szeg˝o transformation. Suppose µ is a symmetric weight function, i.e. a weight function defined on [−π, π] such that µ(θ) > 0 a.e., and µ(−θ) = µ(θ), θ ∈ [−π, π]. Denote by {Φn }n0 the family of monic polynomials orthogonal with respect to µ and consider the para-orthogonal polynomials defined by Bn (z, τ) = Φn (z) + τΦ∗n (z), |τ| = 1. It is known that the zeros of Bn (z, τ) are simple, interlace with the zeros of Bn−1 (z, τ), and lie on the unit circle T (see [23]). Take τ = 1. Notice that, since Φn (1) = Φ∗n (1) 0, z = 1 can not be a zero of Bn (z, τ). Moreover, the coeﬃcients of Bn (z, τ) are real, and then, if n is even, the zeros of Bn (z, τ) are on T in complex conjugated pairs. On the other hand, if n is odd, then the only real zero is z = −1 and the remaining zeros are again on T in complex conjugated pairs. Now, we want to approximate the integral π ˜Iµ ( f ) = f (eiθ )µ(θ)dθ −π

by an n-point Szeg˝o quadrature formula I˜n ( f ) =

(5.1)

n

A j f (z j ),

j=1

where {z j }nj=1 are the zeros of Bn (z, τ) and the weights {A j }nj=1 are positive and given by Aj = −

An (z j , τ) , 2z j B n (z j , τ)

where An (z, τ) = Ωn (z) − τΩ∗n (z) and Ωn (z) is the n-th associated polynomial of the second kind. Notice that if τ = ±1, then An (z, τ) and Bn (z, τ) have real coeﬃcients. If z j is a zero of Bn (z, ±1) and A˜ j is the coeﬃcient corresponding to z¯ j in (5.1), then A˜ j = −

An (¯z j , τ) An (z j , τ) =− = A¯ j , 2¯z j Bn (¯z j , τ) 2¯z j B (z j , τ) n

and, since A j > 0, we have A˜ j = A¯ j = A j . As a conclusion, Proposition 5.1. [5] Let µ(θ) be a symmetric weight function on [−π, π] and let I˜n ( f ) = o quadrature formula, with the zeros of Bn (z, τ) as nodes. j=1 A j f (z j ) be the n-point Szeg˝ Then n

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131 19

(i) If τ = 1, then (a) If n is even: I˜n ( f ) = n/2 z j )]. j=1 A j [ f (z j ) + f (¯ (n−1)/2 ˜ (b) If n is odd: In ( f ) = B f (−1) j=1 A j [ f (z j ) + f (¯z j )]. (ii) If τ = −1, then A j [ f (z j ) + f (¯z j )]. (a) If n is even: I˜n ( f ) = A f (1) + B f (−1) + (n−2)/2 j=1 (n−1)/2 ˜ A j [ f (z j ) + f (¯z j )]. (b) If n is odd: In ( f ) = A f (1) + j=1

All the weights A,B, and A j are positive. Notice that we only have to compute half of the weights and nodes due to the symmetry of µ(θ). 5.1. The Szeg˝o transformation. Let ν be a weight function supported on the interval [−1, 1]. We can define a symmetric weight µ in [−π, π] as µ(θ) = ν(cos θ)| sin(θ)|.

(5.2)

If z = eiθ and x = 12 (z + z−1 ) = cos θ, then 0 +1 G(x)ν(x)dx = − G(cos θ)ν(cos θ) sin θdθ, −1

and setting ξ = −θ,

−π

+1

−1

Thus,

2

+1

G(x)ν(x)dx =

(5.3)

−π

G(cos ξ)ν(cos ξ) sin ξdξ.

0

−1

or, equivalently,

G(x)ν(x)dx =

+1

−1

π

−π

G(cos θ)ν(cos θ)| sin θ|dθ,

G(x)ν(x)dx =

π

−π

f (eiθ )µ(θ)dθ,

with

iθ

e + e−iθ 1 . f (eiθ ) = G 2 2 There is a relation between the orthogonal polynomials associated with a measure ν supported on [−1, 1] and the orthogonal polynomial sequence associated with the measure µ defined by (5.2), which is supported on the unit circle.

Theorem 5.2. [29] Let {Φn }n0 and {Pn }n0 be the sequences of monic polynomials orthogonal with respect to the weights µ(θ) and ν(x), respectively. Moreover, let {Qn }n0 be the sequence of monic polynomials orthogonal with respect to the weight (1 − x2 )ν(x). −1 If x = (z+z2 ) , z = eiθ , then for n 1, (5.4)

Pn (x)

=

(5.5)

Qn (x)

=

1 [z−n Φ2n (z) + zn Φ2n (1/z)], + Φ2n (0)) z−n−1 Φ2n+2 (z) − zn+1 Φ2n+2 (1/z) 1 . 2n (1 − Φ2n+2 (0)) z − z−1 2n (1

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Notice that B2n (z, 1) , zn and by Proposition 5.1, the zeros of B2n (z, 1) are in T and appear in conjugate pairs. Denote these zeros by z1 , . . . , zn , z¯1 , . . . , z¯n . If z j = eiθ j , then the zeros of Pn (x) are z−n Φ2n (z) + zn Φ2n (1/z) =

x j = cos θ j ,

(5.6)

j = 1, . . . , n.

In a similar way, from (5.5) notice that z−n−1 Φ2n+2 (z) − zn+1 Φ2n+2 (1/z) B2n+2 (z, −1) . = n 2 z − z−1 z (z − 1) Again from Proposition 5.1, B2n+2 (z, −1) has zeros at ζ = ±1 and the remaining 2n zeros appear in conjugate pairs on T. Thus, we can obtain the zeros of Qn (x) in the same way. 1 On the other hand, it is known that −1 G(x)ν(x)dx can be approximated by an n-point Gauss-Christoﬀel quadrature rule, i.e. 1 n Iν (G) = G(x)ν(x)dx = λ jG(x j ) + En (G), −1

j=1

where En (G) = 0 if G ∈ P2n−1 , the nodes x j , j = 1, . . . , n, are the zeros of Pn (x) given by (5.6), and the weights λ j are all positive real numbers. There is a connection between Gauss and Szeg˝o quadrature rules as follows. Theorem 5.3. [5] Let In (G) = nj=1 λ jG(x j ) be the n-point Gauss rule for Iν (G) = 1 2n G(x)ν(x)dx. Set x j = cos θ j and define {z j }2n j=1 and {A j } j=1 by −1 zj zn+ j

= eiθ j , = e

−iθ j

A j = λ j, ,

An+ j = λ j ,

j = 1, . . . , n. j = 1, . . . , n.

2n z j )] coincides with the 2n-point Szeg˝o Then, I˜2n ( f ) = 2n j=1 A j f(z j ) = j=1 A j [ f (z j ) + f (¯ π iθ ˜ quadrature rule Iµ ( f ) = −π f (e )dµ(θ), where ν(x) and µ(θ) are related through the Szeg˝o transformation and the nodes are the zeros of B2n (z, 1). π Proof. We need to prove that I˜2n (L) = I˜µ (L) = −π L(eiθ )µ(θ)dθ, for any L ∈ Λ−2n+1,2n−1 or, equivalently, I˜2n (zk ) = I˜µ (zk ), −2n + 1 k 2n − 1. Without loss of generality, assume k 0. From Theorem (5.3), I˜2n (zk ) =

n

A j [zkj + z¯kj ] = 2

j=1

n

A j cos kθ j .

j=1

If we write cos kθ = T k (cos θ), where T k (x) is the kth Chebyshev polynomial of the first kind, we have 1 n n ˜I2n (zk ) = 2 A j T k (cos θ j ) = 2 A j T k (x j ) = 2 T k (x)ν(x)dx, j=1

j=1

−1

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QUADRATURE RULES ON THE UNIT CIRCLE. A SURVEY.

π and, by (5.3), the last integral becomes 2 −π p(eiθ )µ(θ)dθ, where p(eiθ ) = e−iθ )/2) = 12 cos kθ, and therefore π I˜2n (zk ) = cos(kθ)µ(θ)dθ.

133 21 1 iθ 2 T k ((e

+

−π

But we also have I˜µ (zk ) =

π

−π

eikθ µ(θ)dθ =

π

−π

cos(kθ)µ(θ)dθ + i

π

sin(kθ)µ(θ)dθ, −π

and, since µ(θ) is symmetric, the last integral vanishes and therefore I˜2n (zk ) = I˜µ (zk ), for 0 k 2n − 1. Using a similar argument for −2n + 1 k < 0, the proof follows. There is also a converse result. Theorem 5.4. [5] Let I˜2n ( f ) = 2n o quadrature rule for j=1 A j f (z j ) be the 2n-point Szeg˝ I˜µ ( f ), whose nodes are the zeros of Bn (z, 1). Set zn+ j = z¯ j and z j = eiθ j , j = 1, . . . , n. Then, if x j = cos θ j , j = 1, . . . , n, the formula In (G) = nj=1 A j G(x j ) coincides with the n1 point Gauss rule for Iν (G) = −1 G(x)ν(x)dx, where µ(θ) and ν(x) are related by the Szeg˝o transformation. 5.2. Other quadrature formulas. Using other families of para-orthogonal polynomials, connections with other quadrature formulas can be established. It is well known that 1 if preassigned nodes a1 , . . . , am are needed for quadrature rules for Iν (G) = −1 G(x)ν(x), i.e m n Iν (G) = B jG(a j ) + λ jG(x j ) + E˜ n (G) ≡ Jn (G) + E˜ n (G), j=1

j=1

then E˜ n (G) = 0 for any G ∈ P2n+m−1 . Indeed, Theorem 5.5. [5] Jn (G) is exact in P2n+m−1 if and only if (i) Jn (G) is interpolatory in Pn+m−1 . (ii) The nodes x j , j = 1, . . . , n are the zeros of the n-th monic polynomial orthogonal with respect to the weight function r(x)ν(x), with r(x) = (x − a1 ) . . . (x − am ). Moreover, if r(x j ) 0, j = 1, . . . , n, then {B j }mj=1 and {λ j }nj=1 are all positive real numbers. Now consider the Gauss-Lobatto rule, this is, m = 2, a1 = −1 and a2 = 1. In this case, the resulting weight is r(x)ν(x) = (1 − x2 )ν(x), and from Theorem 5.2, the nodes are the zeros of the para-orthogonal polynomial B2n+2 (z, −1). Following a similar argument as in Theorem 5.3, we have Theorem 5.6. [5] Let AG(1) + BG(−1) + nj=1 λ jG(x j ) be the n-point Gauss-Lobatto quadrature rule for Iν (G). Let x j = cos θ j , z j = eiθ j , and A j = λ j , j = 1, . . . , n. Then, 2A f (1) + 2B f (−1) + nj=1 A j [ f (z j ) + f (¯z j )] is the (2n + 2)-point Szeg˝o quadrature rule for I˜µ , where ν and µ are related through the Szeg˝o transformation. Conversely, Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

134 22

L. GARZA AND F. MARCELLÁN

Theorem 5.7. [5] Let I˜2n+2 ( f ) = 2n+2 o quadrature j=1 A j G(z j ) be the (2n + 2)-point Szeg˝ rule for I˜µ ( f ), with the zeros of B2n+2 (z, −1) as nodes. Let z2n+1 = 1, z2n+2 = −1, and zn+ j = z¯ j , j = 1, . . . , n, with z j = eiθ j , θ j −π, 0. Take A = A2n+1 , B = A2n+2 . Then B A G(1) + G(−1) + A jG(x j ), 2 2 j=1 n

Jn (G) =

x j = cos θ j , j = 1, . . . , n,

coincides with the n-point Gauss-Lobatto quadrature rule for Iν ( f ). Finally, consider the Gauss-Radau quadrature rule, when m = 1 and a1 = ±1. It can be shown that, in this case, the corresponding nodes are the zeros of the para-orthogonal polynomials B2n+1 (z, ±1). Proceeding as in the Gauss and Gauss-Lobatto cases, one can prove Theorem 5.8. [5] Let 12 AG(±1)+ nj=1 λ jG(x j ) be the n-point Gauss-Radau quadrature rule for Iν (G). Let x j = cos θ j , z j = eiθ j , and A j = λ j , j = 1 . . . , n. Then, A f (±1) + n z j )] is the (2n+1)-point Szeg˝o rule for I˜µ ( f ), with the zeros of B2n+1 (z, ±1) j=1 A j [ f (z j )+ f (¯ as nodes, and where ν and µ are related through the Szeg˝o transformation. 5.3. Bernstein measures on the interval [−1, 1]. Now we show quadrature rules for Bernstein measures corresponding to rational modifications of the Jacobi measures, dνα,β (x) = (1 − x)α (1 + x)β dx for the parameters α = ± 21 and β = ± 12 , that is, the so called Chebyshev measures of the first kind α = β = − 12 , of the second kind α = β = 12 , of the third kind α = − 12 , β = 12 , and the fourth kind α = 12 , β = − 12 (see [27] and [29]). If q(x) = kr=0 ar xr is a positive polynomial on [−1, 1] with real coeﬃcients, we consider the following rational modifications of the above measures, that is, the Bernstein measures: √ dx 2 1 − x2 dx dν1 (x) = √ , , dν2 (x) = πqk (x) π 1 − x2 qk (x) " " 1 1 + x dx 1 1 − x dx dν3 (x) = (5.7) , dν4 (x) = , π 1 − x qk (x) π 1 + x qk (x) which are positive Borel measures on [−1, 1]. Taking into account the Féjer-Riesz representation (see [22] and [29]) we know that there exists an algebraic polynomial Ak (z) = kr=0 mr zr , with mr ∈ R for r = 0, · · · , k, m0 > 0, without zeros in D = {z : |z| ≤ 1} such that qk (cos θ) = |Ak (eiθ )|2 . Applying suitable transformations, the measures dνi , (i = 1, · · · , 4), become the Bernsteindθ Szeg˝o measure on [−π, π], dµ(θ) = . The transformations are as follows. 2π|Ak (eiθ )|2 1. First transformation. Since x = cos θ transforms the interval [−π, π] into [−1, 1], then we can define a measure µ1 on [−π, π] such that dµ1 (θ) = 12 ν(cos θ)| sin θ|dθ and µ1 ([−π, π]) = 1. It induces another measure on T that we also denote by µ1 for a sake of simplicity. Notice that when ν is the Chebyshev measure of the first kind, then µ1 is the Lebesgue normalized measure. If P(x) = nl=0 al T l (x), with al ∈ R, l = 0, · · · , n, where {T n (x)}n∈N is the sequence of Chebyshev polynomials of the first kind, and Q(z) = nl=0 al zl , then it Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

135 23

QUADRATURE RULES ON THE UNIT CIRCLE. A SURVEY.

is easy to relate the inner products induced by both measures ν and µ1 as follows 1 Q(z) + Q(z−1 ), zl µ1 . 2

P(x), T l (x)ν =

2. Second transformation. The substitution x = cos θ and the multiplication by

1 defines a finite pos4 sin2 θ 1 ν(x) ν(cos θ) dθ if itive Borel measure µ2 on [−π, π] by dµ2 (θ) = dx < +∞. 4| sin θ| 1 − x2 −1 The measure µ2 induces in a natural way another measure on T that we also denote by µ2 . Notice that when ν is the Chebyshev measure of the second kind, then µ2 is the Lebesgue normalized measure. If P(x) = nl=0 bl Ul (x), with bl ∈ R, l = 0, · · · , n, where {Un (x)}n∈N is the sequence of Chebyshev polynomials of the second kind and Q(z) = nl=0 bl zl , then it is easy to obtain the following relation between the inner products corresponding to both measures P(x), Ul (x)ν = Q(z) − Q(z−1 ), zl+1 µ2 .

3. Third transformation. The substitution x = cos θ and the multiplication by

1 4 cos2

defines a finite posi 1 ν(x) 1 θ tive Borel measure µ3 on [−π, π] by dµ3 (θ) = 2 ν(cos θ)| tan 2 |dθ if dx < −1 1 + x +∞. The measure µ3 induces a measure on T that we also denote by µ3 . Notice that when ν is the Chebyshev measure of the third kind, then µ3 is the Lebesgue normalized measure. θ 2

If P(x) = nl=0 cl Wl (x), with cl ∈ R, l = 0, · · · , n, where {Wn (x)}n∈N is the sequence of Chebyshev polynomials of the third kind and Q(z) = nl=0 cl zl , then the inner products corresponding to both measures are related as follows P(x), Wl (x)ν = z 2 Q(z) + z− 2 Q(z−1 ), zl+ 2 µ3 . 1

1

1

4. Fourth transformation. The substitution x = cos θ and the multiplication by

1 allows us to define 4 sin2 2θ a finite positive Borel measure µ4 on [−π, π] by dµ4 (θ) = 12 ν(cos θ)| cot 2θ |dθ if 1 ν(x) dx < +∞. The measure µ4 induces a measure on T that we also denote −1 1 − x by µ4 . Notice that when ν is the Chebyshev measure of the fourth kind, then µ4 is the Lebesgue normalized measure. If P(x) = nl=0 dl Vl (x), with dl ∈ R, l = 0, · · · , n, where {Vn (x)}n∈N is the sequence of Chebyshev polynomials of the fourth kind and Q(z) = nl=0 dl zl , then

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136 24

L. GARZA AND F. MARCELLÁN

we can relate the inner products corresponding to both measures as follows P(x), Vl (x)ν = z 2 Q(z) − z− 2 Q(z−1 ), zl+ 2 µ4 . 1

1

1

Now, if we consider the Bernstein measures dν1 , dν2 , dν3 , and dν4 defined in (5.7) and we transform each dνi by the i − th transformation, i = 1, · · · , 4, then it is easy to prove that dθ we obtain the same Bernstein-Szeg˝o measure dµ(θ) = . 2π|Ak (eiθ )|2 Consider the Bernstein measures given in (5.7) and the Bernstein-Szeg˝o measure obtained using the above transformations. We show next that for each of these Bernstein measures there exists a quadrature rule with a fixed number of nodes and weights, which is exact in the linear space P of polynomials with real coeﬃcients. dx . There exists a quadrature rule using √ π 1 − x2 qk (x) s s and the weights {λi, j }i=1,··· ,s; j=0,··· ,αi −1 , with i=1 αi = k such that it exactly the nodes {zi }i=1 integrates polynomials, i.e., Theorem 5.9. [4] Let dν1 (x) =

∀P ∈ P, P(x) =

M

bl T l (x), with bl ∈ R, we get

l=0

⎛ ⎞( j) ⎛M s α i −1 ⎜⎜⎜ ⎜⎜⎜ bl zl ⎟⎟⎟ ⎜ ⎟⎟ ⎜ ⎜ P(x)dν1 (x) = 2 ⎜⎜⎝ λi, j ⎜⎝ 2 ⎠ −1

1

i=1 j=0

l=0

|z=zi

⎞ ⎟⎟⎟ ⎟⎟⎟ . ⎟⎠

Proof. Using the first transformation the measure dν1 becomes the measure dµ given dθ by dµ(θ) = . 2π|Ak (eiθ )|2 M bl l M z , then P(x), 1ν1 = bl T l (x) with bl ∈ R, l = 0, · · · , M, and Q(z) = l=0 If P(x) = l=0 2 −1 Q(z) + Q(z ), 1µ . Therefore we can write

1

dx P(x) √ = −1 π 1 − x2 qk (x) ⎞ ⎞ π ⎛⎜ π ⎛⎜ M M bl l ⎟⎟⎟⎟ bl l ⎟⎟⎟⎟ dθ dθ ⎜⎜⎜ ⎜⎜ ⎜ z ⎟⎠ + z ⎟⎠ . ⎜⎝ ⎜⎝ 2π|Ak (eiθ )|2 2π|Ak (eiθ )|2 −π l=0 2 −π l=0 2 for z = eiθ . In order to compute these integrals we apply Theorem 4.2. Indeed if z1 , · · · , z s are the zeros s of A∗k (z), which are located in D, and µ1 , · · · , µ s are their multiplicities with i=1 αi = k, then there exist weights {λi, j }i=1,··· ,s; j=0,··· ,αi −1 , such that for every R ∈ Π we get

i dθ = λi, j R( j) (zi ) 2π|Ak (eiθ )|2 i=1 j=0

s α −1

π

R(eiθ ) −π

and

−π

i dθ = λi, j R( j) (zi ). 2π|Ak (eiθ )|2 i=1 j=0

s α −1

π

R(eiθ )

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QUADRATURE RULES ON THE UNIT CIRCLE. A SURVEY.

M bl l z with bl ∈ R, then l=0 2 ⎛ ⎞( j) ⎛M ⎞ π s α M ⎛ i −1 ⎜⎜⎜ ⎜⎜⎜ bl zl ⎟⎟⎟ ⎜⎜⎜ bl (zl + zl ) ⎟⎟⎟ dθ ⎜ ⎟⎟⎠ ⎜ ⎜ ⎜⎝ ⎟⎠ = 2 ⎜⎜⎝ λi, j ⎜⎝ 2 2 2π|Ak (eiθ )|2 −π l=0 i=1 j=0 l=0

Therefore if Q(z) =

|z=zi

⎞ ⎟⎟⎟ ⎟⎟⎟ , ⎟⎠

and, as a consequence, the statement holds.

Remark 5.10. When A∗k (z) has simple zeros z1 , · · · , zk in D, the weights are λ1 , · · · , λk M and the quadrature rule means that if P(x) = l=0 bl T l (x), with bl ∈ R, then ⎞ ⎛ 1 M k ⎜⎜⎜ bl zli ⎟⎟⎟⎟ dx ⎜ ⎟⎟ . P(x) √ = 2 ⎜⎝⎜ λi 2 ⎠ −1 π 1 − x2 qk (x) i=1 l=0 √ 2 1 − x2 dx . There exists a quadrature rule using Theorem 5.11. [4] Let dν2 (x) = πqk (x) s s the nodes {zi }i=1 and weights {λi, j }i=1,··· ,s; j=0,··· ,αi −1 , with i=1 αi = k such that it exactly integrates polynomials, that is, ∀P ∈ P, P(x) =

M

dl Ul (x), with dl ∈ R, we get

l=0

1

−1

⎞ ⎛ M M s α i −1 ⎟⎟ ⎜⎜⎜ l ( j) l+2 ( j) ⎜ P(x)dν2 (x) = (dl z ) (zi ) − λi, j (dl z ) (zi )⎟⎟⎟⎠ . ⎜⎝λi, j i=1 j=0

l=0

l=0

Remark 5.12. If the zeros z1 , · · · , zk , of A∗k (z) are simple, then the weights are λ1 , · · · , λk and the quadrature rule becomes ⎞ ⎛ M ⎞ 1 ⎛⎜ M k ⎜ M ⎟⎟⎟ ⎟⎟ ⎜⎜⎜ ⎜⎜⎜ l+2 ⎟ l ⎟ ⎟⎟⎠ . d U (x) (x)) = d z − λ d z dν λ ⎟⎠ 2 ⎜⎝ ⎜⎝ i l l l i i l i −1

l=0

i=1

l=0

l=0

" 1 1 + x dx Theorem 5.13. [4] Let us consider the positive measure dν3 (x) = π 1 − x qk (x) s and weights supported on [−1, 1]. Then there exists a quadrature rule with nodes {zi }i=1 s {λi, j }i=1,··· ,s; j=0,··· ,αi −1 , with i=1 αi = k, such that it exactly integrates polynomials, i.e., ∀P ∈ P, P(x) =

M

el Vl (x), with el ∈ R, we get

l=0

1

−1

⎞ ⎛ M M s α i −1 ⎟⎟ ⎜⎜⎜ l ( j) l+1 ( j) ⎜⎜⎝λi, j P(x)dν3 (x) = (el z ) (zi ) − λi, j (el z ) (zi )⎟⎟⎟⎠ . i=1 j=0

l=0

l=0

Remark 5.14. If the zeros z1 , · · · , zk of A∗k (z) are simple, then the weights are λ1 , · · · , λk and the quadrature rule is ⎞ ⎛ M ⎞ 1 ⎛⎜ M M k ⎜ ⎟⎟⎟ ⎟ ⎜⎜⎜ ⎜⎜⎜ l+1 ⎟ l ⎟ el zi − λi el zi ⎟⎟⎟⎠ . ⎜⎝ el Vl (x)⎟⎠ dν3 (x) = ⎜⎝λi −1

l=0

i=1

l=0

l=0

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138 26

L. GARZA AND F. MARCELLÁN

" 1 1 − x dx supTheorem 5.15. [4] Let consider the positive measure dν4 (x) = π 1 + x qk (x) s ported on [−1, 1]. Then there exists a quadrature rule with nodes {zi }i=1 and weights s {λi, j }i=1,··· ,s; j=0,··· ,αi −1 , with i=1 αi = k, such that it exactly integrates polynomials, i.e., M

∀P ∈ P, P(x) =

fl Wl (x), with fl ∈ R, we get

l=0

1

−1

⎞ ⎛ M M s α i −1 ⎟⎟ ⎜⎜⎜ l ( j) l+1 ( j) ⎜⎜⎝λi, j P(x)dν4 (x) = ( fl z ) (zi ) + λi, j ( fl z ) (zi )⎟⎟⎟⎠ . i=1 j=0

l=0

l=0

Remark 5.16. If the zeros z1 , · · · , zk of A∗k (z) are simple, then the weights are λ1 , · · · , λk and the quadrature rule is ⎞ ⎛ M ⎞ 1 ⎛⎜ M k ⎜ M ⎟⎟⎟ ⎟ ⎜⎜⎜ ⎜⎜⎜ l+1 ⎟ l ⎟ fl Wl (x)⎟⎠ dν4 (x)) = fl zi + λi fl zi ⎟⎟⎟⎠ . ⎜⎝ ⎜⎝λi −1

l=0

i=1

l=0

l=0

Remark 5.17. In order to compute the nodes of the quadrature rules we propose to degree use the next property. If (x − a)µ is a factor of the polynomial qk (x) of arbitrary √ 2 k, then √ the Joukowski transformation of a with modulus less than one, (a ± a − 1 with |a ± a2 − 1| < 1) is a node of the quadrature rule with multiplicity µ. Therefore, if all the factors are linear with real zeros, then the nodes are in [−1, 1]. Otherwise if the factors are quadratic with complex zeros, we have a pair of complex conjugated nodes. Notice that this fact does not play a relevant role in the method. With this method for determining nodes and the previous one given in Proposition 1 for determining the weights we prepare the method in order to be applied without errors in the calculus. 6. Acknowledgements. The authors thank the referees by the careful revision of the manuscript. Their comments and suggestions have contributed to improve the presentation. References [1] N. I. Akhiezer, The Classical Moment Problem and Some Related Questions in Analysis, Hafner, New York, (1965). [2] M. Alfaro, F. Marcellán, Carathéodory functions and orthogonal polynomials on the unit circle, in Complex Methods in Approximation Theory, A. Martínez Finkelshtein Editor, Universidad de Almería, (1997), 1–22. [3] E. Berriochoa, A. Cachafeiro, and F. Marcellán, A new numerical quadrature formula on the unit circle, Numer. Algorithms 44 (2007), 391-401. [4] E. Berriochoa, A. Cachafeiro, J. García-Amor, and F. Marcellán, New quadrature rules for Bernstein measures on the interval [−1, 1], Electr. Trans. Numer. Anal. 30 (2008), 278-290. [5] A. Bultheel, L. Daruis, and P. González-Vera, A connection between quadrature formulas on the unit circle and the interval [-1, 1], J. Comput. Appl. Math. 132 (2001), 1-14. [6] A. Bultheel, P. González-Vera, E. Hendriksen, and O. Njåstad, Orthogonality and quadrature on the unit circle, in: C. Brezinski, L. Gori, A. Ronveaux (Eds.), Orthogonal Polynomials and their Applications, Vol. 9 of IMACS Annals on Computing and Applied Mathematics, J.C. Baltzer AG, Basel, (1991), 205-210. [7] M. J. Cantero, L. Moral, and L. Velázquez, Measures and paraorthogonal polynomials on the unit circle, East J. Approx. 8 (2002), 447-464. [8] T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978. [9] L. Darius, P. González-Vera, and F. Marcellán, Gaussian quadrature formulae on the unit circle, J. Comput. App. Math. 140 (2002), 159-183. [10] L. Darius, P. González-Vera, and O. Njåstad, Szeg˝o quadrature formulas for certain Jacobi-type weight functions, Math. Comput. 71 (2000), 683-701.

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QUADRATURE RULES ON THE UNIT CIRCLE. A SURVEY.

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[11] L. Daruis, J. Hernández, and F. Marcellán, Spectral transformations for Hermitian Toeplitz matrices, J. Comput. Appl. Math. 202 (2007), 155–176. [12] P. J. Davis, P. Rabinowitz, Methods of Numerical Integration, 2nd Edition, Academic Press, New York, (1984). [13] T. Erdélyi, P. Nevai, J. Zhang,and J. S. Geronimo A simple proof of "Favard’s theorem" on the unit circle, Att. Sem. Mat. Fis. Univ. Modena, 39 (1991), 551–556. [14] G. Freud, Orthogonal Polynomials, Pergamon Press, New York, 1971. [15] L. Garza, J. Hernández, and F. Marcellán, Orthogonal polynomials and measures on the unit circle. The Geronimus transformations, J. Comput. Appl. Math. (2009). doi:10.1016/j.cam.2007.11.023. [16] W. Gautschi, Orthogonal Polynomials: Computation and Approximation. Oxford University Press, Oxford, 2004. [17] Ya. L. Geronimus, Orthogonal polynomials: estimates, asymptotic formulas, and series of polynomials orthogonal on the unit circle and on an interval. Consultants Bureau, New York, 1961. [18] Ya. L. Geronimus, Polynomials orthogonal on a circle and their applications, in Series and Approximation, Amer. Math. Soc. Transl. Series1, Vol 3(1962), Amer. Math. Soc. Providence RI, 1–79. [19] L. Golinskii, Quadrature formula and zeros of paraorthogonal polynomials on the unit circle, Acta Math. Hungar. 96 (2002), 169-186. [20] P. González Vera, J. C. Santos-León, and O. Njåstad, Some results about numerical quadratures on the unit circle, Adv. Comput. Math. 5 (1996), 297-328. [21] W. B. Gragg, Positive definite Toeplitz matrices, the Arnoldi process for isometric operators, and Gaussian quadrature on the unit circle, J. Comput. Appl. Math. 46 (1993), 183-198. [22] U. Grenander, G. Szeg˝o, Toeplitz forms and their applications. University of California Press, Berkeley, 1958. [23] W. B. Jones, O. Njåstad, and W. J. Thron , Moment theory, orthogonal polynomials, quadrature, and continued fractions associated with the unit circle, Bull. London Math. Soc. 21 (1989), 113-152. [24] V. I. Krylov, Approximate Calculation of Integrals, McMillan, New York, 1962. [25] F. Marcellán, Polinomios ortogonales no estándar. Aplicaciones en Análisis Numérico y Teoría de Aproximación, Rev. Acad. Colomb. Ciencias Exactas, Físicas y Naturales, 30 (117) (2006), 563-579 (In spanish). [26] F. Peherstorfer and R. Steinbauer, Characterization of orthogonal polynomials with respect to a functional, J. Comput. Appl. Math. 65 (1995), 339–355. [27] T. Rivlin, The Chebyshev polynomials, John Wiley and Sons, New York, 1974. [28] B. Simon, Orthogonal Polynomials on the Unit Circle, Amer. Math. Soc. Coll. Publ., Vol. 54, Part 2, Amer. Math. Soc., Providence, RI, 2005. [29] G. Szeg˝o, Orthogonal Polynomials, Amer. Math. Soc. Coll. Publ., Vol. 23, 4th ed., Amer. Math. Soc., Providence, RI, 1975. [30] L. Wong, First and second kind paraorthogonal polynomials and their zeros, J. Approx. Theory 146 (2007), 282–293 Universidad Aut´onoma de Tamaulipas, Carretera Sendero Nacional Km. 3, AP2005, Matamoros, Tamaulipas, M´exico. Current address: Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911, Leganés, Spain. E-mail address: [email protected] Departamento de Matem´aticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911, Legan´es, Spain. E-mail address: [email protected]

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http://dx.doi.org/10.1090/conm/507/09957

Contemporary Mathematics Volume 507, 2010

On the multilinear trigonometric problem of moments A. Ibort, P. Linares and J.G. Llavona Abstract. A multilinear generalization of the trigonometric problem of moments is presented and discussed. The moments ck , k ∈ Zn of a regular Borel polymeasure γ on Tn are characterized by means of a norm || · ||w on functions on Zn . Some properties of this norm are analyzed and various examples are presented showing that it is strictly weaker than the Fr´echet norms and the absolute convergence norm. The convolution product of multilinear functionals is deﬁned and an inverse theorem is proved.

To Guillermo L´ opez Lagomasino on his 60th birthday.

1. Introduction The classical trigonometric problem of moments consists in determining if there exists a positive regular Borel measure µ on the circle T such that for a given sequence of complex numbers ck , k ∈ Z we have: e−ikθ µ(dθ), k ∈ Z. (1.1) ck = T

The Fourier–Stieltjes transform of a Borel measure µ ∈ Bo(T) is deﬁned as the map µ ˆ on Z (the dual group of T) such that: µ ˆ(k) = T e−ikθ µ(dθ), for all k ∈ Z. Thus the trigonometric moment problem amounts to determine if a given function c on Z is the Fourier-Stieltjes transform of a positive regular Borel measure µ. Riesz and Herglotz [Ri11, He11] showed that a necessary and suﬃcient condition for the existence of such a measure is that the function c(k) = ck is positive deﬁnite, that is, if for any positive integer N the quadratic form deﬁned on CN +1 by N k,l=0 c(k − ¯ l)ξk ξl is positive (notice that this condition implies that c−k = c¯k ). The problem of moments (1.1) has a natural extension to signed Borel measures. In such case, it is well-known that functions c(k) for which a signed Borel measure exists satisfying eq. (1.1) are ﬁnite linear combinations of positive deﬁnite functions (see for instance [Ru62]). 2000 Mathematics Subject Classiﬁcation. Primary 44A60, Secondary 46G25 . The ﬁrst author was supported in part by Project MTM 2007-62478. The second author was partially supported by FPU-MEC AP-2004-4843 Grant and by the ”Programa de formaci´ on del profesorado universitario del MEC”. The second and third author were supported in part by Project MTM 2006-03531. The authors wish to thank the suggestions and observations of the referee that have helped greatly to shape the ﬁnal form of this article. c 2010 American Mathematical Society c 0000 (copyright holder)

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The extension of the trigonometric moment problem to several variables does not imply further diﬃculties (see for instance the original papers [Hi33], [Ha35], [Ha36]). Thus if we denote by Tn the n–fold cartesian product of the group T, its dual group is Zn . If we are given a sequence of complex numbers ck , k = (k1 , . . . , kn ) ∈ Zn , the multivariate trigonometric moment problem posed by ck consists in determining if there exists a positive regular Borel measure µ on Tn such that: ck = e−ik·θ µ(dθ), k ∈ Zn , θ = (θ1 , . . . , θn ) ∈ Tn . Tn

The solution to the trigonometric multivariate moment problem is provided by a similar positivity condition on the moments ck as in the univariate case. There will exists a regular positive Borel measure µ on Tn whose moments are given by the family ck if and only if the function c(k) = ck deﬁned by them on Zn is positive deﬁnite, that is, if for any ﬁnite set of points k0 , . . . , kN ∈ Zn the quadratic form N ¯ i,j=0 c(ki − kj )ξi ξj is positive. The proof of this can be obtained by using M. Riesz extension principle for positive functionals (see for instance [Ak65]). Even if we had considered the more general situation of signed measures, the solution comes easily by using the Jordan decomposition of signed measures. In all cases, the linear operator Lc deﬁned on the space of trigonometric polynomials p(θ) = ik·θ on Tn by the formula: |k|≤N pk e Lc pk eik·θ = pk c−k , is bounded in the uniform topology, thus it can be extended uniquely to a bounded functional on Tn , hence by using Riesz representation theorem the existence of the searched measure is obtained. Notice that a similar argument will work if we extend the class of measures we consider to complex valued measures. However, the problem above takes an interesting turn when n > 1. In such case the functional Lc above can be considered as acting on the space of trigonometric polynomials on Tn , this is, on the set of variables zk = eiθk , k = 1, . . . , n, or it can be also considered as deﬁned in the n–fold cartesian product of the spaces of trigonometric polynomials on T. This second approach amounts to consider Lc as deﬁning a multilinear functional on the space of trigonometric polynomials in one variable. The natural notion of boundedness for multilinear functionals is that of boundedness on each component. A multilinear functional L deﬁned on the n-fold cartesian product C(T) × · · · × C(T) is bounded if: |L(ϕ1 , . . . , ϕn )| ≤ ||L||||ϕ1 ||∞ · · · ||ϕn ||∞ ,

∀ϕ1 , . . . , ϕn ∈ C(T).

Bounded multilinear functionals (even in the bilinear case) cannot be represented in general by a measure on the corresponding cartesian product space, in our case, Tn . The exploration of this phenomena was started by Fr´echet [Fr15] and fully recognized on its importance by Morse and Transue [Mo49] that coined the name bimeasure for the generalized notion of measure that suited this problem, in other words, a generalized notion of measure that allows for an integral representation of such functionals. A polymeasure is a natural extension of the concept of a measure as a function on the cartesian product of a family of σ-algebras such that it is a measure on each component. Moreover a polymeasure is not necessarily the extension of a product of measures, this is, given measures µk on the σ-algebras Σk , k = 1, . . . , n, there is a natural extension of the cartesian product measure ×k µk Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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143 3

to the σ-algebra generated by the families Σk , but not all polymeasures can be obtained in this way (see for instance [Be84], p. 33). There is however a natural extension of Riesz theorem that relates bounded multilinear functionals on spaces of continuous functions on compact sets and regular Borel polymeasures (see for instance Bombal et al [Bo98, Bo01], and [Bl01] and references therein). Thus we are naturally led to consider an extension of the trigonometric problem of moments to the multilinear case, where given a family of complex numbers ck indexed by the multiindex k ∈ Zn , we would like to know under what conditions there will exist a regular polymeasure γ on the n-dimensional torus Tn such that: e−ik1 θ1 ⊗ · · · ⊗ e−ikn θn γ(dθ), ∀k ∈ Zn , (1.2) ck =

Tn

where Tn e−ik1 θ1 ⊗ · · · ⊗ e−ikn θn γ(dθ) or Tn (ϕ1 , . . . , ϕn ) dγ represents the integral of the family of continuous functions ϕk ∈ C(T) with respect to the polymeasure γ [Do87]. In a recent paper by the authors [Ib08] it was shown that in general it cannot be expected that the Hausdorﬀ problem of moments for a multimoment sequence µk will have a classical solution, that is, a measure µ on [0, 1]n such that µk will be the moments of the measure µ. The classical boundedness conditions on the moments µk must be relaxed considerably to allow for polymeasures on the n–dimensional cube [0, 1]n as solutions to the problem. Thus a weak boundedness condition on the moments was introduced that characterized completely the solutions to the multilinear Hausdorﬀ problem of moments. In this paper we address the question of what are the natural boundedness conditions on the moments µk that guarantee the existence of a polymeasure on Tn . Such condition will be stablished by means of a norm ||·||w on the space of functions on Zn . Some properties of this norm will be analyzed. In particular it will be shown that it is strictly weaker than the supremum norm and strictly smaller than the Fr´echet norm || · ||Fn . The space of functions c on Zn with ﬁnite || · ||w norm constitutes a Banach algebra with respect to the usual product of functions. A new deﬁnition of the convolution product of two bounded multilinear functionals will be given that extends the usual notion of convolution of measures. The space of bounded multilinear functionals becomes a Banach algebra with respect to this convolution product and the Fourier-Stieltjes transform deﬁnes a Banach algebra isomorphism with respect to it. Finally an inversion formula for polymeasures is proven and it is shown that a polymeasure can be represented by a trigonometric series that is convergent in the weak*–topology to the given polymeasure γ. Some examples in the linear and bilinear case are discussed that illustrates the ideas and results discussed in the paper. 2. Polymeasures and a multilinear Riesz representation theorem We will brieﬂy review ﬁrst the basic deﬁnitions and terminology concerning polymeasures as well as the analogue of Riesz representation theorem for them. Let Σk be a σ-algebra of subsets of the set Sk , k = 1, . . . , n. A polymeasure γ on the σ-algebras Σ1 , . . . , Σn is a separately valued σ-additive function on the cartesian product of Σ1 , . . . , Σn (see for instance Dobrakov [Do87] ). If we ﬁx a family of measurable sets Fl ∈ Σl , l = k, the polymeasure γ induces a measure on Σk by means of the formula γk (·) = γ(F1 , . . . , Fk−1 , ·, Fk+1 , . . . , Fn ). As well as in the case of ordinary measures (n = 1), we can deﬁne the variation of the polymeasure Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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γ to be the set function v(γ) : Σ1 × · · · × Σn → [0, +∞] given by: r rn 1 k1 kn ··· |γ(A1 , . . . , An )| , v(γ)(A1 , . . . , An ) = sup k1

kn

where the supremum is taken over all ﬁnite partitions {Akl l }rkll =1 of the sets Al ∈ Σl . Following [Bo98] we must consider also the semivariation function ||γ|| : Σ1 × · · · × Σn → [0, +∞] associated to the polymeasure γ deﬁned by: r rn 1 k1 k1 kn kn ··· a1 · · · an γ(A1 , . . . , An ) , (2.1) ||γ||(A1 , . . . , An ) = sup k1

kn

where the supremum is taken over all ﬁnite partitions {Akl l }rkll =1 of the sets Al ∈ Σl and all collections {akl l }rkll =1 such that |akl l | ≤ 1. In the linear case n = 1 the semivariation and variation of a measure coincides, however this is not the case for n > 1 [Bo01]. We will deﬁne the semivariation of the polymeasure γ to be the number ||γ||(S1 , . . . , Sn ). The semivariation is a norm, also called the Fr´echet norm (even though we reserve the later terminology for the norm discussed in section 4, eq. (4.2)) and the space of polymeasures with ﬁnite semivariation is a Banach space. The semivariation of γ will be simply denoted as ||γ|| in what follows. If the polymeasure γ has ﬁnite semivariation, then for any family of bounded Σk -measurable scalar functions fk , we can deﬁne the integral: (f1 ⊗ · · · ⊗ fn ) γ, (also denoted as (f1 , . . . , fn )γ) by taking the limits of the integrals of n-tuples of simple functions uniformly converging to the fk ’s [Do87]. If we denote by Bo(Kl ) the Borel σ-algebra on the compact space Kl , a polymeasure γ on the product of the σ-algebras Bo(K1 ) × · · · × Bo(Kn ) is said to be regular if for any Borel subsets Al ⊂ Kl , l = k, the set function: γk (A) = γ(A1 , . . . , Ak−1 , A, Ak+1 , . . . , An ) is a Radon measure on Kk , k = 1, . . . , n. The space of regular countably additive polymeasures on Bo(K1 ) × · · · × Bo(Kn ) will be simply denoted by BP(K1 , . . . , Kn ) which is a Banach space equipped with the semivariation norm. We will denote by Ln (C(K1 ), . . . , C(Kn )) the space of continuous scalar n-linear maps on C(K1 ) × · · · × C(Kn ). Then we have the following theorem [Bo98]: Theorem 2.1. Let K1 , . . . , Kn be Hausdorﬀ compact topological spaces. There exists an isometric isomorphism between Ln (C(K1 ), . . . , C(Kn )) and the space of regular countably additive Borel polymeasures BP(K1 , . . . , Kn ) deﬁned on the product of the Borel σ-algebras of the Hausdorﬀ compact spaces Kl equipped with the semivariation norm. 3. Multilinear functionals and the Fourier-Stieltjes transform We will summarize here some properties of functionals on C(T) from the point of view of harmonic analysis that will be useful to address the multilinear trigono of metric problem of moments. The space of unitary irreducible representations T the group T is again an abelian group whose elements are the characters χk (z) = z k ,

with the abelian z ∈ T, k ∈ Z. Moreover χk χl = χk+l , k, l ∈ Z, hence we identify T Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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145 5

group Z. In the following we will use the parametrization of the unit circle given by z = eiθ , θ ∈ [0, 2π), thus we have χk (θ) = eikθ . Riesz representation theorem allow us to identify regular Borel measures µ on T with continuous linear functionals L on the Banach space C(T) of continuous functions on T by means of L(f ) = T f (z)dµ(z). We will say that the linear functional L is unitary if L(1) = 1. Any functional L with L(1) = 0 can be made unitary by scaling it by the complex number L(1). We will say that a linear functional L is T–invariant if L(f w ) = L(f ) for all w ∈ T, where f w denotes the function f translated by w, that is, f w (z) = f (wz), f ∈ C(T). There is a unique unitary functional which is T–invariant and that corresponds to the normalized Haar measure µ0 on T. ˆ k ) = L(χ ¯k ) The Fourier-Stieltjes transform of the functional L is deﬁned as L(χ ˆ with Z, we will consider L ˆ to be a and because of the natural identiﬁcation of T function on Z. Such function will be also denoted in what follows as µ ˆ if µ is the measure on T deﬁned by L. We can introduce a structure of Banach algebra in the space of bounded functionals on C(T) by deﬁning the convolution product L M of the functionals L and M as follows: (3.1)

(L M )(f ) = (L ⊗ M )(∆f ),

∀f ∈ C(T),

where the map ∆ : C(T) → C(T × T), given by ∆f (z, w) = f (zw) is the dual of the composition law in the abelian group T. Notice that the tensor product L ⊗ M of ˆ π C(T), which is strictly contained the functionals L and M is just deﬁned on C(T)⊗ on C(T × T), however in the case of the tensor product of two linear functionals, both its projective and injective norms coincide and L ⊗ M is continuous in the injective topology on C(T) ⊗ C(T), hence it has a unique continuous extension to ˆ C(T) ∼ C(T)⊗ = C(T × T) (see for instance [Ry02], chaps. 2,3). Moreover from eq. (3.1) we get immediately that ||L M || ≤ ||L||||M ||. If f ∈ L1 (T) we can deﬁne Lf as the linear bounded functional on C(T) deﬁned by the measure f µ0 , this is, Lf (g) = T f (θ)g(θ) µ0 (dθ). Then we deﬁne f g by the formula Lf g = Lf Lg . A simple computation shows that f g is given by the usual convolution product formula (f g)(θ) = T f (φ − θ)g(φ) µ0 (dφ). We can also easily check that (3.2)

M , L M =L

in fact:

M )(χk ) L M (χk ) = (L ⊗ M )(∆χk ) = (L ⊗ M )(χ ¯k ⊗ χ ¯ k ) = (L because ∆χk (z, w) = χk (zw) = χk (z)χk (w), hence, (3.3)

∆χk = χk ⊗ χk .

and formula (3.2) is proven. If L now denotes a continuous multilinear functional on C(T) its FourierStieltjes transform is simply deﬁned in a similar manner to the linear case as the

n ∼ function on T = Zn given by (3.4)

k , . . . , χk ) = L(χ ¯ k1 , . . . , χ ¯kn ) = L(e−ik1 θ1 , · · · , e−ikn θn ). L(χ 1 n

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If in addition there is a polymeasure γ such that L = Lγ , its Fourier–Stieltjes transform will be written as:

k , . . . , χk ) = (3.5) L(χ (e−ik1 θ1 ⊗ · · · ⊗ e−ikn θn ) γ(dθ), 1 n Tn

for all k = (k1 , . . . , kn ) ∈ Z . The Fourier–Stieltjes transform of the polymeasure γ will be also denoted as γˆ . Because trigonometric polynomials are dense in C(T), the Fourier-Stieltjes transform is an injective mapping on the space of bounded multilinear functionals on C(T). Thus, as it was discussed in the introduction, the weak multilinear trigonometric problem of moments consists in determining under what conditions there exists a regular polymeasure γ in T such that ck = Tn (e−ik1 θ1 ⊗ . . . ⊗ e−ikn θn ) γ(dθ), for all k = (k1 , . . . , kn ) ∈ Zn , for a given ndimensional sequence ck , k ∈ Zn , or, in other words, such that ck = γˆ (k). Because of the previous observations, the multilinear trigonometric problems of moments is determined, that is, if there exists a solution it is unique. Hence solving the multilinear trigonometric problem of moments amounts to determine which functions c on Zn are the Fourier–Stieltjes transform of a polymeasure. To solve this question we introduce a new notion of boundedness for functions on Zn . n

4. The weak multilinear trigonometric problem of moments As it was discussed at the end of the previous section, solving the weak multilinear trigonometric problem for the sequence of moments ck amounts to determine ˆ It is clear that the existence of a bounded multilinear functional L such that c = L. n the set of functions c on Z solving the weak multilinear trigonometric problem can be equipped with a norm by importing directly the norm of the corresponding ˆ = c. bounded multilinear functionals, this is we can deﬁne ||c||w = ||L|| where L We denote such space by BPn (Z) and it represents the space of functions on Zn which are Fourier-Stieltjes transforms of Borel regular polymeasures on Tn . In the following paragraphs we provide an alternative deﬁnition of the norm || · ||w deﬁned solely in terms of the function c and that permits an easy comparison with other natural norms on the space of functions on Zn such as the Fr´echet norm || · ||Fn (see deﬁnition 4.2 below) that could be consider as a natural candidate to characterize the elements on BPn (Z). Given a function c on Zn , we will deﬁne deﬁne a real number ||c||w as: (4.1) ||c||w = sup a · · · a c(k , . . . , k ) k1 kn 1 n , |k|≤N ||ϕl ||∞ ≤ 1, l = 1, . . . , n ϕl (θ) = akl e−ikl θ , N ∈ N with |k| = nl=1 |kl |. We shall denote by BPn (Z) the set of functions c on Zn such that ||c||w < ∞ and we will say that c ∈ BPn (Z) is weakly bounded. Moreover a given n-dimensional sequence of moments ck , k ∈ Zn , will be said to be weakly bounded if c(k) = ck ∈ BPn (Z). Next theorem shows us that the function || · ||w deﬁnes a norm and we will compare it with the norms || · ||L∞ , || · ||L1 and || · ||Fn , the Fr´echet norm deﬁned as (4.2) ||c||Fn = sup ak1 · · · akn c(k1 , . . . , kn ) . |ak |≤1,l=1,...,n |k|≤N Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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Fr´echet norms are the natural norms generalizing the notion of semivariation norm on the space of polymeasures eq, (2.1), (see [Bl01] for an exhaustive discussion on the subject). Theorem 4.1. The function || · ||w is a norm on BPn (Z) and (BPn (Z), || · ||w ) is a closed subspace of c0 (Zn ). Moreover ||c||L∞ ≤ ||c||w ≤ ||c||Fn ≤ ||c||L1 and the inequalities are strict. Proof. Notice that given j1 , . . . , jm , if we choose the family of numbers akl = δjl ,kl , k ∈ Zn , then the trigonometric polynomials ϕl (θ) = kl akl e−ikl θ are such ak1 · · · akn ck1 ···kn = cj1 ···jn . Thus ||c||L∞ ≤ ||c||w . that ||ϕl ||∞ = 1 and Let us assume that ||c||w = 0, then because of the previous observation we have that 0 ≤ ||c||L∞ ≤ ||c||w = 0 which implies that c = 0. Moreover it is obvious that ||λc||w = |λ|||c||w as well as the triangle inequality for || · ||w . Moreover it is clear that BPn (T) ⊂ c0 (Zn ) is a closed subspace. It is enough to show it when n = 1, then the polynomials φ(θ) = 2N1+1 |k|≤N eikθ are such that ||ϕ||∞ = 1 and |k|≤N ak ck = 2N1+1 |k|≤N ck will not converge unless ck goes to zero. Hence, again because of the inequality ||c||L∞ ≤ ||c||w , any Cauchy sequence on the norm || · ||w will be convergent componentwise, hence the sequence converges in BPn (T). To prove the inequality ||c||w ≤ ||c||Fn we will observe that if the trigonometric polynomial ϕ(θ) = |k|≤N ak eikθ is such that ||ϕ||∞ ≤ 1, then necessarily |ak | ≤ 1. Hence: sup a · · · a c ||c||w ≤ k1 kn k1 ···kn = ||c||Fn . |akl |≤1,N ∈N |k|≤N The last equality follows immediately from the deﬁnition of the Fr´echet norm ||·||Fn . Theorem 4.2. A sequence of multimoments ck , k ∈ Zn is a solution of the weak trigonometric multilinear moment problem if and only if ||c||w < ∞. Moreover ˆ ||c||w = ||L|| where L is a bounded multilinear functional such that c = L. Proof. If c = (ck ) is a solution of the weak multilinear trigonometric problem, then there will exist a polymeasure γ on BP(T, . . . , T) such that: ck = (e−ik1 θ1 ⊗ · · · ⊗ e−ikn θn ) γ(dθ), k = (k1 , . . . , kn ). Tn

The functional Lγ : C(T)×· · ·×C(T) → C associated to γ by Riesz multilinear theorem, Thm. 2.1, is bounded and ||Lγ || = ||γ|| where ||γ|| denotes the semivariation norm of γ. Now we have: (4.3) a · · · a c k1 kn k = |k|≤N

N N = Lγ ak1 e−ik1 θ1 , . . . , akn e−ikn θn k1 =−N

kn =−N

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N −ikl θl with ϕal = the corresponding trigonometric polynomials dekl =−N akl e ﬁned by the coeﬃcients akl . But the polynomials ϕal , l = 1, . . . , n are such that ||ϕal ||∞ ≤ 1, then ||c||w ≤ ||γ|| and ck is weakly bounded. Conversely, if c is weakly bounded, then it is easy to check that if ϕl = Nl Nl ikl θ , l = 1, · · · , n is a family of trigonometric polynomials, then: kl =−Nl akl e

(4.4)

N N1 Nn |Lc (ϕ1 , . . . , ϕn )| ≤ ak1 · · · akn Lc (eik1 θ1 , . . . , eikn θn ) = k=−N N 1 n aN aN k1 kn ··· c−k1 ···−kn ||ϕ1 ||∞ · · · ||ϕn ||∞ ≤ ≤ ||ϕ1 ||∞ ||ϕn ||∞ k=−N

≤ ||c||w ||ϕ1 ||∞ · · · ||ϕn ||∞ l because the coeﬃcients cN = aNl /||ϕl ||∞ are such that the associated trigonometric Nl kl Nl kl−ikl θ , l = 1, . . . , n, satisfy ||ψl ||∞ ≤ 1. polynomials ψl = kl =−Nl ckl e

Now it is clear from the proof of the previous theorem that if ck is weakly bounded, γ is the polymeasure deﬁned by it, ck = γˆ (k), and Lγ denotes the corresponding bounded functional, we have that ||Lγ || = ||γ|| = ||ˆ γ ||w = ||c||w , hence the Fourier–Stieltjes transform is an isometry of Banach spaces BP(Tn ) and BPn (Z). 5. Convolution product of multilinear functionals We will close the discussion started in the previous section by extending the convolution product on linear functionals discussed in §3 to the multilinear case, and proving that the multilinear Fourier–Stieltjes transform is in fact an isomorphism of Banach algebras. Before proceeding to the deﬁnition of the convolution product for multilinear functionals we will introduce some deﬁnitions and notations. Given a n–multilinear functional L : C(T) × · · · × C(T) → C, we will denote its natural extension to the tensor product C(T) ⊗ · · · ⊗ C(T) by L⊗ , this is, L⊗ (f1 ⊗ · · · ⊗ fn ) = L(f1 , . . . , fn ), for all fl ∈ C(T), l = 1, . . . , n. Now let σ ∈ Sn be an arbitrary permutation, then we will denote by Lσ⊗ the functional on C(T) ⊗ · · · ⊗ C(T), deﬁned as Lσ⊗ (f1 ⊗ · · · ⊗ fn ) = L(fσ(1) , . . . , fσ(n) ). It is clear that the multilinear functional L is bounded if and only if Lσ⊗ is continuous in the projective topology for all σ ∈ Sn . Let us consider now two continuous n–multilinear functionals L1 , L2 on T. We deﬁne L1 L2 by following the idea expressed on eq. (3.1). We will deﬁne L1 L2 as the n–multilinear functional in C(T) given by: (5.1)

(L1 L2 )⊗ (f1 ⊗ · · · ⊗ fn ) = (L1 × L2 )σ⊗ (∆(f1 ) ⊗ · · · ⊗ ∆(fn )),

for all fl ∈ C(T), where σ is the permutation in S2n given by (1 n + 1 2 n + 2 3 n + 3 · · · n 2n). However the previous expression eq. (5.1) will make sense only if, as it happens in the linear case (see the observations after eq. (3.1)), the linear ˆ C(T) on the ﬁrst two factors, functional (L1 × L2 )σ⊗ can be extended to C(T)⊗ and the same on the third and four factors, etc., so its action on ∆(f1 ), ∆(f2 ), etc. will be deﬁned. The following lemma prove that this is indeed the case. Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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Lemma 5.1. Let L and M two continuous multilinear functionals on the Banach space E. Then the linear operator (L × M )σ⊗ deﬁned on E ⊗2n with σ ∈ S2n , the permutation (1 n + 1 2 n + 2 3 n + 3 · · · n 2n), satisﬁes: |(L × M )σ⊗ (f ⊗ g ⊗ u)| ≤ ||L||||M ||||f ⊗ g|| ||u||π , for all f , g ∈ E and u = u1 ⊗ . . . ⊗ u2n−2 an homogeneous element in E ⊗2n−2 , and the norms || · || , || · ||π denote the usual injective and projective topology norms in the respective tensor product spaces. Proof. It is clear from the deﬁnitions that: |(L × M )σ⊗ (f ⊗ g ⊗ u)| = |L⊗ (f ⊗ u )M ⊗ (g ⊗ u )|, where u = u1 ⊗ u3 ⊗ . . . and u = u2 ⊗ u4 ⊗ . . .. Notice that u ⊗ u = uσ and ||u ||π ||u ||π = ||uσ ||π = ||u||π . If we denote by Lu the continuous linear functional on E deﬁned by Lu (f ) = L⊗ (f ⊗ u ) and Mu the continuous linear functional on E deﬁned in a similar way: Mu (g) = M ⊗ (g ⊗ u ), we have: (5.2)

|L⊗ (f ⊗ u )M ⊗ (g ⊗ u )| = |Lu ⊗ Mu (f ⊗ g)| ≤ ||Lu ||||Mu ||||f ⊗ g|| .

Moreover, a simple computation shows that: ||Lu || = sup |Lu (v)| ≤ ||L||||u ||π , ||v||=1

and similarly for ||Mu ||, hence substituting these expressions on eq. (5.2), we conclude with the desired result: |(L × M )σ⊗ (f ⊗ g ⊗ u)| ≤ ||L||||M ||||f ⊗ g|| ||u ||π ||u ||π . The application of Lemma 5.1 to two continuous multilinear functionals L1 , L2 ˆ C(T) ⊗ ˆ C(T) ⊗ ˆ π C(T)⊗ ˆπ ··· ∼ on C(T) allows to extend (L1 L2 )⊗ to C(T)⊗ = 2 ˆ 2 ˆ C(T )⊗π · · · ⊗π C(T ) thus making expression eq. (5.1) meaningful. The convolution product of two continuous multilinear functionals L1 , L2 could have also been written using the integral representation oﬀered by Thm. 2.1 and their corresponding polymeasures γ1 and γ2 as, (5.3) L1 L2 (f1 , . . . , fn ) = f1 (θ1 − θ1 ) ⊗ · · · ⊗ fn (θn − θn ) γ1 (dθ) γ2 (dθ ), Tn ×Tn

however the deﬁnition we have chosen and the simple results above dispense us with the cumbersome task of ﬁlling up the details of the construction of the integral calculus for polymeasures that would allow to make precise the formula above eq. (5.3). We will explore now the relation between the convolution product eq. (5.1) and the Fourier–Stieltjes transform of two multilinear functionals. Lemma 5.2. If L1 and L2 are two n–multilinear functionals on C(T) such that ˆ 1 and L ˆ 2 exist, then L L 1 L2 exists and (5.4)

L 1 L 2 = L1 · L2 .

Proof. We have: ¯ k1 , . . . , χ ¯kn ) = (L1 × L2 )σ⊗ (∆χ ¯k1 ⊗ · · · ⊗ ∆χ ¯kn ), L 1 L2 (k1 , . . . , kn ) = L1 L2 (χ Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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however we have already proved, see eq. (3.3), that ∆χk = χk ⊗ χk , hence: σ⊗ L ((χ ¯ k1 ⊗ χ ¯ k1 ) · · · ⊗ (χ ¯ kn ⊗ χ ¯kn )) = 1 L2 (k1 , . . . , kn ) = (L1 × L2 ) ¯ k1 ⊗ χ ¯ k2 ⊗ · · · ⊗ χ ¯ kn ⊗ χ ¯ k1 ⊗ χ ¯ k2 ⊗ χ ¯ k3 ⊗ · · · ⊗ χ ¯ kn ) = = (L1 ⊗ L2 )(χ ˆ ˆ ¯k ⊗ · · · ⊗ χ ¯k )L2 (χ ¯k ⊗ · · · ⊗ χ ¯k ) = L1 (k1 , . . . , kn )L2 (k1 , . . . , kn ). = L1 ( χ 1

n

1

n

Theorem 5.3. If L1 and L2 are two bounded n–multilinear functionals on C(T), then L1 L2 is a bounded n-multilinear functional on C(T) and ||L1 L2 || ≤ ||L1 ||||L2 ||. ˆ 1 ||w and ||L ˆ 2 ||w are Proof. If L1 and L2 are bounded we have that both ||L ﬁnite. On the other hand it is easy to see that ||c1 · c2 ||w ≤ ||c1 ||w ||c2 ||w for all functions c1 and c2 on Zn . In fact we have: sup ak1 · · · akn (c1 · c2 )k1 ···kn = ||c1 · c2 ||w = |k|≤N ||ϕl ||∞ ≤ 1, l = 1, . . . , n −ikl θ ϕl (θ) = akl e ,N ∈ N = sup ak1 · · · akn (c1 )k1 ···kn × |k|≤N,|l|≤N ||ϕl ||∞ ≤ 1, l = 1, . . . , n ϕl (θ) = akl e−ikl θ , N ∈ N

≤

×

sup ||ϕl ||∞ ≤ 1, l = 1, . . . , n ϕl (θ) = akl e−ikl θ , N ∈ N sup ||ϕj ||∞ ≤ 1, j = 1, . . . , n ϕj (θ) = blj e−ilj θ , N ∈ N

× δk1 ,l1 · · · · · · δkn ,ln (c2 )l1 ···ln | ≤ ak1 · · · akn (c1 )k1 ···kn × |k|≤N b · · · b (c ) l1 ln 2 l1 ···ln = ||c1 ||w ||c2 ||w . |l|≤N

ˆ ˆ ˆ ˆ Hence we have that ||L 1 L2 ||w = ||L1 L2 ||w ≤ ||L1 ||w ||L2 ||w < ∞ and because of Thm. 4.2 there exists a polymeasure associated to L1 L2 and L1 L2 is bounded. As in the linear case, to any function f ∈ L1 (Tn )we can associate the bounded multilinear functional Lf deﬁned as Lf (g1 , . . . , gn ) = Tn g1 (θ1 ) · · · gn (θn )f (θ)µ0 (dθ) where µ0 denotes now the Haar measure on Tn . Again as in the linear case we can deﬁne f γ as the polymeasure such that Lf γ = Lf Lγ and a simple computation shows that Lγf (g) = Lγ (f g). 6. An inversion formula In this section we will study the inverse map of the Fourier-Stieltjes transform on polymeasures and we will prove that the sequence of Ces` aro means of the Fourier series of a weakly bounded sequence of moments converges in the weak*– topology to a polymeasure. In fact, we recall from section 2, Thm. 2.1, that the Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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151 11

space of polymeasures on a family of compact sets Kl , l = 1, . . . , n can be idenπ ˆπ ···⊗ ˆ π C(Kn ), tiﬁed with the dual topological space of 1≤l≤n C(Kl ) = C(K1 )⊗ π hence BP(K1 , . . . , Kn ) inherits a weak topology from the subspace 1≤l≤n C(Kl ) ⊂ π ∗∗ 1≤l≤n C(Kl ) . We shall call this topology the w∗–topology on BP(K1 , . . . , Kn ). Given a bounded multilinear functional L on Tn and its moment function c on n Z , we can deﬁne the Fourier series of L (or γ) on Tn as: c(k)χk . (6.1) S(L) = k∈Zn

The series above is not pointwise convergent in general, however the Fourier series eq. (6.1) allows to reconstruct the polymeasure γ associated to the functional L in the sense that the ﬁnite approximations provided by the Ces` aro means σN (γ) =

N −1 1 SM (γ), N M =0

of the partial sums

SM (γ) =

c(k)χk ,

|k|≤M

of the Fourier series of γ, converge to γ in the weak*–topology in the space of polymeasures. More precisely, for each N ∈ N, we can deﬁne the measure on Tn , given by N −1 1 µN = c(k)χk dθ1 · · · dθn , (2π)n N M =0 |k|≤M

and we prove that µN → γ, N → ∞, in the weak*–topology. Theorem 6.1. Given a bounded multilinear functional L on Tn and its corresponding polymeasure γ, the sequence of Ces` aro means σN (γ) of L converges in the w∗–topology on BP(Tn ) to the polymeasure γ. Proof. Let γ be the polymeasure corresponding to the multilinear functional L, this is: L(f1 , . . . , fn ) = (f1 ⊗ · · · ⊗ fn ) γ. Tn

Now we consider the F´ejer summability kernel KN , N ∈ N: KN (θ) =

N −1 1 DM (θ), N M =0

where DM (θ) denotes the n-dimensional Dirichlet kernel χk (θ). DM (θ) = |k|≤M

Then a simple computation shows that: LµN = LγKN = LσN (γ) , or, abusing the notation, we can write it in the most common way: KN L =

N −1 1 SM (γ). N M =0

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Then given functions f1 , . . . , fn ∈ C(T), we can compute |(Lγ −LσN (γ) )(f1 , . . . , fn )| to get: |(Lγ − LσN (γ) )(f1 , . . . , fn )|

=

⊗ |L⊗ γ (f1 ⊗ · · · ⊗ fn ) − LσN (γ) (f1 ⊗ · · · ⊗ fn )| =

=

⊗ |L⊗ γ (f1 ⊗ · · · ⊗ fn ) − LγKN (f1 ⊗ · · · ⊗ fn )| =

=

⊗ |L⊗ γ (f1 ⊗ · · · ⊗ fn ) − Lγ (KN (f1 ⊗ · · · ⊗ fn ))| =

=

|L⊗ γ (f1 ⊗ · · · ⊗ fn − KN (f1 ⊗ · · · ⊗ fn ))| ≤

=

||γ|| ||f1 ⊗ · · · ⊗ fn − KN (f1 ⊗ · · · ⊗ fn )||,

however, because of F´ejer theorem, KN f → f in norm || · ||L∞ when N → ∞ and the convergence is proved. 7. Some examples 7.1. The linear case. Let L be a complex bounded linear functional on C(T), then we shall denote as before by c(k) the Fourier-Stieltjes transform of L. Because of Riesz representation theorem, there exists a complex measure µ on T such that L(f ) = T f dµ. Any complex measure has ﬁnite total variation ||µ||, hence we have ||L|| = ||µ|| = ||c||w . It is clear that in the linear case ||c||F1 = ||c||L1 where || · ||F1 denotes the Fr´echet norm with n = 1 deﬁned in eq. (4.2). Hence there are examples of bounded functionals such that ||c||F1 = ∞. Let us consider for instance the functional deﬁned by L(f ) = k∈S fˆ(k), where S ⊂ Z is an inﬁnite Sidon subset of Z. If S is a Sidon set, then the restriction algebra AS (T) and CS (T) coincide and in consequence |LS (f )| = | k∈S fˆ(k)| ≤ k∈S |fˆ(k)| ≤ ||fˆ||L1 and the functional ˆ k ) = 1 if k ∈ S L will be bounded. However a simple check shows that c(k) = L(χ and 0 otherwise. Thus for any inﬁnite Sidon set S we have ||c||F1 = ∞. 7.2. The bilinear case: a simple example. Let us consider a simple example that illustrates most of the precedent results. We shall consider the bilinear functional on C(T), given by: L(f, g) = fˆ(k)ˆ g (k), k∈S

where S ⊂ Z is an inﬁnite subset of the integers. It is clear that L is bounded in the norm L2 , |L(f, g)| ≤ ||f ||2 ||g||2 , thus L is bounded with respect to the uniform topology. Hence, because of the multilinear Riesz theorem, Thm. 2.1, L deﬁnes ˆ B (k). ˆ A (k)1 a regular Borel bimeasure on T given explicitely by γ(A, B) = k∈S 1 Then a simple computation shows: ˆ cmn = L(m, n) = (e−imθ ⊗ e−inψ ) γ(d(θ, ψ)) = δk+m δk+n , T2

k∈S

this is, cmn = 1 if m = n ∈ S and cmn = 0 otherwise. Because of Thm. 4.2, the function cmn on Z2 has ﬁnite weak norm || · ||w . In fact it can be shown that ||c||w = 1, however it is obvious that cmn is unbounded in the Fr´echet norm || · ||F2 . (Notice that if we select a increasing family of ﬁnite subsets SN ⊂ S, with the ˆ SN (m, n), then ||c||F2 ≥ N property SN ⊂ SN +1 , we can choose numbers amn = 1 for all N ). ˆ 1 , k2 ) = The bilinear functional L is idempotent, LL = L. Clearly because L(k 2 2 ˆ ˆ = 1∆ , 1∆S , where ∆S = {(k1 , k2 ) ∈ Z | k1 = k2 ∈ S}, we have L = L S Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

ON THE MULTILINEAR TRIGONOMETRIC PROBLEM OF MOMENTS

153 13

and because of Lemma 5.2, we conclude that L L = L. Moreover, if the family cmn were the Fourier-Stieltjes transform of a measure on T2 , such measure will be idempotent, but idempotent measures have support on the coset ring of Z2 [Ru62]. Thus any choice of an inﬁnite S that is not in the coset ring of Z2 will correspond to a bimeasure on T2 that cannot be the extension of a measure. It should be stressed that even if the sequence of moments above cmn is the Fourier-Stieltjes transform of a measure, and we take for instance S = 2Z, then its F2 norm will be inﬁnite. The arguments above can be easily extended to the n-linear case, n > 2. References Ak65. N.I. Akhiezer, The Classical Moment Problem Oliver and Boyd Ltd. 1965. Be84. C. Berg, J.P.R. Christensen, P. Ressel. Harmonic analysis on semigroups. Theory of positive deﬁnite and related functions. Springer, Berlin (1984). Bl01. R. Blei. Analysis in Integer and Fractional Dimensions. Cambridge Studies in Advanced Mathematics 71. Cambridge Univ. Press (2001). Bo98. F. Bombal, I. Villanueva, Multilinear operators on spaces of continuous functions, Funct. Approx. Comment. Math. XXVI (1998), 117-126. Bo01. F. Bombal, I. Villanueva. Integral operators on the product of C(K) spaces. J. Math. Anal. Appl., (2001) 264, 107–121. Do87. I. Dobrakov. On integration in Banach spaces VIII (polymeasures). Czech. Math. J. (1987) 37, 487–506. Fr15. M. Fr´ echet. Sur les fonctionnelles bilineares. Trans. Am. Math. Soc., (1915) 16, 215–234. Ha35. E.K. Haviland. On the momentum problem for distribution functions in more than one dimension. Am. J. Math., (1935) 57, 562-568. Ha36. E.K. Haviland. On the momentum problem for distribution functions in more than one dimension. II. Am. J. Math., (1936) 58, 164-168. ¨ He11. G. Herglotz. Uber Potenzreihen mit positivem reellen Teil im Einheitskreis. Ber. Ver. S¨ achs. GEs. d. Wiss. Leipzig., 63 501-511 (1911). Hi33. T.H. Hildebrandt, I.J. Shoenberg. On linear functional operations and the moment problem for a ﬁnite interval in one and several dimensions. Ann. Maths. (1933) 34, 317-328. Ib08. A. Ibort, P. Linares, J.G. Llavona. On the multilinear Hausdorﬀ problem of moments. Preprint (2008). Mo49. M. Morse, W. Transue. Integral representation of bilinear functionals. Proc. Natl. Acad. Sci. USA (1949) 35, 136-143. ´ Norm., 28, Ri11. F. Riesz. Sur certains syst` emes singuliers d’´ equations int´ egrales. Ann. Ec. 33-62 (1911). Ru62. W. Rudin. Fourier Analysis on Groups. Interscience Tracts in Pure and Applied Mathematics, vol 12. Inters. Publ. New York (1962). Ry02. R. Ryan. Introduction to Tensor Products of Banach Spaces. Springer Monographs in Mathematics. Springer-Verlag London (2002). ´ticas, Universidad Carlos III de Madrid, Avda. de la Departamento de Matema es, Spain Universidad 30, 28911 Legan´ E-mail address: [email protected] ´ lisis Matema ´tico, Facultad de Matema ´ticas, Universidad ComDepartamento de Ana plutense de Madrid, 28040 Madrid, Spain E-mail address: jl [email protected], [email protected]

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http://dx.doi.org/10.1090/conm/507/09958

Contemporary Mathematics Volume 507, 2010

Multiple orthogonal polynomial ensembles Arno B. J. Kuijlaars Dedicated to Guillermo L´ opez Lagomasino, on the occasion of his 60th birthday

Abstract. Multiple orthogonal polynomials are traditionally studied because of their connections to number theory and approximation theory. In recent years they were found to be connected to certain models in random matrix theory. In this paper we introduce the notion of a multiple orthogonal polynomial ensemble (MOP ensemble) and derive some of their basic properties. It is shown that Angelesco and Nikishin systems give rise to MOP ensembles and that the equilibrium problems that are associated with these systems have a natural interpretation in the context of MOP ensembles.

1. Introduction Multiple Orthogonal Polynomials (MOPs) were introduced and studied for problems in analytic number theory (irrationality and transcendence proofs). Later they appeared in approximation theory, most notably in the theory of Hermite-Pad´e approximation and in this context they are also called Hermite-Pad´e polynomials [2, 4, 21, 34, 35, 42, 43, 44, 54, 62]. MOPs were also studied from the point of view of new special functions [6, 19, 23, 51, 66, 69]. See the books [46, 61] and the survey papers [3, 8, 67, 68] for these aspects of MOPs. Further developments in these directions are reported in e.g. [7, 10, 16, 22, 24, 25, 38, 39, 52, 56]. Recently MOPs also appeared in a natural way in probability theory and mathematical physics in certain models coming from random matrix theory and nonintersecting paths. The connection was ﬁrst observed in [14] where MOPs were used in a random matrix model with external source. In the Gaussian case, the external source model has an equivalent interpretation in terms of non-intersecting Brownian motions. The external source model was further analyzed with the use of multiple Hermite and multiple Laguerre polynomials in [5, 9, 15, 17, 32, 45, 48, 63, 55, 57, 58, 71], see also [6, 9, 16, 31, 53]. A related non-intersecting path model was studied in [49] using MOPs for modiﬁed Bessel weights that were introduced earlier in [23]. The biorthogonal polynomials arising in the two matrix model were identiﬁed as MOPs in [50]. For a special case they were asymptotically 1991 Mathematics Subject Classiﬁcation. Primary 60C05; Secondary 31A15, 42C05. The author was supported in part by FWO-Flanders project G.0427.09, by K.U. Leuven research grant OT/08/33, by the Belgian Interuniversity Attraction Pole P06/02, by the European Science Foundation Program MISGAM, and by grant MTM2008-06689-C02-01 of the Spanish Ministry of Science and Innovation. 1

c 2010 American Mathematical Society

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ARNO B. J. KUIJLAARS

analyzed in [37, 59]. The Cauchy two matrix model and their associated Cauchy biorthogonal polynomials have a number of similar features [12, 13]. MOPs were generelized to MOPs of mixed type in [1, 27, 28, 29]. Asymptotic results were mainly obtained from an analysis of the RiemannHilbert problem for MOPs, formulated by Van Assche et al. [70] as an extension of the Riemann-Hilbert problem for orthogonal polynomials [41]. The application of the Deift/Zhou steepest descent analysis [30] to the Riemann-Hilbert problem for MOPs presents several interesting new features that however we will not discuss here. It is the aim of this paper to give an introductory account of MOPs from the point of view of determinantal point processes. After discussing the deﬁnition and some of the basic properties of MOPs we discuss a multiple integral representations for the type II MOPs, which is essentially taken from [14]. Under a suitable constant sign condition the formula can be interpreted as the expectation value of the random polynomial j=1 (z − xj ) with roots x1 , . . . , xn from a determinantal point process (called a MOP ensemble) on the real line. The constant sign condition holds in particular for Angelesco and Nikishin systems. For both of these systems we show that the joint probability density function (p.d.f.) of the associated MOP ensemble takes on a particular nice form. In the large n limit it allows for a natural probabilistic interpretation of the vector equilibrium problems that are associated with Angelesco and Nikishin systems. 2. Multiple orthogonal polynomials 2.1. Deﬁnitions. Given weight functions w1 , . . . , wp on R and a multi-index n = (n1 , . . . , np ) ∈ Np , the type II MOP is a monic polynomial Pn of degree |n| = n1 + · · · + np such that ∞ Pn (x)xk wj (x)dx = 0, k = 0, . . . , nj − 1, j = 1, . . . , p. (2.1) −∞

Throughout we will write n = |n| = n1 + · · · + np . The conditions (2.1) give a system of n linear equations for the n free coeﬃcients of the polynomial Pn (recall that Pn is monic). If the system has a unique solution we say that the multi-index n is normal (with respect to the weights w1 , . . . , wp ). In this paper we mainly deal with the type II MOP, but at times it is useful (j) to consider the dual notion of type I MOPs as well. These are polynomials An , (j) j = 1, . . . , p, of degrees deg An ≤ nj − 1, such that the linear form Qn (x) =

(2.2)

p

(j)

An (x)wj (x)

j=1

satisﬁes

∞

(2.3) −∞

xk Qn (x)dx = 0,

k = 0, 1, . . . , n − 2.

If we supplement this with the normalizing condition ∞ xn−1 Qn (x)dx = 1, (2.4) −∞

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then again we have a system of n = |n| linear equations for the in total n coeﬃcients (j) of the polynomials An , j = 1, . . . , p. 2.2. Determinantal expressions. Let ∞ (j) ck = xk wj (x)dx −∞

denote the kth moment of the weight wj , and let (j) (j) Hm,n = ck+l

k=0,...,m,l=0,...,n

be the (m+1)×(n+1) Hankel matrix with the moments of wj . The conditions (2.3) and (2.4) give rise to a linear system whose matrix has the block Hankel structure (1) (2) (p) (2.5) Mn = Hn−1,n . H · · · H n−1,n n−1,n −1 −1 −1 1 2 p Therefore the type I MOPs uniquely exist if and only if (1) (2) (2.6) Dn := det Mn = Hn−1,n Hn−1,n2 −1 · · · 1 −1

(p) Hn−1,np −1 = 0.

The linear system arising from the type II conditions (2.1) has a matrix which is the transpose of (2.5). Therefore the non-vanishing of the determinant (2.6) also guarantees the existence and uniqueness of the type II MOP. Suppose Dn = 0. Then it is easy to see that the type II MOP has the determinantal formula 1 x 1 (1) (2) (p) (2.7) Pn (x) = Hn,n2 −1 · · · Hn,np −1 x2 . H Dn n,n1 −1 .. . xn Indeed, the right-hand side of (2.7) is a monic polynomial of degree n. If we multiply the right-hand side of (2.7) by xk wj (x) and integrate with respect to x, we can perform these operations in the last column to obtain a determinant with two equal columns if k ≤ nj − 1. This proves the type II orthogonality conditions (2.1). The type I MOPs have a similar determinantal expression. For j = 1, . . . , p we have 1 (j) (2.8) An (x) = × Dn (j−1) H (1) n−2,n1 −1 · · · Hn−2,nj−1 −1 0 ··· 0

Hn−2,nj −1

(j)

Hn−2,nj+1 −1

(j+1)

···

1 x · · · xnj −1

0

···

(p) Hn−2,np −1 . 0

These and similar determinantal formulas have recently been considered from the point of view of integrable systems in [1, 11]. Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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2.3. Multiple integral representation. For what follows it is convenient to write j Nj = ni , N0 = 0 i=1

and to introduce two sequences of functions f1 , . . . , fn and g1 , . . . , gn by fj (x) = xj−1 ,

(2.9)

j = 1, . . . , n

and (2.10)

gi+Nj−1 (x) = xi−1 wj (x),

i = 1, . . . , nj ,

j = 1, . . . , p.

Then the block Hankel matrix (2.5) can be written as ∞

mj,k = fj (x)gk (x) dx (2.11) Mn = mj,k j,k=1,...,n , −∞

and Dn = det Mn = det

∞ −∞

fj (x)gk (x) dx j,k=1,...,n .

For general m and n = |n| we also write

(2.12) Mm,n = mj,k j=1,...,m,k=1,...,n , so that we have by (2.7) (2.13)

1 Pn (x) = Mn+1,n Dn

1 x .. . xn

and by (2.8) and (2.10) (2.14)

Qn (x) =

p

(j) An (x)wj (x)

j=1

1 = Dn

Mn−1,n g1 (x) g2 (x) · · · gn (x) .

The following lemma is standard, see e.g. [47, Proposition 2.10] where it is called a generalized Cauchy-Binet identity. Lemma 2.1. We have ∞ n

1 ∞ (2.15) Dn = ··· det fj (xk ) j,k=1,...,n · det gj (xk ) j,k=1,...,n dxk . n! −∞ −∞ k=1

Proof. Expanding the two determinants on the right-hand side of (2.15) we get n

(−1)sgn σ+sgn τ fσ(k) (xk )gτ (k) (xk ) det fj (xk ) j,k · det gj (xk ) j,k = σ

τ

k=1

where the sums are for σ and τ over the symmetric group Sn . By (2.11) the right-hand side of (2.15) is equal to (2.16) n n −1 1 1 (−1)sgn σ+sgn τ mσ(k),τ (k) = (−1)sgn(σ◦τ ) mσ◦τ −1 (k),k . n! σ τ n! σ τ k=1

k=1

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For any ﬁxed σ, we have that σ ◦ τ −1 runs through Sn as τ runs through Sn . Hence

(2.17)

(−1)sgn(σ◦τ

−1

τ

)

n

mσ◦τ −1 (k),k = det Mn = Dn .

k=1

The equality (2.15) follows from (2.16) and (2.17).

There is a similar multiple integral representation for the type II MOPs, which was stated for a special case in [14], see also [33]. We emphasize that it is important here that fj (x) = xj−1 . Proposition 2.2. Assume Dn = 0. Then the type II MOP has the multiple integral representation (2.18)

∞

−∞

1 × Dn · n! n n

(z − xk ) · det fj (xk ) j,k=1,...,n · det gj (xk ) j,k=1,...,n dxk .

Pn (z) = ···

∞

−∞ k=1

k=1

Proof. Since fj (x) = x minant, and therefore n

j−1

we have that det [fj (xk )] is a Vandermonde deter-

(z − xk ) · det fj (xk ) j,k=1,...,n = det fj (xk ) j,k=1,...,n+1

k=1

where we have put fn+1 (x) = xn ,

and

xn+1 = z.

Thus, by expanding the determinant we have n n

(z − xk ) · det fj (xk ) j,k=1,...,n = (−1)sgn σ fσ(k) (xk ) · fσ(n+1) (z) σ∈Sn+1

k=1

k=1

and similarly n

det gj (xk ) j,k=1,...,n = (−1)sgn τ gτ (k) (xk ). τ ∈Sn

k=1

Integrating the product of the two above expressions with respect to x1 , . . . , xn we obtain (2.19) ∞ −∞

···

∞

n

n

(z − xk ) · det fj (xk ) j,k=1,...,n · det gj (xk ) j,k=1,...,n dxk

−∞ k=1

=

k=1

(−1)sgn σ+sgn τ

σ∈Sn+1 τ ∈Sn

=

n

mσ(k),τ (k) · fσ(n+1) (z)

k=1

τ ∈Sn σ∈Sn+1

(−1)sgn(σ◦τ

−1

)

n

mσ◦τ −1 (k),k · z σ(n+1)−1 ,

k=1

where we used the deﬁnition of mj,k as given in (2.11) also for j = n + 1. For each ﬁxed τ ∈ Sn we have that the sum over σ in (2.19) is equal to the determinant in the right-hand side of (2.13) and the proposition follows. Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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In an analogous way we ﬁnd the following multiple integral representation for the linear form of type I MOPs, which is due to Desrosiers and Forrester [33]. Proposition 2.3. Assume Dn = 0. Then the linear form of type I MOPs satisﬁes ∞ Qn (x) 1 dx = × (2.20) z − x D n · n! −∞ ∞ ∞ n n

··· (z − xk )−1 · det fj (xk ) j,k=1,...,n · det gj (xk ) j,k=1,...,n dxk . −∞

−∞ k=1

k=1

Proof. Here we use the property n

(z − xk )−1 · det fj (xk ) j,k=1,...,n = det fj (xk ) j,k=1,...,n+1

k=1

where now we put 1 , and xn+1 = z. z−x The rest of the proof follows along the same lines as the proof of Proposition 2.2. We omit the details, see also [33]. fn+1 (x) =

3. MOP ensembles 3.1. Probabilistic interpretation. The multiple integral representations (2.15), (2.18) and (2.20) have a natural probabilistic interpretation in case the product of determinants

det fj (xk ) j,k=1,...,n · det gj (xk ) j,k=1,...,n is of a ﬁxed sign for (x1 , . . . , xn ) ∈ Rn . That is, if it is always ≥ 0 or always ≤ 0. Indeed, in that case it follows by (2.15) that 1 det [fj (xk )]j,k=1,...,n · det [gj (xk )]j,k=1,...,n Zn is a probability density function on Rn , where (3.1)

P(x1 , . . . , xn ) =

(3.2)

Zn = Dn n!

is the normalizing constant (also called partition function in statistical mechanics literature), so that · · · P(x1 , . . . , xn )dx1 · · · dxn = 1. The multiple integral representations (2.18) and (2.20) then show that

n Pn (z) = E (3.3) (z − xk ) , z ∈ C, k=1 n

(3.4)

Qn (z) = E

(z − xk )

−1

,

z ∈ C \ R,

k=1

where the mathematical expectation is taken with respect to the p.d.f. (3.1). Thus Pn (z) is the average of the polynomials nk=1 (z − xk ) where the roots x1 , . . . , xn are distributed according to (3.1). In cases where the distribution (3.1) can be interpreted as the eigenvalue distribution of a random matrix ensembles, one would call Pn the average characteristic polynomial. Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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161 7

3.2. Biorthogonal ensembles. A biorthogonal ensemble, see [18], is a probability density function on Rn of the form (3.1) with certain given functions f1 , . . . , fn and g1 , . . . , gn , not necessarily of the form (2.9) and (2.10). The p.d.f. is invariant under permutations of variables. We think of the ensemble as giving us n random points or particles xj on the real line, and so it is a random point process. A biorthogonal ensemble is a special case of a determinantal point process, see e.g. [47, 65], This means that there is a correlation kernel Kn (x, y) so that P(x1 , . . . , xn ) =

1 det [Kn (xj , xk )]j,k=1,...,n n!

and so that marginal densities (m point correlation functions) are determinants ∞ ∞ (n − m)! det [Kn (xj , xk )]j,k=1,...,m . ··· P(x1 , . . . , xn )dxm+1 · · · dxn = n! −∞ −∞ n−m times

Taking for example m = 1 we have that n1 Kn (x, x) is the mean density of points, that is 1 b Kn (x, x)dx n a is the expected fraction of points lying in the interval [a, b]. In a biorthogonal ensemble, the correlation kernel can be written as a bordered determinant f1 (x) .. −1 Mn . (3.5) Kn (x, y) = det Mn fn (x) g1 (y) · · · gn (y) 0 where Mn is the matrix

Mn = mj,k

, j,k=1,...,n

mj,k =

∞

−∞

fj (x)gk (x) dx

In the formulation of the biorthogonal ensemble (3.1), we have some freedom in choosing the functions f1 , . . . , fn . and g1 , . . . , gn . Indeed, if φ1 , . . . , φn and ψ1 , . . . , ψn are functions with the same linear span as the fj ’s and gj ’s, respectively, then we could use these functions instead. A particular nice form appears if the functions φj and ψk are biorthogonal, i.e., ∞ φj (x)ψk (x)dx = δj,k . −∞

Then the representation (3.5) reduces to In Kn (x, y) = − (3.6) ψ1 (y) · · · ψn (y)

φ1 (x) n .. . = φj (x)ψj (y). φn (x) j=1 0

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3.3. OP ensembles. If fj (x) = gj (x) = xj−1 non-negative weight function w on R, then (3.7)

1 1 det [fj (xk )] · det [gj (xk )] = Zn Zn

w(x), j = 1, . . . , n for some

(xk − xj )2 ·

1≤j 0, such that uniformly in n and for ξ with dist (ξ, J) ≤ L/n,

(1.11) Kn(1,0) (ξ, ξ) = o n2 . If we assume also w is continuous in J, there is the additional equivalence: (V) For each ﬁxed j, we have uniformly in ξ ∈ J, (1.12)

˜ n (ξ, ξ) = 1. lim (tn,j+1 (ξ) − tn,j (ξ)) K

n→∞

Thus, universality is equivalent to asymptotics for derivatives of the repro(1,0) ducing kernels, or just weak growth estimates on Kn . Moreover, when w is continuous, universality is equivalent to “clock spacing” of zeros of the reproducing kernel, in the terminology of Barry Simon. See the papers [13], [36], [42] for more details on the relation between universality and varying assumptions on zero spacing. We prove Theorem 1.3 in Section 2. 2. Proofs Throughout, we assume the hypotheses of Theorem 1.3. In the sequel C, C1 , C2 , . . . denote constants independent of n, x, y, s, t. The same symbol does not necessarily denote the same constant in diﬀerent occurrences. We shall write C = C (α) or C = C (α) to respectively denote dependence on, or independence of, the parameter α. We use ∼ in the following sense: given real sequences {cn }, {dn }, we write cn ∼ dn if there exist positive constants C1 , C2 with C1 ≤ cn /dn ≤ C2 . Similar notation is used for functions and sequences of functions. Let {ξn } denote a sequence in J, and for n ≥ 1, Kn ξn + K (ξa ,ξ ) , ξn + K (ξb ,ξ ) n n n n n n (2.1) fn (a, b) = . Kn (ξn , ξn ) As noted above, the equivalence of (I) and (II) is the main result of [22]. We shall prove (II) ⇐⇒ (III); (I) ⇐⇒ (IV) and (II) ⇐⇒ (V). We begin by summarizing some results from [22]. Recall too that the exponential type A of an entire function g is

log max|z|=r |g (z)| A = lim sup , r r→∞ if this number is ﬁnite. Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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Lemma 2.1. (a) {fn }∞ n=1 is uniformly bounded in compact subsets of C. Let f (a, b) be the limit of some subsequence {fn (·, ·)}n∈S of {fn (·, ·)}∞ n=1 . It is an entire function of exponential type in a, b, that satisﬁes for all complex a, b, |f (a, b)| ≤ C1 eC2 (|Im a|+|Im b|) .

(2.2) (b) For all a ∈ C,

∞

2

|f (a, s)| ds ≤ f (a, a ¯) .

(2.3) −∞

(c) Let σ be the exponential type of f (a, ·). This is independent of a ∈ R, and (2.4)

σ = π sup f (x, x) . x∈R

(d) For real a, the function f (a, ·)has only real zeros. Proof. (a) This is Lemma 5.2(a) and (b) in [22]. (b) This is Lemma 5.3(b) in [22]. (c) This is Lemmas 6.1 and 6.4 in [22]. (d) This is Lemma 5.2(c) in [22]. Proof of (II) ⇒ (III). This is similar to Corollary 1.3 in [19], and generalizes that corollary. Expanding fn as a double Taylor series gives

r

s ∞ (r,s) Kn (ξn , ξn ) a b . fn (a, b) = n (ξn , ξn ) n (ξn , ξn ) r!s!Kn (ξn , ξn ) K K r,s=0 By using the Maclaurin series of sin and the binomial theorem, we see that ∞ r s (aπ) (bπ) sin π (a − b) = τr,s . π (a − b) r!s! r,s=0

Thus ∞ ar bs sin π (a − b) = fn (a, b) − π (a − b) r!s! r,s=0

(r,s)

Kn

(ξn , ξn )

n (ξn , ξn )r+s Kn (ξn , ξn ) K

−π

r+s

τr,s

.

π(a−b) Since our hypothesis is that fn (a, b) − sinπ(a−b) converges uniformly to 0 for a, b in compact subsets of the plane, we deduce that for each ﬁxed r, s ≥ 0, (r,s)

lim

n→∞

Kn

(ξn , ξn ) = π r+s τr,s . r+s Kn (ξn , ξn ) Kn (ξn , ξn )

Since {ξn } is any sequence in J, we have shown that uniformly for ξ ∈ J, (r,s)

lim

n→∞

Kn

(ξ, ξ)

n (ξ, ξ)r+s Kn (ξ, ξ) K

˜ n(r,s) (ξ, ξ) K = π r+s τr,s . n→∞ K n (ξ, ξ)r+s+1

= lim

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Proof of (III) ⇒ (II). We essentially retrack the steps of the proof of sin π(a−b) is a sequence of analytic (II) ⇒ (III). By Lemma 2.1(a), fn (a, b) − π(a−b) n≥1

functions that is uniformly bounded in compact subsets of the plane. Moreover our hypothesis is that individual Maclaurin series coeﬃcients in the double series converge to 0 as n → ∞. Classical complex analysis then shows that sin π (a − b) lim fn (a, b) − =0 n→∞ π (a − b) uniformly in compact subsets of the plane. Since {ξn } in the deﬁnition of fn is any sequence in J, we obtain the stated uniformity in ξ in (1.6). We note that (III) does not immediately imply (IV) because the latter involves points that lie outside J. Proof of (I) ⇒ (IV). Since we assumed that w = µ ∼ 1 uniformly in a neighborhood of J, a standard estimate for Christoﬀel functions [30, Theorem 20, p. 116] gives uniformly for ξ in a neighborhood of J, (2.5)

n (ξ, ξ) ∼ Kn (ξ, ξ) ∼ n. K

We may then reformulate our hypothesis (1.5) as lim fn (a, a) = 1,

n→∞

uniformly for a in compact subsets of the real line. By the uniform boundedness in Lemma 2.1(a), this extends to uniformity for a in compact subsets of the plane. Then also d lim fn (a, a) = 0, n→∞ da uniformly for a in compact subsets of the plane, that is uniformly in such a, (1,0) ξn + K (ξa ,ξ ) , ξn + K (ξa ,ξ ) 2Kn n n n n n n = 0. lim n→∞ Kn (ξn , ξn ) Kn (ξn , ξn ) The result then follows from (2.5) and the fact that {ξn } is any sequence in J, while we may allow |a| ≤ L. Proof of (IV) ⇒ (I). Let |a| ≤ L with a real. By the Mean Value Theorem, for some t between ξ and ξ + na ,

Kn ξ + na , ξ + na 1 a ∂ a 1 −1 = (Kn (t, t)) = 2K (1,0) (t, t) = o (1) , Kn (ξ, ξ) Kn (ξ, ξ) n ∂t Kn (ξ, ξ) n n uniformly in ξ ∈ J by our hypothesis and (2.5). It then follows that for some R > 0, we have lim fn (a, a) = 1, n→∞

uniformly for |a| ≤ R. In view of the uniform boundedness of {fn }, convergence continuation theorems gives this for all real (and even complex) a. Since {ξn } in the deﬁnition of {fn } is any sequence in J, (1.5) follows uniformly for ξ ∈ J. The most diﬃcult equivalence concerns the spacing of the zeros: Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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Proof of (II) ⇒ (V). It is now a well established fact, ﬁrst observed by the ﬁrst author in [15], that the universality (1.7) implies “clock spacing” of zeros. The latter is a phrase coined by Barry Simon. It has been analyzed in a number of contexts, and in weaker and stronger forms, by Last, Simon and others (see [13], [36]). Although established in [15] only for the zeros of the orthogonal polynomials (that is the special case where ξ is a zero of pn ), the exact same proof works for the case stated here. The main idea is that the uniform convergence in (1.6) and Hurwitz’s theorem imply that as n → ∞, the zeros of Kn ξn , ξn + K (ξz ,ξ ) n

n

n

converge to those of sinπzπz . Because {ξn } is any sequence in J, we obtain the stated uniformity in ξ. To prove (V) ⇒ (II), we shall need: Lemma 2.2. Assume the hypotheses of Theorem 1.2. Assume also that if {ξn } is a sequence in J, then for each ﬁxed j, tn,j+1 (ξn ) − tn,j (ξn ) (2.6) lim = 1. n→∞ tn,1 (ξn ) − tn,0 (ξn ) Let f be as in the previous lemma. Then ∞ π π sin (σz − kπ) sin (σw − kπ) (2.7) f (z, w) = . f k ,k σ σ σz − kπ σw − kπ k=−∞

Moreover, if for each k ∈ Z, (2.8)

π π = 1, f k ,k σ σ

then (2.9)

f (z, w) =

sin π (z − w) . π (z − w)

Remark. Barry Simon calls the limit (2.6) “weak clock” behavior. Proof. Let {ρj }j=0 denote the zeros of f (0, z) in increasing order, and let ρ0 = 0. By Hurwitz’ Theorem, ρj =

lim

n→∞,n∈S

ρj,n ,

where {ρj,n } are the zeros of fn (0, z), appropriately ordered. Note that with an appropriate ordering, ˜ n (ξn , ξn ) (tj,n − ξn ) . ρj,n = K Then our spacing assumption (2.6) gives, perhaps with a reindexation of the zeros, ρj+1 − ρj tn,j+1 (ξn ) − tn,j (ξn ) = 1. = lim n→∞,n∈S ρ 1 − ρ0 tn,1 (ξn ) − tn,0 (ξn ) Thus setting ∆ = ρ1 , and recalling ρ0 = 0, we have ρj = j∆, j ∈ Z. Note too that if j = , then fn (ρj,n , ρ,n ) =

Kn (tj,n , t,n ) =0 Kn (ξn , ξn )

so for j = , (2.10)

f (j∆, ∆) = f (ρj , ρ ) = 0.

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The spacing assumption (2.6) ensures that all zeros {ρj }j=0 = {j∆}j=0 are simple zeros of f (0, z). Next, let π sin ∆ z . g (z) = f (0, z) / π z ∆ This is entire, and has no zeros, and satisﬁes g (0) = f (0, 0) = 1. Moreover, it is a ratio of entire functions of exponential type, so has exponential type. By the Hadamard factorization theorem, it must have the form g (z) = eCz , for some constant C. Since g is real valued on the real line, C must be real. But then for all j ∈ Z, 1 eC∆(j+ 2 ) f 0, ∆ j + 1 = . 2 π j + 1 2

This contradicts the fact that f (0, ·) is bounded on the real axis unless C = 0. Thus sin π z f (0, z) = π ∆ , ∆z π and in particular, the exponential type of f (0, ·), which we called σ, equals ∆ . π By Lemma 2.1(c), for any real a, f (a, ·) then has exponential type ∆ . Since also f (a, ·) ∈ L2 (R), (recall Lemma 2.1(b)), we can apply the cardinal series expansion [39, p. 91] π

∞ sin ∆ z − kπ f (a, z) = . f (a, k∆) π ∆ z − kπ k=−∞

π In turn, f (·, k∆) is an entire function of exponential type ∆ that belongs to L2 (R), so applying the cardinal series expansion again, gives ⎡ π

⎤

π ∞ ∞ sin ∆ a − jπ z − kπ sin ∆ ⎣ ⎦ f (a, z) = . f (j∆, k∆) π π ∆ a − jπ ∆ z − kπ j=−∞ k=−∞

In view of (2.10), we obtain for all real a, and all complex z, π

π ∞ sin ∆ a − kπ sin ∆ z − kπ f (a, z) = . f (k∆, k∆) π π ∆ a − kπ ∆ z − kπ k=−∞

By analytic continuation, this extends to all complex a as well. Recalling that π ∆ = σ, we obtain (2.7). Finally if (2.8) holds, then ∞ sin (σa − kπ) sin (σz − kπ) sin σ (a − z) = , f (a, z) = σa − kπ σz − kπ σ (a − z) k=−∞

by applying this identity to the special function all real x, f (x, x) = 1,

sin σ(a−z) σ(a−z) .

In particular, then for

so by (2.4), σ = π supx∈R f (x, x) = π and (2.9) also follows.

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Proof of (V) ⇒ (II). Our hypothesis (1.12) implies the weak clock spacing (2.6)

of Lemma 2.2. The result then follows if we can show that for all k, f k πσ , k πσ = 1. Our hypothesis shows that for each sequence {ξn } in J, and each ﬁxed j, ˜ n (ξn , ξn ) = 1. lim (tn,j+1 (ξn ) − tn,j (ξn )) K

(2.11)

n→∞

Fix an integer , and set ξn = tn, (ξn ) . It then follows that, as sets, {tjn (ξn )}j = {tjn (ξn )}j . Our hypothesis (1.12) gives ˜ n (ξ , ξ ) = 1, lim (tn,1 (ξn ) − tn,0 (ξn )) K n n

n→∞

or equivalently ˜ n (ξn , ξn ) = 1. lim (tn,+1 (ξn ) − tn, (ξn )) K

n→∞

Together, this and (2.11) give ˜ n (tn, (ξn ) , tn, (ξn )) ˜ n (ξn , ξn ) K K = lim = 1, ˜ n (ξn , ξn ) ˜ n (ξn , ξn ) n→∞ n→∞ K K lim

or equivalently w (tn, (ξn )) fn (ρ,n , ρ,n ) = 1, n→∞ w (ξn ) lim

and hence for each , f (ρ , ρ ) = 1. Since in our earlier notation, ρj = j∆ = j πσ , we have (2.8). As the limit function π(z−w) f (z, w) = sinπ(z−w) is independent of the subsequence {fn }n∈S from which f was formed, the result now follows from the previous lemma. References [1] A. Avila, J. Last, and B. Simon, Bulk universality and clock spacing of zeros for ergodic Jacobi matrices with a.c. spectrum, submitted [2] J. Baik, T. Kriecherbauer, K. T-R. McLaughlin, P.D. Miller, Uniform Asymptotics for Polynomials Orthogonal with respect to a General Class of Discrete Weights and Universality Results for Associated Ensembles, Princeton Annals of Mathematics Studies, 2006. [3] P. Deift, Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach, Courant Institute Lecture Notes, Vol. 3, New York University Pres, New York, 1999. [4] P. Desrosiers, P. J. Forrester, Hermite and Laguerre β-ensembles: asymptotic corrections to the eigenvalue density, Nuclear Phys. B 743 (2006), no. 3, 307–332. [5] M. Findley, Universality for Regular Measures satisfying Szeg˝ o’s Condition, J. Approx. Theory, 155 (2008), 136–154. [6] P. J. Forrester, Spacing distributions in random matrix ensembles, in Recent perspectives in random matrix theory and number theory, London Math. Soc. Lecture Note Ser., 322, Cambridge Univ. Press, Cambridge, 2005, pp. 279–307. [7] P. J. Forrester, Log-gases and Random matrices, online book, http://www.ms.unimelb.edu.au/˜matpjf/matpjf.html . [8] G. Freud, Orthogonal Polynomials, Pergamon Press/ Akademiai Kiado, Budapest, 1971. [9] T. M. Garoni, P. J. Forrester, N. E. Frankel, Asymptotic corrections to the eigenvalue density of the GUE and LUE, J. Math. Phys., 46 (2005), no. 10, 103301, 17 pp.

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[10] A. R. Its, A. B. Kuijlaars, and J. Ostensson, Critical edge behavior in unitary random matrix ensembles and the thirty fourth Painlev´ e transcendent, International Maths. Research Notices, 2008, article ID rnn017, 67 pages. [11] A. B. Kuijlaars and M. Vanlessen, Universality for Eigenvalue Correlations from the Modiﬁed Jacobi Unitary Ensemble, International Maths. Research Notices, 30(2002), 1575– 1600. [12] A. B. Kuijlaars and M. Vanlessen, Universality for eigenvalue correlations at the origin of the spectrum, Comm. Math. Phys. 243 (2003), 163–191. [13] Y. Last and B. Simon, Fine Structure of the Zeros of Orthogonal Polynomials IV. A Priori Bounds and Clock Behavior, Comm. Pure Appl. Math. 61 (2008), 486–538. [14] Eli Levin and D. S. Lubinsky, Universality Limits for Exponential Weights, Constructive Approximation, 29 (2009), 247–275. [15] Eli Levin and D. S. Lubinsky, Applications of Universality Limits to Zeros and Reproducing Kernels of Orthogonal Polynomials, Journal of Approximation Theory, 150(2008), 69–95. [16] Eli Levin and D. S. Lubinsky, Universality Limits Involving Orthogonal Polynomials on the Unit Circle, Computational Methods and Function Theory, 7(2007), 543–561. [17] Eli Levin and D. S. Lubinsky, Universality Limits in the Bulk for Varying Measures, Advances in Mathematics, 219(2008), 743–779. [18] Eli Levin and D. S. Lubinsky, Universality Limits at the Soft Edge of the Spectrum via Classical Complex Analysis, manuscript. [19] D. S. Lubinsky, A New Approach to Universality Limits involving Orthogonal Polynomials, to appear in Annals of Mathematics. [20] D. S. Lubinsky, A New Approach to Universality Limits at the Edge of the Spectrum, Contemporary Mathematics (60th Birthday of Percy Deift), 458(2008), 281–290. [21] D. S. Lubinsky, Mutually Regular Measures have Similar Universality Limits, (in Proceedings of Twelfth Texas Conference on Approximation Theory (eds. M. Neamtu, L. Schumaker), Nashboro Press, Nashville 2008, pp. 256–269. [22] D. S. Lubinsky, Universality Limits in the Bulk for Arbitrary Measures with Compact Support, J. d’ Analyse de Mathematique, 106(2008), 373–394. [23] D. S. Lubinsky, Universality Limits at the Hard Edge of the Spectrum for Measures with Compact Support, International Mathematics Research Notices, (2008) 2008: rnn099-39. [24] D. S. Lubinsky, Universality Limits for Random Matrices and de Branges Spaces of Entire Functions, Journal of Functional Analysis, 256 (2009), 3688–3729. [25] A. Mate, P. Nevai, V. Totik, Szego’s Extremum Problem on the Unit Circle, Annals of Math., 134(1991), 433–453. [26] K. T.-R. McLaughlin, Asymptotic analysis of random matrices with external source and a family of algebraic curves, Nonlinearity 20 (2007), 1547–1571. [27] K. T.-R. McLaughlin and P. Miller, The ∂¯ steepest descent method and the asymptotic behavior of polynomials orthogonal on the unit circle with ﬁxed and exponentially varying nonanalytic weights, Intern. Math. Res. Papers, 2006, Article ID 48673, pages 1–78. [28] K. T.-R. McLaughlin and P. Miller, The ∂¯ steepest descent method for orthogonal polynomials on the real line with varying weights, Intern. Math. Res. Notices, 2008, Article ID rnn075, pages 1–66. [29] M. L. Mehta, Random Matrices, 2nd edn., Academic Press, Boston, 1991. [30] P. Nevai, Orthogonal Polynomials, Memoirs of the AMS no. 213 (1979). [31] P. Nevai, Geza Freud, Orthogonal Polynomials and Christoﬀel Functions: A Case Study, J. Approx. Theory, 48(1986), 3–167. [32] L. Pastur, From random matrices to quasi-periodic Jacobi matrices via orthogonal polynomials, J. Approx. Theory 139 (2006), 269–292. [33] L. Pastur and M. Shcherbina, Universality of the local eigenvalue statistics for a class of Unitary Invariant Random Matrix Ensembles, J. Statistical Physics, 86(1997), 109–147. [34] B. Simon, Orthogonal Polynomials on the Unit Circle, Parts 1 and 2, American Mathematical Society, Providence, 2005. [35] B. Simon, Two Extensions of Lubinsky’s Universality Theorem, Journal d’Analyse de Mathematique, 105 (2008), 345–362.

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ELI LEVIN AND DORON S. LUBINSKY

[36] B. Simon, The Christoﬀel-Darboux Kernel, Perspectives in PDE, Harmonic Analysis and Applications, Proceedings of Symposia in Pure and Applied Mathematics, 79 (2008), 295– 335. [37] A. Soshnikov, Universality at the Edge of the Spectrum in Wigner Random Matrices, Comm. Math. Phys., 207(1999), 697–733. [38] H. Stahl and V. Totik, General Orthogonal Polynomials, Cambridge University Press, Cambridge, 1992. [39] F. Stenger, Numerical Methods Based on Sinc and Analytic Functions, Springer, New York, 1993. [40] T. Tao, V. Vu, From the Littlewood-Oﬀord Problem to the Circular Law: Universality of the Spectral Distribution of Random Matrices, Bull. Amer. Math. Soc., 46 (2009), 377–396. [41] V. Totik, Asymptotics for Christoﬀel Functions for General Measures on the Real Line, J. d’Analyse Math., 81(2000), 283–303. [42] V. Totik, Universality and ﬁne zero spacing on general sets, to appear in Arkiv for Matematik. [43] C. Tracy, H. L. Widom, Universality of the distribution functions of random matrix theory, in Statistical physics on the eve of the 21st century, Ser. Adv. Statist. Mech., 14, World Sci. Publ., River Edge, NJ, 1999, pp. 230–239. [44] C. Tracy, H. L. Widom, Universality of the distribution functions of random matrix theory, in Integrable systems: from classical to quantum (Montr´eal, QC, 1999), CRM Proc. Lecture Notes, 26, Amer. Math. Soc., Providence, RI, 2000, pp.251–264. [45] C. Tracy, H. L. Widom, The Pearcey process, Communications in Mathematical Physics, 263 (2006), no. 2, 381–400. [46] M. Vanlessen, Strong Asymptotics of Lageurre-type Orthogonal Polynomials and Applications in Random Matrix Theory, Constr. Approx., 25(2007), 125–175. [47] H. Widom, Toeplitz determinants, random matrices and random permutations, in Toeplitz matrices and singular integral equations (Pobershau, 2001), Birkh¨ auser, Basel, 2002, pp. 317–328. Mathematics Department, The Open University of Israel, P.O. Box 808, Raanana 43107, Israel E-mail address: [email protected] School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA E-mail address: [email protected]

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http://dx.doi.org/10.1090/conm/507/09960

Contemporary Mathematics Volume 507, 2010

Greedy energy points with external ﬁelds A. L´opez Garc´ıa Dedicated to my father, G. L´ opez Lagomasino, on the occasion of his 60th birthday.

Abstract. In this paper we introduce several extremal sequences of points on locally compact metric spaces and study their asymptotic properties. These sequences are deﬁned through a greedy algorithm by minimizing a certain energy functional whose expression involves an external ﬁeld. Some results are also obtained in the context of Euclidian spaces Rp , p ≥ 2. As a particular example, given a closed set A ⊂ Rp , a lower semicontinuous function f : Rp → (−∞, +∞] and an integer m ≥ 2, we investigate (under suitable conditions on A and f ) sequences {ai }∞ 1 ⊂ A that are constructed inductively by selecting the ﬁrst m points a1 , . . . , am so that the functional 1≤i 0, s = 0.

We shall use the notations Is (µ), Is,f (µ) and Usµ to denote, respectively, the energy (1.3), weighted energy (1.5) and potential (1.4) of a measure µ ∈ M(Rp ) Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

´ A. LOPEZ GARC´IA

194 6

with respect to the Riesz s-kernel. We will also use the symbols ws (A) and caps (A) to denote the Wiener s-energy and s-capacity of a set A ⊂ Rp in this setting. The paper is organized as follows. In Section 2 we present our results and in Section 3 we provide their proofs. 2. Statement of results Our ﬁrst result is the following generalization of Theorem 1.2. Theorem 2.1. Let k : X × X → R ∪ {+∞} be an arbitrary kernel on a locally compact metric space X, A ⊂ X be a compact conductor, and f : X → R∪{+∞} be ∗ an external ﬁeld. Assume that the Gauss variational problem is solvable. If {ωN } is a sequence of optimal weighted N -point conﬁgurations on A, then ∗ Ef (ωN ) = Vf . N →∞ N2

(2.1)

lim

Furthermore, if the Gauss variational problem has a unique solution µ ∈ Mf (A), then 1 ∗ (2.2) δx −→ µ , N → ∞, N ∗ x∈ωN

where δx is the unit Dirac measure concentrated at x. Remark 2.2. As the proof of Theorem 2.1 shows, without assuming the uniqueness of the equilibrium measure one can deduce that any convergent subsequence of (1/N ) x∈ω∗ δx converges weak-star to an equilibrium measure. This observation N is also applicable to the following result concerning greedy conﬁgurations. Theorem 2.3. Let k : X × X → R ∪ {+∞} be a symmetric kernel on a locally compact metric space X, A ⊂ X be a closed set, and f : X → R ∪ {+∞} be an external ﬁeld satisfying (1.11) in case that X is not compact. Assume that the f Gauss variational problem is solvable and µ ∈ Mf (A) is a solution. Let {αN,µ } be a weighted greedy (f, µ)-energy sequence on A. Then (i) the following limit (2.3)

lim

N →∞

f Ef (αN,µ )

N2

= Vf

holds. (ii) If the equilibrium measure µ ∈ Mf (A) is unique, it follows that 1 ∗ (2.4) δa −→ µ , N → ∞, N f a∈αN,µ

(2.5)

Unf (an ) = Vf − n→∞ n lim

f dµ ,

where an is the n-th element of the weighted greedy (f, µ)-energy sequence. Conditions (2.3)-(2.5) are related in the following way. Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

195 7

GREEDY ENERGY POINTS WITH EXTERNAL FIELDS

Proposition 2.4. Let k : X × X → R be a real-valued symmetric kernel on a locally compact metric space X, A ⊂ X be a closed set, and f : X → R ∪ {+∞} be an external ﬁeld. Assume that the Gauss variational problem is solvable and ∗ µ ∈ Mf (A) is a solution. Suppose that {bn }∞ n=1 ⊂ Sµ is a sequence of points such that N 1 ∗ (2.6) δb −→ µ , N → ∞, N n=1 n and set Tnf (x) :=

n−1

k(x, bi ) + (n − 1)f (x) ,

x ∈ A,

n ≥ 2.

i=1

If the following limit Tnf (bn ) = Vf − n→∞ n

(2.7)

lim

f dµ

holds, then (2.8)

lim

N →∞

Ef ({b1 , . . . , bN }) = Vf . N2

Theorem 2.3 can be extended to the following class of weighted greedy sequences. Definition 2.5. Let m ≥ 2 be a ﬁxed integer. Under the same assumptions of Deﬁnition 1.4, suppose that the Gauss variational problem is solvable and µ ∈ Mf (A) is an equilibrium measure. A sequence (an = an,m,f,µ )∞ n=1 ⊂ A is called a weighted greedy (m, f, µ)-energy sequence on A if it is generated inductively in the following way: • The ﬁrst m points a1 , . . . , am are selected so that {a1 , . . . , am } is an optimal weighted m-point conﬁguration on Sµ∗ , i.e. Ef ({a1 , . . . , am }) ≤ Ef ({x1 , . . . , xm })

(2.9)

for all (x1 , . . . , xm ) ∈ Sµ∗ × · · · × Sµ∗ . • Assuming that a1 , . . . , amN have been selected, where N ≥ 1 is an integer, the next set of m points {amN +1 , . . . , am(N +1) } ⊂ Sµ∗ are chosen to minimize the energy functional (2.10) (f,m)

UmN (x1 , . . . , xm ) :=

mN m i=1 l=1

k(xi , al )+

k(xi , xj )+((N +1)m−1)

1≤i 0 , lim f (x) = +∞ .

|x|→∞

Using the same arguments employed to prove Theorem I.1.3 in [11] (which concerns the case p = 2 and s = 0) and the fact that ks is positive deﬁnite (see [6, Theorem 1.15]), it is not diﬃcult to see that the Gauss variational problem on A in the presence of f has a unique solution λ = λs,f ∈ Mf (A). Furthermore, the inequality λ (2.16) Us (x) + f (x) ≤ Vs,f − f dλ is valid for all x ∈ supp(λ), where Vs,f := Is,f (λ) denotes the minimal energy constant (1.6), and (2.17) Usλ (x) + f (x) ≥ Vs,f − f dλ holds q.e. on A (relative to the s-capacity of sets). Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

GREEDY ENERGY POINTS WITH EXTERNAL FIELDS

197 9

We remark that if p = 2 and s = 0 then these properties hold if (2.15) is replaced by the condition lim (f (x) − log |x|) = +∞ .

(2.18)

|x|→∞

The following result holds. Lemma 2.9. Let p ≥ 2 and p − 2 ≤ s < p. Assume that A ⊂ Rp is closed and f satisﬁes the conditions (2.14) and (2.15) (or (2.18) in the case p = 2, s = 0). Let λ = λs,f be the equilibrium measure solving the Gauss variational problem on A in the presence of f . If {x1 , . . . , xn } ⊂ Rp is an arbitrary collection of points and n 1 (2.19) + nf (x) ≥ M for q.e. x ∈ supp(λ) , |x − xi |s i=1 then for all x ∈ Rp , (2.20)

n i=1

1 ≥ M − n(Ws,f (λ) − Usλ (x)) , |x − xi |s

where Ws,f (λ) is deﬁned in (1.8) and Usλ is the potential associated to λ. Moreover, (2.19) implies that n 1 (2.21) + nf (x) ≥ M for q.e. x ∈ A . |x − xi |s i=1 Remark 2.10. The case p = 2, s = 0 of Lemma 2.9 (the logarithmic kernel is employed in this case) is known as the generalized Bernstein-Walsh lemma and was proved by H. Mhaskar and E. Saﬀ in [9]. Corollary 2.11. Assume that all the assumptions of Lemma 2.9 hold. Let (an = an,f )∞ n=1 be a weighted greedy f -energy sequence on A constructed using the Riesz kernel ks for s ∈ [p − 2, p). Then this sequence is well-deﬁned and an ∈ Sλ∗ for all n ≥ 2. Moreover, all the asymptotic properties in Theorem 2.3 are applicable f f by αN = {a1 , . . . , aN } and µ by λ). to this sequence (replacing αN,µ Corollary 2.12. Let m ≥ 2 and assume that all the assumptions of Lemma 2.9 hold. Let (an = an,m,f )∞ n=1 be a weighted greedy (m, f )-energy sequence on A obtained using the Riesz kernel ks for s ∈ [p − 2, p). Then this sequence is well-deﬁned and an ∈ Sλ∗ for all n ≥ 1. Furthermore, all the asymptotic properties in Theorem (f,m) (f,m) 2.7 are applicable to this sequence (replacing αmN,µ by αmN = {a1 , . . . , aN } and µ by λ). We remark that the problem of ﬁnding an explicit representation of the solution of a Gauss variational problem in Rp is a diﬃcult task in general. However, there are certain assumptions on f that could alleviate the diﬃculty of this problem, as the following result shows in the case of Newtonian potentials. Proposition 2.13. Let p ≥ 3 and s = p − 2. Assume that f is a radially symmetric function (i.e. f (x) = f (|x|) for all x ∈ Rp ) satisfying (2.15). Assume further that, as a function of R+ , f has an absolutely continuous derivative and obeys one of the following conditions: (i) r p−1 f (r) is increasing on (0, ∞); (ii) f is convex on (0, ∞). Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

´ A. LOPEZ GARC´IA

198 10

Let r0 be the smallest number for which f (r) > 0 for all r > r0 , and let R0 be the smallest solution of R0p−1 f (R0 ) = p − 2 (it is easy to see that r0 < R0 and R0 is ﬁnite). If λp−2,f is the solution of the Gauss variational problem on A = Rp with f as the external ﬁeld, then supp(λp−2,f ) = {x ∈ Rp : r0 ≤ |x| ≤ R0 } , and λp−2,f is given by (2.22)

dλp−2,f (x) =

1 (r p−1 f (r)) dr dσp−1 (x) , p−2

x = rx ,

r = |x| ,

where dσp−1 denotes the normalized surface area measure of the unit sphere S p−1 (σp−1 (S p−1 ) = 1) in Rp . Moreover, (2.23)

Wp−2,f (λp−2,f ) =

and

1 + f (R0 ) , R0p−2

⎧ 1/R0p−2 + f (R0 ) − f (r0 ) , if |x| ≤ r0 , ⎪ ⎪ ⎪ ⎪ ⎨ λp−2,f (x) = Up−2 1/R0p−2 + f (R0 ) − f (x) , if r0 < |x| < R0 , ⎪ ⎪ ⎪ ⎪ ⎩ if |x| ≥ R0 . 1/|x|p−2 ,

(2.24)

Remark 2.14. The case p = 2, s = 0 was analyzed by Saﬀ and Totik in [11]. 3. Proofs Proof of Theorem 2.1. Our ﬁrst goal is to show that (3.1)

lim sup N →∞

∗ Ef (ωN ) ≤ Vf . 2 N

N Let ν ∈ Mf (A) be arbitrary, and consider the measure λ := j=1 ν on the product space X N . Deﬁne the function h : X N → R ∪ {+∞} by h(x1 , . . . , xN ) := ∗ Ef ({x1 , . . . , xN }). Therefore, Ef (ωN ) ≤ h(x1 , . . . , xN ) for all (x1 , . . . , xN ) ∈ AN . Integrating with respect to λ it follows that ∗ Ef (ωN )≤ h(x1 , . . . , xN ) dλ(x1 , . . . , xN ) AN

=

k(xi , xj ) dλ(x1 , . . . , xN ) + 2(N − 1)

AN 1≤i=j≤N

=

k(xi , xj ) dν(xi ) dν(xj ) + 2(N − 1)

1≤i=j≤N

= N (N − 1)

f (xi ) dλ(x1 , . . . , xN )

AN i=1

N

A2

i=1

f (xi ) dν(xi )

A

f (x) dν(x) = N (N − 1)If (ν) .

k(x, y) dν(x) dν(y) + 2 A2

N

A

∗ ) ≤ N (N − 1)Vf , and Taking the inﬁmum over ν ∈ Mf (A) we obtain that Ef (ωN therefore (3.1) holds. Next we show that ∗ Ef (ωN ) (3.2) Vf ≤ lim inf 2 N →∞ N

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199 11

GREEDY ENERGY POINTS WITH EXTERNAL FIELDS ∗ and at the same time we verify (2.2). Let ωN = {x1 , . . . , xN } and deﬁne

νN :=

N 1 δx . N i=1 i

Assume that gn : A × A → R is a sequence of non-decreasing continuous functions that converges pointwise to k on A. We ﬁx n. Then (3.3) gn (x, y) dνN (x) dνN (y) + 2 f dνN 1 gn (xi , xi ) + N 2 i=1

N

=

1 (gn (xi , xi ) + 2f (xi )) + N 2 i=1 =

1 N2

N

N

f (xi )

i=1

1≤i=j≤N

N

≤

gn (xi , xj ) + 2N

k(xi , xj ) + 2(N − 1)

N

f (xi )

i=1

1≤i=j≤N

∗ (gn (xi , xi ) + 2f (xi )) + Ef (ωN ) .

i=1

Let C := inf{k(x, y) : (x, y) ∈ A2 } and D := inf{f (x) : x ∈ A}. Both C and D are ﬁnite since A is compact and k and f are lower semicontinuous. Using ∗ Ef (ωN ) ≤ N (N − 1)Vf we obtain ND ≤

(3.4)

N

f (xi ) ≤

i=1

N (Vf − C) . 2

By the compactness of A and the continuity of gn , there exists a constant Mn > 0 such that N |gn (xi , xi )| ≤ N Mn . i=1

In particular,

N

i=1 gn (xi , xi ) N2

(3.5)

−→ 0,

N −→ ∞ .

From (3.4) and (3.5) we conclude that N i=1 (gn (xi , xi ) + 2f (xi )) −→ 0, N −→ ∞ . (3.6) N2 Let ν ∈ M1 (A) be a cluster point of the sequence {νN } in the weak-star topology. Then there exists a subsequence {νN }N ∈N that converges weak-star to ν (cf. [5, Lemma 1.2.1]). Therefore (3.7) gn (x, y) dν(x) dν(y) + 2 f (x) dν(x) ≤ lim inf N ∈N

gn (x, y) dνN (x) dνN (y) + 2

f (x) dνN (x) .

Now we apply (3.7), (3.3), (3.6) and (3.1) to obtain gn (x, y) dν(x) dν(y) + 2 f (x) dν(x) ≤ Vf .

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´ A. LOPEZ GARC´IA

200 12

From the monotone convergence theorem we conclude that gn (x, y) dν(x) dν(y) + 2 f (x) dν(x) ≤ Vf . If (ν) = lim n→∞

Therefore ν = µ, the equilibrium measure. Since µ is the only cluster point of {νN }, (2.2) follows. Using (3.3) we have gn (x, y) dµ(x) dµ(y) + 2 f (x) dµ(x)

1 1 ∗ ∗ (gn (xi , xi ) + 2f (xi )) + Ef (ωN ) = lim inf 2 Ef (ωN ), ≤ lim inf 2 N →∞ N N →∞ N i=1 N

from which (3.2) follows. Finally, (2.1) is a consequence of (3.2) and (3.1).

Lemma 3.1. Let k : X × X → R ∪ {+∞} be a symmetric kernel on a locally compact metric space X, A ⊂ X be a compact set, and f : X → R ∪ {+∞} be an external ﬁeld. Assume that the Gauss variational problem is solvable and µ ∈ Mf (A) is a solution. Let {τn } ⊂ M1 (Sµ∗ ) be a sequence of measures that converges to µ in the weak-star topology. Then (3.8) lim f dτn = f dµ . n→∞

µ

Proof. Since f and U are lower semicontinuous we have f dµ ≤ lim inf f dτn , n→∞

lim sup n→∞

(Wf (µ) − U ) dτn ≤ µ

(Wf (µ) − U µ ) dµ .

In addition, for x ∈ Sµ∗ the inequality f (x) ≤ Wf (µ) − U µ (x) holds, and therefore lim sup f dτn ≤ lim sup (Wf (µ) − U µ ) dτn . n→∞

n→∞

By (1.9) and (1.10), f = Wf (µ) − U µ q.e. on Sµ , and since µ has ﬁnite energy this equality holds µ-a.e. Thus f dµ = (Wf (µ) − U µ ) dµ ,

and (3.8) follows.

Proof of Theorem 2.3. To prove this result we follow closely ideas from chapter V of [11]. By deﬁnition, Unf (an ) ≤ Unf (x)

for all x ∈ Sµ∗ ,

n ≥ 2.

We have, for any x ∈ Sµ∗ , f ) Ef (αN,µ

=2

k(ai , aj ) + 2(N − 1)

1≤i 0, t ∈ (α, β) . A dt A maximal horizontal arc is called a horizontal trajectory (or simply a trajectory) of n . Analogously, trajectories of −n are called orthogonal or vertical trajectories of n ; along these curves ⇔

Vn (z)/A(z) (dz)2 > 0

Re ξn (z) = const .

We can deﬁne a conformal invariant metric associated with the quadratic dif ferential , given by the length element |dξn | = | Vn /A|(z)|dz|; the n -length of a curve γ is Vn 1 (z) |dz| ;

γ n = A π γ

(observe that this deﬁnition diﬀers by a normalization constant from the deﬁnition 5.3 in [19]). Furthermore, if D is a simply connected domain not containing singular points of n , we can introduce the n -distance by dist(z1 , z2 ; , D) = inf{ γ n : z1 , z2 ∈ γ¯ , γ ⊂ D} . Trajectories and orthogonal trajectories are in fact geodesics (in the n -metric) between any two points they contain (see [13, Thm. 8.4] for a more precise statement). A simply connected domain D not containing points from A ∪ Vn is called a n -rectangle if it is delimited by two horizontal and two vertical arcs of n ; in other words, if ξn (D) is a (euclidean) rectangle [a, b] × [c, d], and D → ξn (D) is a one-to-one conformal mapping. We call the value d − c the n -height, b − a the n -length, and 2(b − a + d − c) the n -perimeter of D. Obviously, these deﬁnitions are consistent with the freedom in the selection of the natural parameter ξn . Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

A. MART´INEZ-FINKELSHTEIN AND E.A. RAKHMANOV

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3.3. Local asymptotics. Theorems on local asymptotics of solutions are well-known (see [1] and [11]); using the local estimates that appear in [11, Ch. VI, Theorem 11.1] it follows that if D is a n -rectangle such that function gn deﬁned in (3.4) is holomorphic in ξn (D) = [a, b] × [c, d], then the diﬀerential equation (3.3) has in [a, b]×[c, d] two linearly independent holomorphic solutions wj , j = 1, 2, of the form (3.5) wj (ξ) = exp (−1)j+1 λn ξ (1 + εj (ξ)) , and such that (3.6)

|εj (ξ)| ≤ exp

1 |λn |

ξ

|gn (t)| |dt|

− 1,

ξ ∈ [a, b] × [c, d] .

sj

The integrals here are taken following the progressive paths (according to the terminology of [11]), i.e. contours along which Re(ξ) is non-decreasing (for j = 1) or non-increasing (for j = 2). In the case of the rectangle we may take s1 = a + ic and s2 = b + id, so that the whole rectangle is reachable by progressive paths. Taking advantage of the fact that the coeﬃcients of the original equation (1.2) are polynomials, we can estimate the total variation M (D) = max |gn (t)| |dt| , γ

γ

where γ is any horizontal or vertical segment in [a, b] × [c, d]. This yields the following result: be an Euclidean rectangle Proposition 3.1. With the assumptions above, let D and ξn can be continued holomorphically to D in such a way such that ξn (D) ⊂ D that ξn (A ∪ Vn ) ∩ D = ∅. Let dn = dist(∂ D, ξn (D)) be the Euclidean distance from Then we can replace the estimates (3.6) by ξn (D) to the boundary of D. K − 1, ξ ∈ R. (3.7) |εj (ξ)| ≤ exp dn |λn | In a n -rectangle D we can select a single valued branch of the function Hn introduced in (2.5). Then a direct consequence of the proposition above is Corollary 3.2. Let D be a -rectangle. Then a general solution of (1.2) in D has the form

(3.8) y(z) = Hn (z) κ1 ζnλn (z) (1 + ε1 (z)) + κ2 ζn−λn (z) (1 + ε2 (z)) , with ζn deﬁned in (2.2). We have (3.9)

|εj (z)| ≤ exp

K dn |λn |

− 1,

z ∈ D,

where dn is the Euclidean distance deﬁned in Proposition 3.1. 3.4. Global asymptotic formula away from zeros. The result above shows that if we stay away from the singularities A ∪ Vn , we can control the errors in the WKB approximation uniformly. This motivates the following deﬁnition. For ε > 0 and t ∈ C and subset K ⊂ C we denote def def Dε (t) = {z ∈ C : |z − t| < ε} , Dε (K) = D(t, ε) , t∈K

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ON ASYMPTOTICS OF HEINE-STIELTJES AND VAN VLECK POLYNOMIALS

215 7

and def

Dn,ε = Dε (A ∪ Vn ) . Let D be a n -rectangle in C \ Dn,ε containing inﬁnity. We can select the branch of ζn and the initial point z0 in (2.2) in such a way that |ζn (z)| ≥ 1 + δ for a δ > 0, and there is a constant κ such that

(3.10) Qn (z) = Hn (z) ζnλn (z) (1 + ε1 (z)) + κ ζn−λn (z) (1 + ε2 (z)) . It follows that if we continue all functions analytically in the maximal domain containing D such that |ζn (z)| ≥ 1 + δ, then formula (3.10) still holds. Next, standard arguments show that in a domain D, if Qn = 0 for all suﬃciently large n, then there exists a dominant term in (3.8): Lemma 3.3. Let D = {z ∈ C : a < Re(ξn ) < b, c < Im(ξn ) < d} be a simply connected n -rectangle in C \ Dn,ε , with π . d−c> |λn | If Ω does not contain zeros of a solution y of (1.2) then for any choice of the lower limit of integration and with an appropriate choice of the branch of the square root in (2.2) we have y(z) = Hn (z) ζnλn (z) (1 + εn (z)) .

(3.11)

Moreover, there exists a constant M = M (ε, D), independent of n, such that for any n -rectangle D ⊂ D, |εn (z)| ≤

(3.12)

M , |λn |

t ∈ D .

Now let us observe what happens if we have a union of two n -rectangles, D(1) and D(2) , such that D(1) ∩ D(2) has an interior point, z0 . Assume ﬁrst that we take this z0 as the lower limit of integration in (2.2) and (2.5). From Lemma 3.3 it follows that if y ≡ 0, y(z) = κ(j) Hn (z) ζnλn (z) 1 + εn(j) (z) , z ∈ D (j) , j = 1, 2, for certain non zero constants κ(1) and κ(2) . Evaluating at z = z0 we get (2) (2) κ(1) 1 + ε(1) 1 + ε (z ) = κ (z ) , z ∈ D (1) ∩ D(2) , 0 0 n n so that, (1)

κ(2) = κ(1)

1 + εn (z0 ) 1+

(2) εn (z0 )

= κ(1) (1 + εn (z)) ,

|εn (z)| ≤

M . |λn |

In the intersection (neighborhood of z0 ) both expressions for y should match, hence we have that in D(1) ∩ D(2) , ±λn ζnλn (z) 1 + ε(1) (z) (1 + εn (z)) 1 + ε(2) n (z) = ζn n (z) . Since D(1) ∩ D(2) is an open set, necessarily the same branch of ζn has been taken in both sides of the previous identity. This argument shows that Lemma 3.3 is valid in the union D(1) ∪ D(2) , eventually with a diﬀerent constant in the right hand side of (3.12). Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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Now, modifying the lower limit of integration only changes the normalization constant κ in (3.11). That means that in the representation (3.11) we can choose any lower bound z0 ∈ D(1) ∪ D(2) , as long as we understand the function ζn as the analytic continuation along a path in D(1) ∪ D(2) . The discussion above motivates the following deﬁnition: Definition 3.4. Let D(j) , j = 1, . . . , k, be a ﬁnite set of n -rectangles with n -height greater than |λn | and bounded n -perimeters. If each intersection D(j) ∩ D(j+1) , j = 1, . . . , k − 1, has an interior point, then their union ∪kj=1 D(j) is called a ﬁnite n -chain (see Figure 1).

D(3)

D(2) D(1)

Figure 1. A n -chain. Thus, we have proved the following result: ⊂ C \ Dn,ε be a domain, and D = ∪k D(j) ⊂ D a ﬁnite Lemma 3.5. Let D j=1 If a solution y of (1.2) does not vanish in D and simply connected n -chain in D. then for any choice of the lower limit of integration in D, and with an appropriate choice of the branch of the square root in (2.2) there exists a constant C such that (3.11)–(3.12) holds in D. The branch of ζn is obtained by analytic continuation in D. In other words, asymptotic representation can be continued along the chains of n -rectangles, as long as we stay away from the zeros of the solution and of the sets A and Vn . 3.5. Zeros of a solution of the diﬀerential equation. Let z0 ∈ C \ Dn,ε . Denote by Ω be a maximal simply connected n -rectangle in C \ Dn,ε containing z0 , and by γ the vertical trajectory in Ω passing through z0 (z0 is a regular point of the quadratic diﬀerential n , so that γ is well deﬁned). Assume that y is a nontrivial solution of (1.2) in Ω and z0 is a zero of y. Since in Ω the expression (3.8) is valid, we have that κ2 1 + ε1 (z) mod (2πi) . (3.13) y(z) = 0 ⇔ 2λn ξn (z) = log − κ1 1 + ε2 (z) Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

ON ASYMPTOTICS OF HEINE-STIELTJES AND VAN VLECK POLYNOMIALS

Hence, from the assumption y(z0 ) = 0 it follows that κ2 1 + ε1 (z0 ) 2λn ξn (z0 ) = log − κ1 1 + ε2 (z0 )

217 9

mod (2πi) .

Assume that z1 is another point on γ that satisﬁes the following condition: λn z1 |dξn (t)| = λn γ(z0 , z1 ) n ∈ N , (3.14) π z0 where γ(z0 , z1 ) is the arc of γ joining z0 and z1 . Definition 3.6. Let z0 ∈ C \ Dn,ε , and let γ be the largest connected vertical arc in C \ Dn,ε of the quadratic diﬀerential n passing thorough z0 . The set ωj ω= j

is a necklace in C\Dn,ε corresponding to z0 if all ωj , called beads, are n -rectangles of the form ωj = {z ∈ Ω : | Re(ξn (z) − ξn (zj ))| < δ, | Im(ξn (z) − ξn (zj ))| < δ} , where each zj satisﬁes condition (3.14), there exists a constant M such that λ2n δ ≤ M , and all ωj ⊂ C \ Dn,ε . The vertical arc γ is the string of the necklace.

z1 z0

Figure 2. Necklace corresponding to z0 . A direct consequence of Rouche’s theorem is the following statement: Lemma 3.7. Let Ω be a simply connected n -rectangle in C \ Dn,ε , and z0 ∈ Ω be a zero of a nontrivial solution y of (1.2) in Ω. If ω is the necklace in C \ Dn,ε corresponding to z0 , then each bead wj ⊂ C \ Dn,ε contains one and only one zero of y, and y(z) = 0 for z ∈ Ω \ ω. So far we have not assumed anything special about the solution of (1.2). Now we concentrate on the n-th degree Heine-Stieltjes polynomial. The main diﬀerence is that we know that it has exactly n zeros, with account of multiplicity. This and Lemma 3.7 immediately yield the following Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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Proposition 3.8. Assume that z0 ∈ C \ Dn,ε is a zero of a Heine-Stieltjes polynomial Qn , and let γ be the largest connected vertical arc in C \ Dn,ε of the quadratic diﬀerential n passing thorough z0 . Then n+2 .

γ n ≤ λn In particular, every string of a zero-carrying necklace has a ﬁnite n -length, and starts and ends at Dn,ε , or is a closed curve. Indeed, by Lemma 3.7, every necklace carries at least λn γ ϕn − 2 zeros of Qn , where γ is the string of the necklace. Summarizing, if Qn is a Heine-Stieltjes polynomial, there exist a ﬁnite number of zero-carrying necklaces with the corresponding strings γn,j such that all the zeros belong to the union of these necklaces or lie in Dn,ε . 4. Asymptotics of Van Vleck polynomials As it is clear from Theorem 2.1, the zeros of the Van Vleck polynomial Vn are the key parameters in the asymptotic expression for the Heine-Stieltjes polynomials. Hence, as a next step we derive a set of equations that will characterize their positions. We start with a formal argument and postpone to next section a more detailed discussion about consistency and meaning of these equations. Theorem 4.1. With the assumptions and notations above, let y = Qn be a Heine-Stieltjes polynomial (solution of (1.2)) corresponding to Vn , and let γn,k be the set of arcs deﬁned in Theorem 2.1. If there exists an ε > 0 such that all arcs γk,n are disjoint and the n -distance between them is > ε, then the following system of equations is satisﬁed: Vn (t) 1 ρk /2 + δk,n mk (4.1) + , k = 1, . . . , p − 1, mk ∈ Mn , dt = πi γk,n A(t) λn λn where the index set Mn is a ﬁnite subset of N ∪ {0}. If for γk,n we denote by ηk the set of endpoints of γk,n that belong to A, then B(a) def + 1 − card(ηk ). ρk = A (a) a∈η k

Furthermore, there exists a constant C = C(ε) such that C |δk,n | ≤ , n so that δk,n is the error term in (4.1). This theorem is a simple consequence of the asymptotic formula (2.4), which is valid in a neighborhood of γk,n . Observing its increment along a closed Jordan curve γ k,n encircling γk,n in the positive direction and using the argument principle we get the formulas above. The system of equation (4.1) consists of p − 1 equations (since we have ﬁxed the residue at inﬁnity, equation k = p is dependent from the other p − 1 ones). More exactly, (4.1) presents a collection of systems of equations. In order to deﬁne it completely we need to specify: (a) the combinatorics: points from A ∪ Vn are arranged in p pairs (endpoints of γk,n ). Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

ON ASYMPTOTICS OF HEINE-STIELTJES AND VAN VLECK POLYNOMIALS

219 11

(b) the homotopic types of curves γk,n (once the combinatorics is ﬁxed); and (c) the range of values the integer parameters mk in the right hand side of (4.1) may take. Observe that by (4.1), 1 Vn (t) mk ρk /2 1 dt = , k = 1, . . . , p − 1, mk ∈ N. + +O πi γk,n A(t) λn λn n2 This system allows eventually to ﬁnd the position of “almost all” zeros of Van Vleck polynomials, with an error smaller than the distance between zeros. In most part of the range this error is O(n−2 ). A complete analysis however is cumbersome and contains a combination of analytic, geometric and combinatorial arguments. We restrict our presentation here to the case p = 2 (three points), which at least has a trivial combinatorics and a rather simple geometry, with additional remarks on the case p = 3. 4.1. Case of p = 2. We have A = {a0 , a1 , a2 } (in general, non-collinear) and want to discuss the issues (a)-(c) raised above in order to deﬁne completely the set of equations determining the position of the zero of the Van Vleck polynomial. The ﬁrst ingredient we need is the solution of the classical minimal capacity problem posed in the class of all continua in C containing A (the Chebotarev’s problem). It was proved by Gr¨otzsch [3] and Lavrentiev [6, 7] that there exists a unique Γ∗ = Γ∗ (A) satisfying (4.2)

cap(Γ∗ ) = min{cap(F ) : F a continuum containing A},

where cap(·) denotes the logarithmic capacity. If aj ’s are not collinear, then Γ∗ is a union of three arcs, γ0∗ , γ1∗ , and γ2∗ , connecting a point v ∗ = v ∗ (A) with points aj ’s, respectively (see also [5] and [12]). We call this Γ∗ the Chebotarev’s compact or Chebotarev’s continuum corresponding to A, and point v ∗ is the Chebotarev’s center of the set A. If we deﬁne t − v ∗ 1/2 1 |dt|, k = 0, 1, 2, (4.3) Mj = π γk A(t) then M0 + M1 + M2 = 1. Observe that each Mj is the -length of γj∗ , where ∗

∗ =

def

v∗ − z (dz)2 . A(z)

Next we deﬁne three analytic functions (elements) wk (v), k = 0, 1, 2. We describe ﬁrst w0 as a germ of an analytic function at v = a0 , which allows unlimited analytic continuation to C \ A. For all v in a suﬃciently small neighborhood of a0 def let ∆0 be the segment [a0 , v], and ∆1 = [a1 , a2 ]. Denote Ω = C \ (∆0 ∪ ∆1 ). With √ def R(z) = (z − v ∗ )/A(z) we consider in Ω the single-valued branch of R given by the asymptotic condition limz→∞ z R(z) = 1, and deﬁne 1 1 w0 (v) = R(t) dt = R(t) dt, πi ∆0 2πi ∂∆0 where ∂∆0 is the doubly-connected component of ∂Ω √ (the Carath´eodory boundary of Ω), with the boundary values of the branch of R speciﬁed above. It is clear Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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that w0 is analytic at v = a0 , and may be continued as a multi-valued analytic function to C \ A. Similarly, we deﬁne w1 and w2 , starting from v = a1 and v = a2 , respectively. The proof of the following statement can be found, e.g. in [8]: Proposition 4.2. For k = 0, 1, 2 there exist an analytic arc k , connecting ak with v ∗ , deﬁned by k = {v ∈ C : wk (v) ∈ [0, Mk ]}. Furthermore, for k = 0, 1, 2, wk (v) is univalent in a neighborhood Lk of k (which is mapped by wk onto a neighborhood of the interval [0, Mk ]). Now we are ready to formulate a theorem about the asymptotic location of the zeros of Van Vleck polynomials for p = 2. Theorem 4.3. Let y = Qn be a Heine-Stieltjes polynomial (solution of (1.2)) with A = {a0 , a1 , a2 }, corresponding to Vn (z) = z − vn . Assume that there exists an ε > 0 such that dist(vn , v ∗ ) > ε (where v ∗ is the Chebotarev’s center). Then for all suﬃciently large n, there exist an index k ∈ {0, 1, 2} and an integer value 0 ≤ mk ≤ [nMn ] such that the following equation is satisﬁed: (4.4)

wk (vn ) =

δk,n mk 1 B(ak ) + + , λn λn 2A (ak ) λn

and there exists a constant C = C(ε) such that C |δk,n | ≤ . n Obviously, this statement does not cover, roughly speaking, εn zeros out of n + 1 possible zeros of Van Vleck polynomials, but it provides an error estimate of order C(ε)n−2 . The exceptional set of “missed” Van Vleck zeros may be made smaller (up to a constant) for the price of relaxing the error estimate; also the technical details become more cumbersome. 4.2. Case of p = 3. Now we turn to the problems (a) and (b) related to the system (4.1), namely, we discuss the combinatorics and the homotopic type of curves γk,n from Theorem 4.1; we leave the issue (c) of the range of values of mk in the equations to the following Section. Here we illustrate the situation considering the points aj from A at vertices of a rectangle whose height (vertical size) is smaller than length (horizontal size), see Fig. 3, with the results of some numerical experiments, as well as Fig. 4, where the corresponding Chebotarev’s continuum is depicted. In this case Vn (z) = (z − v1,n )(z − v2,n ); its zeros are determined completely by a system of two independent equations of the form (4.1). We claim that in this situation there are exactly 8 + 1 homotopically diﬀerent groups of systems, see Figure 5. In eight of them we integrate along two arcs γk,n , each connecting a point from A with a zero of Vn . In the remaining case (Figure 5, bottom right) one of the curves will connect both zeros of Vn . For the case depicted in Figure 5, upper left, system (4.1) is written as ⎧ v1 ⎪ Vn (t) mk,1 δk,n,1 1 1 B(a1 ) ⎪ ⎪ dt = + + , ⎪ ⎨w1 (c1 , c2 ) = πi A(t) λn λn 2A (a1 ) λn a1 v2 ⎪ ⎪ Vn (t) mj,2 δj,n,2 1 1 B(a2 ) ⎪ ⎪w2 (c1 , c2 ) = dt = + + , ⎩ πi a2 A(t) λn λn 2A (a2 ) λn Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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ON ASYMPTOTICS OF HEINE-STIELTJES AND VAN VLECK POLYNOMIALS

Figure 3. Zeros of Heine-Stieltjes (small dots) and of the corresponding Van Vleck polynomials (fat dots) for aj ’s at vertices of a rectangle (p = 3). where (mk,1 , mj,2 ) is a pair of integers. We postpone the discussion about the possible range of variation of these constants until next section. In order to claim that C |δk,n,j | ≤ n we need to leave aside again a fraction of zeros of Van Vleck polynomials (for which conditions of Theorem 2.1 are not satisﬁed), which is typically o(n2 ) (recall that the cardinality of the set of all Van Vleck polynomials corresponding to a Heine-Stieltjes polynomial of degree n is σ(n) = O(n2 )). 5. The Van Vleck set (outline of the proof ) This section is essentially based on the results of our recent paper [8], and we will be regularly referring the reader to some parts of this work for further details. We have seen that even in the simplest cases the geometry behind the set of equations (4.1) is quite involved; furthermore, we have not clariﬁed yet the selection Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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a1

a0

v1∗

v2∗

a2

a3

Figure 4. Chebotarev’s compact corresponding to 4 points forming a rectangle. of the index set Mn . However, the situation becomes much more clear when we take limit as n → ∞. We start by considering an arbitrary p for the price of omitting some nonessential details; a more thorough analysis will be carried out at the end for the particular (but non-trivial) case p = 2. 5.1. Zero distribution of Heine-Stieltjes polynomials and critical measures. It is known (see e.g. [16]) that the zeros of the Heine-Stieltjes polynomials accumulate on the convex hull of A. Hence, without loss of generality we may assume that there is a subsequence Λ ⊂ N such that the Van Vleck polynomials have a limit, (5.1)

lim Vn (z) = V (z) =

n∈Λ

p−1

(z − vj ) .

j=1

This limit induces the following rational quadratic diﬀerential on the Riemann sphere, V (z) (5.2) =− (dz)2 . A(z) Let us consider the corresponding sequence of Heine-Stieltjes polynomials Qn , n ∈ Λ; from Proposition 3.8 it follows that the zeros of Qn ’s lie asymptotically on critical trajectories of , and that these trajectories have the total -length 1. Further results can be obtained using the electrostatic interpretation of these zeros; it has been proved in [8] that any weak-* limit of the normalized zero-counting measure for Qn ’s is a continuous critical measure with respect to the set A of ﬁxed points on the plane. Let us recall the deﬁnition in its simplest form, suﬃcient for what follows. With every (real-valued) Borel measure µ on C we associate its (continuous) logarithmic energy 1 def (5.3) E(µ) = dµ(x)dµ(y) . log |x − y| Any smooth complex-valued function h in the closure Ω of a domain Ω containing A generates a local variation of Ω by z → z t = z + t h(z), t ∈ C, and consequently, Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

ON ASYMPTOTICS OF HEINE-STIELTJES AND VAN VLECK POLYNOMIALS

a1

* a0

*

a1

223 15

* a0

*

v1 v1 v2 v2 a2

*

* a3

a2

*

* a3

a1 *

* a0

a1 *

* a0

v1

v2

v1

v2

a2 *

* a3

a2 *

* a3

a1 *

* a0

a1

* a0

*

v1

v2 v2

a2 *

a1

* v1

a2

v1

* a3

a2

*

* a3

* a0

a1

*

* a0

v2

v1

* a3

* a1

a2

*

* a3

* * a0

* v1

a2

v2

v2

* a3

Figure 5. Possible homotopic classes of curves γn,k for the case p = 3; compare with the examples depicted in Figure 3.

def

a variation of sets e → et = {z t : z ∈ e}, and (signed) measures: µ → µt , deﬁned by µt (et ) = µ(e). We say that a signed measure µ supported in Ω is a continuous A-critical if for any h smooth in Ω \ A such that hA ≡ 0, (5.4)

E(µt ) − E(µ) d E(µt )t=0 = lim = 0. t→0 dt t

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A-critical measures appear quite frequently in many problems of approximation theory; any such a measure µ can be characterized in terms of its logarithmic potential 1 def U µ (z) = dµ(t) log |z − t| as follows: Lemma 5.1 ([8], Section 5.3). The logarithmic potential of an A-critical measure µ satisﬁes the following properties: (i) if supp(µ) = γ1 ∪ · · · ∪ γs , where γj are the connected components of supp(µ), then U µ (z) = wj = const,

z ∈ γj ,

j = 1, . . . , s.

(ii) at any regular point z ∈ supp(µ) (that is, such that locally at z, supp(µ) is a smooth Jordan arc), (5.5)

∂U µ ∂U µ (z) = (z) , ∂n+ ∂n− where n± are the normal vectors to supp(µ) at z pointing in the opposite directions. Additionally, if z ∈ supp(µ) \ A is not regular, then

(5.6)

grad U µ (z) = 0.

Reciprocally, assume that a ﬁnite real measure µ, whose support supp(µ) consists of a union of a ﬁnite set of analytic arcs, supp(µ) = γ1 ∪ · · · ∪ γs , satisﬁes conditions (i) and (ii) above. Then µ is A-critical. In other words, (i) says that A-critical measures are in fact equilibrium measures in a piece-wise constant external ﬁeld, exhibiting additionally (see (ii)) the so-called S-property, introduced ﬁrst by Stahl [17] and in a more general context, by Gonchar and Rakhmanov [2] (where it was used to establish the well-known “1/9”-conjecture in approximation theory). A ﬁrst rigorous proof of the connection of a critical measure with the S-property appeared in [14]. With a polynomial Q(z) = nk=1 (z − ζk ) of degree n we associate its (normalized) zero counting measure ν(Q) =

n 1 δζk . n k=1

The diﬀerential equation (1.2) is an expression of the fact that the zeros of Qn sit in equilibrium (zeros of the gradient of the discrete total energy) in presence of an external ﬁeld depending from the residues ρk in (1.4). The intensity of this ﬁeld decays with n proportionally to 1/n, so that in the limit we get Proposition 5.2 ([8], Section 7). Let νn = ν(Qn ) be a zero-counting measure corresponding to a sequence of Heine-Stieltjes polynomials Qn . Then any weak-* limit point µ of νn is a unit continuous A-critical measure. In other words, weak-* limits of the normalized zero counting measures of Qn ’s are unit positive A-critical measures. The inverse inclusion (that any unit positive A-critical measure is a weak-* limit of the normalized zero counting measures of Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

ON ASYMPTOTICS OF HEINE-STIELTJES AND VAN VLECK POLYNOMIALS

225 17

Heine-Stieltjes polynomials) is also valid, but it cannot be established within the framework of this paper. Continuous critical measures can be also related with rational quadratic differentials on C by means of a variational argument. Our WKB analysis allows to establish this link, at least partially, more directly: Theorem 5.3. Let y = Qn be a Heine-Stieltjes polynomial (solution of (1.2)) corresponding to Vn , and assume that (5.1) holds. Then V = − (z) dz 2 A is a closed or Jenkins-Striebel quadratic diﬀerential (all its trajectories are either critical or closed), and there is a probability A-critical measure µ on C such that the normalized zero counting measures ν(Qn ) converge (in a weak-* sense and along the subset Λ) to µ. The support Γ = supp(µ) consists of criticaltrajectories of , C \ Γ is connected, and we can ﬁx the single valued branch of V /A there by limz→∞ z V (z)/A(z) = 1. With this convention, z V 1/n = exp Re (5.7) lim |Qn (z)| (t) dt n A locally uniformly in C \ Γ, where a proper normalization of the integral in the right hand side is chosen, so that z V (t) dt − log |z| = 0. lim Re z→∞ A The analysis above yields the following addendum, that we state using the notation and assumptions of Theorems 2.1 and 5.3: Proposition 5.4. The support Γ = supp(µ) is comprised of p analytic arcs γk , such that lim Γn = Γ. n∈Λ

5.2. Structure of the family of positive critical measures. Theorem 5.3 is essential for understanding the asymptotics of Heine-Stieltjes polynomials, although this result is in a certain sense implicit, since it depends on the limit V of the Van Vleck polynomials Vn , that constitute therefore the main parameters of the problem. We must complement this description with the study of the set of all possible limits V . To this end it is convenient to consider a correspondence between closed quadratic diﬀerentials and A-critical measures, independently of their origin. In general, this is not a one-to-one correspondence, since many critical measures may correspond to the same closed quadratic diﬀerential. It was pointed out in [8] that the bijection between closed quadratic diﬀerentials of the form (5.2) and signed A-critical measures is restored if we restrict ourselves to signed measure with a connected complement of the support. Indeed, any such a measure is supported on a ﬁnite union of analytic arcs, and these arcs are necessarily critical trajectories of a quadratic diﬀerential like in (5.2). Reciprocally, from (5.2) we can construct explicitly (using the Sokhotsky-Plemelj formulas) a measure µ that satisﬁes (5.4). Thus, any closed quadratic diﬀerential uniquely generates an A-critical measure µ with a connected complement to its support. In general, such a µ is a signed Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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measure, and for our purposes one has to select those quadratic diﬀerentials associated with positive measures. One of the ways to solve this problem goes through describing the global structure of the trajectories of closed rational quadratic differentials with ﬁxed denominators on the Riemann sphere, and the corresponding parameters (numerators), and later extracting from this set the subset giving rise to positive unit A-critical measures. This way is implemented in for p = 2 at the end of this paper. For an arbitrary p such an investigation would be more diﬃcult to carry out, in particular due to the heavy combinatorics involved. Below we present a direct construction from [8], based on the v-local coordinates that we introduce in the space of the A-critical measures, as well as on the topological structure of the Chebotarev’s compact for A. We start by introducing the v-coordinates. Let us recall the notation. We have the ﬁxed set A = {a0 , a1 , . . . , ap } of distinct points on C, A(z) = pj=0 (z − aj ). For any signed A-critical measure µ there exists a rational quadratic diﬀerential on the Riemann sphere C given by (5.8)

def

(z) = −R(z) (dz)2 ,

R(z) =

V (z) , A(z)

def

V (z) =

p−1

(z − vj ),

j=1

such that supp(µ) = Γµ = Γ = γ1 ∪ · · · ∪ γp is a union of trajectories of . We begin by introducing the local coordinates under the assumption that measures are in general position; this notion of genericity (as opposed to some more special or coincidental cases that are possible) means in our context that Γ is comprised of exactly p disjoint arcs. In consequence, zeros vj ’s of V are simple, and def vector v = {v1 , . . . , vp−1 } ∈ Cp−1 can be used as a local coordinate. Deﬁne also def

V = {v : is closed}. Next we use these local coordinates to introduce the period mapping for our quadratic diﬀerentials. The Carath´eodory boundary of C \ Γ consists of p compodef nents γ k = γk+ ∪ γk− , with a positive orientation with respect to C \ Γ. We can consider γ k as cycles in C \ Γ enclosing the endpoints of γk . Part of R over C \ Γ splits into two disjoint sheets, so we may consider γ k as cycles on R. Let us deﬁne def 1 (5.9) wk (v) = wk (v, Γ) = R(z)dz, k = 1, . . . , p, 2πi γk √ √ where Rγ are the boundary values of the branch of R in C \ Γ deﬁned by k √ limz→∞ z R(z) = 1. Clearly, the boundary values ( R)± on γk± are opposite in √ sign.√Therefore, with any choice of orientation of γk and a proper choice of R = ( R)+ on γk , we will have 1 R(z)dz, k = 1, . . . , p. (5.10) wk (v) = πi γk By the Cauchy residue theorem we have that w1 + · · · + wp = 1 for any v ∈ def Cp−1 . Thus, we can restrict the mapping v → w to p − 1 components of w = p−1 (w1 , . . . , wp−1 ) ∈ C . In this way, we have deﬁned the mapping (5.11)

P(·, Γ) : Cp−1 → Cp−1

such that

P(v, Γ) = w(v, Γ).

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Each component function wj (v1 , . . . , vp−1 ) is analytic in each coordinate vk (even if vk is at one of the endpoints of γk ). Once deﬁned by the integral in (5.10), this analytic germ allows an analytic continuation along any curve in C \ A. Arcs γk are not an obstacle for the continuation since the integral in (5.10) depends only on the homotopic class of Γ in C \ (A ∪ v). The homotopy of Γ is a continuous modiﬁcation of all components simultaneously in such a way that they remain disjoint in all intermediate positions. Under this assumption we can continuously √ modify the selected branch of R in C \ Γ along with the motion of Γ. Proposition 5.5. Mapping w = P(v) is locally invertible at any v(v1 , . . . , vp−1 ) ∈ (C \ A)p−1 with vi = vj for i = j. This statement concerns a very classical object, and may be already known (directly or not), although we are not aware of an explicit reference. The proof is on the surface, and it is based on the fact that the Jacobian matrix of this mapping is, up to a√selection of an appropriate basis, the Riemann matrix of the Riemann surface of AV . Furthermore, we have Proposition 5.6. Let µ0 be an A-critical measure such that Γ = supp(µ0 ) has connected components γ10 , . . . , γp0 , and C \ Γ is connected. Let 0 = R0 (z)(dz)2 be the quadratic diﬀerential associated with µ0 , where R0 = V0 /A, and v 0 = 0 (v10 , . . . , vp−1 ) is the vector of zeros of V0 . Assume that 0 is in general position (that is, all vk ’s are pairwise distinct and disjoint with A). Then for an ε > 0 and any mj ∈ R, j ∈ {1, . . . , p − 1}, satisfying |mj − µ0 (γj0 )| < ε,

j = 1, . . . , p − 1,

there exists a unique solution v ∈ V of the system j = 1, . . . , p − 1. p−1 The quadratic diﬀerential = −R(z)(dz)2 , R(z) = k=1 (z − vk )/A(z), is closed, and the associated A-critical measure µ satisﬁes µ(γj ) = mj , j = 1, . . . , p−1, where supp(µ) = γ1 ∪ · · · ∪ γp and supp(µ) is homotopic to supp(µ0 ). (5.12)

wj (v, Γ) = mj ,

This Proposition introduces a topology in the set of A-critical measures in general position. We call a cell any connected component of this topological space. A measure in general position preserves sign along any connected component of its support. This sign is subsequently preserved when we homotopically modify the measure within its cell. In particular, if µ is a positive A-critical measure, then all measures in the same cell with µ are positive. In v-coordinates a cell G = G(Γ) in V is a subspace of Cp−1 R2p−2 , which is a manifold of the real dimension p − 1, deﬁned by p − 1 real equations of the form 1 (5.13) Im wj (v) = Im R(t) dt = 0, j = 1, . . . , p − 1, 2πi γj =γ where Γ 1 ∪ · · · ∪ γ p−1 ∪ γ p is a union of Jordan contours γ k on R (double arcs) depending on v, but mutually homotopically equivalent for values of v from the same cell. Practically, any v0 ∈ G has a neighborhood of v satisfying (5.13) with constant γ k ’s. Next, let V+ be the set of positive A-critical measures; it will be comprised of 3p−1 cells, as it is described below. Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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The Chebotarev’s compact Γ∗ = Γ∗ (A), associated with A = {a0 , . . . , ap } (that is, the continuum solving the extremal problem (4.2)), consists of critical trajectories of = −R∗ dz 2 , R∗ = V ∗ /A. Assume that the set A satisﬁes the p−1 condition that A and V ∗ do not have common zeros and V ∗ (z) = k=1 (z − vk∗ ) ∗ does not have multiple zeros. Thus, the critical set A∗ = A∪{v1∗ , . . . , vp−1 } consists ∗ of 2p diﬀerent points, and Γ is comprised of 2p−1 arcs, that are critical trajectories of . Each trajectory joins two diﬀerent points from A∗ . Now we come to the procedure of selection of combinatorial (rather than homotopic) types of cells; once the combinatorial type is ﬁxed, the homotopic one will be determined from the Chebotarev’s continuum, as described next. Each zero v ∗ = vk∗ , k = 1, . . . , p − 1, is connected by component arcs of Γ∗ with three other points, say a∗1 , a∗2 , a∗3 ∈ A∗ . We select one of these three arcs (for deﬁniteness, [v ∗ , a∗1 ]) and join two other arcs to make a single arc [a∗2 , a∗3 ], bypassing v ∗ (we think that the arc [a∗2 , a∗3 ] still follows the two arcs from Γ∗ , but without touching v ∗ , instead passing inﬁnitely close to it). This procedure, carried out at each zeros vk∗ of V ∗ , creates a compact set Γ, and consequently, a cell G(Γ) of corresponding + . measures µ ∈ V The selection of Γ, and hence, of the cell G(Γ), is made by choosing one of the three connections for each vk∗ ; there are 3p−1 ways to make the choice. Any choice splits Γ∗ into p “disjoint” arcs Γ∗ = γ1 ∪ · · · ∪ γp ; out of them we select p − 1 arcs (to make a homology basis for C \ Γ∗ ) and then consider the corresponding cycles γ k , as described above. Finally, we describe the cell G(Γ) in terms of the mapping P. Let w = w(v) = the cell G(Γ) is completely deﬁned by the system P(v, Γ); (5.14)

wj (v) = µj ∈ R+ ,

j = 1, . . . , p − 1;

more precisely, there exists a domain M (Γ) = {(µ1 , . . . , µp−1 ) ∈ Rp−1 + } such that for any point (µ1 , . . . , µp−1 ) ∈ M (Γ) system (5.14) has a unique solution v ∈ Cp−1 . Moreover, the corresponding measure µ = µv satisﬁes µ(γj ) = µj , and supp(µ) = γ1 ∪ · · · ∪ γp = Γv is homotopic to Γ. + of unit positive A-critical meaSummarizing, a rough description of the set V sures may be made as follows. The set V+ is a union of 3p−1 of closed bounded cells G(Γ) (Γ = γ1 ∪· · ·∪γp may be selected in 3p−1 ways). The interior G(Γ) of each cell consists of measures µ in general position with supp(µ) homotopic to Γ. Interiors of diﬀerent cells are disjoint. Chebotarev’s measure µ∗ (Robin measure of Γ∗ ) is the ∗ only common point of all boundaries: µ = Γ ∂G(Γ). A graphical description of all these cells for p = 3, obtained from the construction just described, is contained in Figure 5. 5.3. Case p = 2. In order to clarify the construction above let us discuss in more detail the simplest (but far from trivial) case of p = 2. Let us introduce the following set of the plane. For the quadratic diﬀerential (5.15)

v =

v−z 2 dz A(z)

deﬁne def

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229 21

as well as the Van Vleck set def

V+ = {v ∈ C : v is an accumulation point of the zeros of Van Vleck polynomials} . A direct consequence of Theorem 5.3 is that V+ ⊂ V. As it was mentioned for the general case, this inclusion is proper. As in Subsection 4.1, we denote by v ∗ the Chebotarev’s center of A (the value of v in (5.15) such that v∗ corresponds to the Chebotarev’s continuum for A), and let A∗ = A ∪ {v ∗ }. Theorem 5.7 ([8], Section 8.3). The set V is a union of a countable number of analytic arcs k , k ∈ Z, each connecting v ∗ and ∞. Two arcs from V are either identical or have v ∗ as the only ﬁnite common point. The homotopic type of the critical trajectories of v in C \ A remains invariant on each arc k \ A∗ . There are three distinguished arcs k , k ∈ {0, 1, 2}, such that (i) k connects v ∗ with inﬁnity and passes through ak ; (ii) for every v ∈ k the homotopic class of trajectories of the closed quadratic diﬀerential v is trivial. In the terminology introduced in the previous subsection, this theorem says that each cell (connected component) in the topological space of A-critical measures is homeomorphic to an analytic arc with endpoints either at v ∗ , A or inﬁnity. This settles the problem of identiﬁcation of all closed quadratic diﬀerentials ϕv . Using the arguments described above we single out the diﬀerentials corresponding to positive measures: Theorem 5.8. Let k , k ∈ {0, 1, 2}, be the distinguished arcs in V described in + is the union of the sub-arcs + of each k , k ∈ {0, 1, 2}, Theorem 5.7. The set V k connecting ak with the Chebotarev’s center v ∗ (and lying in the convex hull of A). + , k ∈ {0, 1, 2}, then there is a critical trajectory Furthermore, if v ∈ k ∩ V γ(v) of µv connecting v with the pole ak and such that (5.16)

0 ≤ µv (γ(v)) ≤ Mk ,

where Mk is deﬁned in (4.3). In this case both critical trajectories of v that constitute the support of µv are homotopic to a segment. The bijection µv (γ(v)) ↔ v + by points of the interval [0, mk ]. is a parametrization of the set k ∩ V In other words, the only three cells corresponding to positive A-critical measures are homeomorphic to three analytic arcs, joining the Chebotarev’s center v ∗ + and the Chebotarev’s continuum Γ∗ have the with each pole ak ∈ A. Thus, both V same structure (they contain three analytic arcs joining v ∗ with the respective poles ak ) and are metrically close, but not coincident. If v travels an arc + k , the boundary of the cell is reached when either endpoint of this arc is met. If we continue further along the same arc, a new cell is entered, corresponding to sign-changing A-critical measures (see an illustration of the correspondence between the position of v on V and the trajectories of v in Figure 6). Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

A. MART´INEZ-FINKELSHTEIN AND E.A. RAKHMANOV

230 22

Furthermore, by Theorem 5.8, the limit zero v of V (z) = z − v satisﬁes an equation of the form v 1 V (t) dt = β ∈ [0, Mk ], πi ak A(t) which is consistent with our construction in Subsection 4.1. We can go back now to the set of equations (4.1) and add the following result: Theorem 5.9. For any ε > 0, equations (5.17)

wk ( vj,k ) =

j ρk /2 + , λn λn

j = 0, . . . , Mk = [mk n(1 − ε)],

with k = 0, 1, 2, uniquely deﬁne n = (1 − ε)n ± 1 points vj,k . There exists a constant C = C(ε) > 0 such that any Van Vleck zero lies in a C/n2 distance from a point vj,k . In this way, each zero of Vn “belongs” to a O(n−2 ) neighborhood of one of the points deﬁned by the equations above. We need to show additionally that (again, up to a small neighborhood of the Chebotarev’s center v ∗ ) all neighborhoods of this form contain at least one Van Vleck zero (establishing in this form a bijection between points vj,k and the set Vn ). However, the techniques developed in this paper do not allow to complete this proof, so we formulate it as an open question: Conjecture 5.10. There exists a constant C = C(ε) > 0 such that to each point vj,k deﬁned by equations (5.17) it corresponds at least one zero v of a Van Vleck polynomial such that |v − vj,k | ≤ C/n2 . Acknowledgments The authors gratefully acknowledge the help of Dar´ıo Ramos L´ opez with the numerical experiments yielding Figure 3. We wish to acknowledge also the anonymous referee for the careful reading of the manuscript and useful remarks. References [1] M. V. Fedoryuk, Asymptotic analysis. linear ordinary diﬀerential equations, Springer-Verlag, 1993. [2] A. A. Gonchar and E. A. Rakhmanov, Equilibrium measure and the distribution of zeros of extremal polynomials, Mat. Sbornik 125 (1984), no. 2, 117–127, translation from Mat. Sb., Nov. Ser. 134(176), No.3(11), 306-352 (1987). ¨ [3] H. Gr¨ otzsch, Uber ein Variationsproblem der konformen Abbildungen, Ber. Verh.- S¨ achs. Akad. Wiss. Leipzig 82 (1930), 251–263. [4] E. Heine, Handbuch der kugelfunctionen, 2nd. ed., vol. II, G. Reimer, Berlin, 1878. [5] G.V. Kuz’mina, Moduli of families of curves and quadratic diﬀerentials, Proc. Steklov Inst. Math. 139 (1982). [6] M. Lavrentieﬀ, Sur un probl` eme de maximum dans la repr´ esentation conforme, C. R. 191 (1930), 827–829. , On the theory of conformal mappings, Trudy Fiz.-Mat. Inst. Steklov. Otdel. Mat. 5 [7] (1934), 159–245, (Russian). [8] A. Mart´ınez-Finkelshtein and E. A. Rakhmanov, Critical measures, quadratic diﬀerentials, and weak limits of zeros of Stieltjes polynomials, Arxiv: 0902.0193, 2009. [9] A. Mart´ınez-Finkelshtein and E. B. Saﬀ, Asymptotic properties of Heine-Stieltjes and Van Vleck polynomials, J. Approx. Theory 118 (2002), no. 1, 131–151. MR 2003j:33031 [10] J. Nuttall, Asymptotics of generalized Jacobi polynomials, Constr. Approx. 2 (1986), no. 1, 59–77. MR MR891770 (88h:41029)

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231 23

[11] F. W. J. Olver, Asymptotics and special functions, Computer Science and Scientiﬁc Computing, Academic Press, New York, 1974. [12] J. Ortega-Cerd` a and B. Pridhnani, The P´ olya-Tchebotar¨ ov problem, Preprint arxiv:0809.2483. [13] Ch. Pommerenke, Univalent functions, Vandenhoeck & Ruprecht, G¨ ottingen, 1975. [14] E. A. Rakhmanov and E. A. Perevozhnikova, Variations of the equilibrium energy and Sproperty of compacta of minimal capacity, Preprint, 1994. [15] A. Ronveaux (ed.), Heun’s diﬀerential equations, The Clarendon Press Oxford University Press, New York, 1995, With contributions by F. M. Arscott, S. Yu. Slavyanov, D. Schmidt, G. Wolf, P. Maroni and A. Duval. [16] B. Shapiro, Algebro-geometric aspects of Heine–Stieltjes polynomials, Arxiv: 0812.4193, 2008. [17] H. Stahl, Extremal domains associated with an analytic function. I, II, Complex Variables Theory Appl. 4 (1985), no. 4, 311–324, 325–338. MR 88d:30004a [18] T. J. Stieltjes, Sur certains polynˆ omes que v´ eriﬁent une ´ equation diﬀ´ erentielle lin´ eaire du second ordre et sur la teorie des fonctions de Lam´ e, Acta Math. 6 (1885), 321–326. [19] K. Strebel, Quadratic diﬀerentials, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 5, Springer-Verlag, Berlin, 1984. MR 86a:30072 [20] G. Szeg˝ o, Orthogonal polynomials, fourth ed., Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, RI, 1975. Department of Statistics and Applied Mathematics University of Almer´ıa, SPAIN, ´ rica y Computacional, Granada University, SPAIN and Instituto Carlos I de F´ısica Teo E-mail address: [email protected] Department of Mathematics, University of South Florida, USA E-mail address: [email protected]

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A. MART´INEZ-FINKELSHTEIN AND E.A. RAKHMANOV

v ak

v = ak

ak

i) µv (γ(v)) < 0

ii) µv (γ(v)) = 0

iii) 0 < µv (γ(v)) < mk

v

ak

iv) µv (γ(v)) = mk

v = v∗

ak

v) µv (γ (v)) < 0

v

Figure 6. Position of v on 0 ∪1 ∪2 (left) and the corresponding trajectories of the diﬀerential v in (5.15).

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http://dx.doi.org/10.1090/conm/507/09962

Contemporary Mathematics Volume 507, 2010

REMARKS ON RELATIVE ASYMPTOTICS FOR GENERAL ORTHOGONAL POLYNOMIALS E. B. SAFF

Abstract. Using a nonlinear integral characterization of orthogonal polynomials in the complex plane, we provide a simple method for deducing a weak form of relative asymptotics exterior to the convex hull of the common support of the generating measures. The simplicity of the approach makes it a natural precursor for the presentation of Szeg˝ o theory.

Dedicated to Guillermo L´ opez Lagomasino on the occasion of his 60th birthday 1. Introduction Let µ denote a ﬁnite positive Borel measure with compact support Sµ := supp(µ) in the complex plane C and consider the associated inner product and norm < f, g >µ := f (t)g(t)dµ(t), f µ := < f, f >µ . Sµ

If Sµ contains at least N + 1 points, we denote by (1.1)

Pn (z) = Pn (z; µ) = κn z n + · · · , κn > 0,

n = 0, 1, . . . , N,

the unique sequence of polynomials of respective degrees n with positive leading coeﬃcients that are orthonormal with respect to dµ; that is, < Pm , Pn >µ = δm,n . We remark that when Sµ ⊂ R, the Pn ’s satisfy a three-term recurrence relation, which is a useful tool in establishing asymptotic properties (as n → ∞) of these polynomials when Sµ has inﬁnite cardinality. Furthermore, if Sµ is a subset of the unit circle T := {z ∈ C : |z| = 1|}, then the Pn ’s satisfy a two-term recurrence relation (cf. [6]) involving the inverse polynomials Pn∗ (z) := z n Pn (1/z). However, ﬁnite-term recurrences of ﬁxed length do not in general hold for the Pn ’s when Sµ is not a subset of a circle or straight line (cf. [4], [3] for the case when µ is area measure over a bounded Jordan region). The goal of the present note is to show how a simple nonlinear characterization of orthogonal polynomials can be used to establish relative asymptotics (more precisely, comparison estimates) for general sequences of orthonormal polynomials. For comparison estimates for orthogonal polynomials on an interval, see, for example, results of J. Korous described in [6], Section 7.1. 2000 Mathematics Subject Classiﬁcation. Primary 42C05 ; Secondary 26C05 . Key words and phrases. Orthogonal polynomials, Relative asymptotics, Ratio asymptotics. The research of the author was supported, in part, by U.S. National Science Foundation grant DMS-0808093. 1

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234 2

E. B. SAFF

2. A Nonlinear Characterization We begin with the simple observation that for any polynomial Q of degree at most n, and any point z0 ∈ C, we have (2.1)

Q(zt/z0 ) − Q(z) dµ(t) = 0, Pn (t) t − z0 Sµ

since the expression in brackets is a polynomial in t of degree at most n − 1. Consequently, if z0 ∈ / Sµ and deg(Q) ≤ n, (2.2)

Q(z) Sµ

Pn (t) dµ(t) = t − z0

Sµ

Pn (t)Q(zt/z0 ) dµ(t). t − z0

Introducing notation for the Cauchy kernel, 1 k(t, z) := , t−z we deduce for z0 = z that Q(z) < k(·, z), Pn >µ =< Qk(·, z), Pn >µ ,

(2.3)

z∈ / Sµ ,

deg(Q) ≤ n,

which, for Q = Pn , yields Pn (z) < k(·, z), Pn >µ =< k(·, z), |Pn |2 >µ ,

(2.4)

z∈ / Sµ .

Combining (2.3) and (2.4), we further obtain (for Q not identically zero) (2.5)

Pn (z) < Qk(·, z), Pn >µ =< k(·, z), |Pn |2 >µ , Q(z)

z∈ / Sµ ,

deg(Q) ≤ n.

We remark that the restriction“z ∈ / Sµ ” in formulas (2.3)-(2.5) is imposed only to ensure that the integrals in these formulas exist; hence for certain measures µ such as area measure this restriction can be removed, i.e. these formulas hold for all z ∈ C. We now show that the identity (2.4) characterizes the orthonormal polynomial Pn . Proposition 2.1. Let qn (z) = τn z n + · · · , τn > 0, be a polynomial of degree n normalized so that qn µ = 1. If, for |z| large, qn (z) < k(·, z), qn >µ =< k(·, z), |qn |2 >µ ,

(2.6) then qn = Pn (·; µ).

Proof. Expanding k(t, z) = 1/(t − z) in powers of 1/z we deduce from (2.6) that −

∞ k=0

1 1 1 < k(·, z), |qn |2 >µ = O( n+1 ) < ·k , qn >µ = z k+1 qn (z) z

as

z → ∞.

Consequently, < ·k , qn >µ = 0 for k = 0, . . . , n − 1, which, taking into account the imposed normalizations, establishes the claim of the proposition. We note that Bender and Ben-Naim [1] consider similar nonlinear characterizations of Pn (z; µ); however, they do not mention the more general identities (2.2) Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

235 RELATIVE ASYMPTOTICS FOR GENERAL ORTHOGONAL POLYNOMIALS REMARKS ON RELATIVE ASYMPTOTICS FOR GENERAL ORTHOGONAL POLYNOMIALS3

and (2.3). Using (2.4) we provide a short proof of the well-known result of Fej´er (see, e.g. [2]) concerning the zeros of Pn (z; µ). Proposition 2.2. Pn (z; µ) has no zeros that lie outside the convex hull Co(Sµ ) of Sµ . / Co(Sµ ). Without loss of generality we Proof. Suppose Pn (z0 ) = 0 for some z0 ∈ assume that z0 ∈ R+ and that the imaginary axis separates Sµ from z0 . Then, from (2.4), we have (2.7)

< k(·, z0 ), |Pn |2 >µ = 0.

But Re k(t, z0 ) < 0 for all t ∈ Sµ , and so < k(·, z0 ), |Pn |2 >µ has negative real part, which contradicts (2.7). In view of (2.4), the same proof gives Corollary 2.3. For Pn = Pn (·; µ) we have (2.8)

χn (z) :=< k(·, z), Pn >µ = 0,

z∈ / Co(Sµ ).

Remark 2.4. If z ∈ Co(Sµ ), then (2.8) need no longer be true. Indeed, if we take dµ(t) = (2π)−1 dθ on the unit circle T, then Pn (z) = z n , and Pn (t) |dt| 1 1 t¯n dµ(t) = |dt| = = 0, |z| < 1, χn (z) = n (t − z) t − z 2π t − z 2π t T T T for each n ≥ 0. 3. Applications to Relative Asymptotics Hereafter we assume that the supports of the measures of orthogonality contain inﬁnitely many points. In this section we show how formula (2.5) can be utilized to obtain results on the comparative growth of polynomials that are orthonormal with respect to varying weights. We begin with Proposition 3.1. Let qn (z) = λn,n z n + · · · , λn,n > 0, be orthonormal with respect to the ﬁnite positive measure νn and pn (z) = τn,n z n + · · · , τn,n > 0, be orthonormal with respect to ρn (z)dνn (z), where the supports Sνn are all contained in a compact set K ⊂ C. If there exist positive constants m1 , m2 such that m1 ≤ ρn (z) ≤ m2 for all z ∈ Sνn , n = 0, 1, . . . , then for any closed set E ⊂ C \ Co(K) there exist positive constants c1 , c2 such that (3.1)

c1 ≤ |pn (z)/qn (z)| ≤ c2 ,

z ∈ E,

n = 0, 1, . . . .

Proof. First we observe that the assumption on the ρn ’s implies that (3.2)

√ √ λn,n m1 ≤ ≤ m2 , τn,n

n = 0, 1, . . . ;

indeed, λn,n /τn,n =< qn , pn >ρn νn , from which it follows by the Cauchy-Schwarz inequality that √ √ λn,n /τn,n ≤ qn ρn νn pn ρn νn ≤ m2 qn νn = m2 . Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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E. B. SAFF

Similarly, from the fact that τn,n /λn,n =< pn , qn >νn we deduce the lower bound in (3.2). From (2.5) with Pn = pn , Q = qn and µ = ρn νn we obtain that, for z ∈ / K, (3.3)

pn (z) < qn k(·, z), pn >ρn νn =< k(·, z), |pn |2 >ρn νn . qn (z)

Select a point z0 in K and deﬁne the functions fn (z) :=< (z0 − z)qn k(·, z), pn >ρn νn ,

gn (z) :=< (z0 − z)k(·, z), |pn |2 >ρn νn .

Observe that each fn and each gn is analytic in C \ K, even at inﬁnity. Moreover, for each n, we have gn (∞) = 1 and, from (3.3), it follows that fn (∞) = λn,n /τn,n . It is evident from the deﬁnition of the gn ’s that these functions are uniformly bounded on any closed set F ⊂ C\K by the constant MF := max{|(z0 −z)/(t−z)| : z ∈ F, t ∈ K} and thus they form a normal family of analytic functions in C \ K. The same is true for the fn ’s since, by the Cauchy-Schwarz inequality we have, for z ∈ F, the following estimate: √ |fn (z)| ≤ (z0 − z)qn k(·, z)ρn νn ≤ MF qn ρn νn ≤ MF m2 . Finally, we observe from Proposition 2.2, Corollary 2.3 and (3.3) that the fn ’s and gn ’s are zero-free in the domain C \ Co(K). Hence, by Hurwitz’s theorem, any limit function of these normal families is either identically zero or never zero in C \ Co(K). But, in view of (3.2) and of the previously noted values of fn (∞) and gn (∞), the former possibility cannot occur. Consequently, from (3.3), we deduce that the ratios pn (z)/qn (z) form a normal family of zero-free analytic functions in C \ Co(K) and that every limit function of this family is zero-free in this domain, from which the conclusion (3.1) follows. 3.0.1. Measures supported on the unit circle and on the disk. In this subsection we consider polynomials that are orthonormal with respect to varying measures that are either supported on the unit circle T or are absolutely continuous with respect to area measure over the open unit disk D. Suppose that for n = 0, 1, . . . , the polynomial pn (z) = τn,n z n +· · · is orthonormal with respect to the measure νn supported on the unit circle. Identity (2.5) with Q(z) = z n and Pn = pn becomes (2.5) (3.4)

pn (z) < ·n k(·, z), pn >νn =< k(·, z), |pn |2 >νn . zn

Setting (3.5)

fn (z) :=< −z ·n k(·, z), pn >νn ,

gn (z) :=< −zk(·, z), |pn |2 >νn ,

we observe that fn (z) can be written in terms of the inverse polynomial p∗n as fn (z) =< −zk(·, z), p∗n >νn . Using the fact that |pn (t)| = |p∗n (t)| for t ∈ Sνn we deduce that the fn ’s form a normal family of nonzero analytic functions in C \ D provided the total masses νn are uniformly bounded. Moreover, from (3.4), we see that fn (∞) = 1/τn,n , which will be uniformly bounded below by a positive constant provided the τn,n ’s are bounded above. Clearly the gn ’s form a normal family in C \ D , and so by Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

237 RELATIVE ASYMPTOTICS FOR GENERAL ORTHOGONAL POLYNOMIALS REMARKS ON RELATIVE ASYMPTOTICS FOR GENERAL ORTHOGONAL POLYNOMIALS5

arguing as above we obtain the following result for polynomials orthogonal on the unit circle with varying weights. Proposition 3.2. If pn (z) = τn,n z n + · · · , τn,n > 0, is orthonormal with respect to the measure νn with Sνn ⊂ T and the sequences {νn } and {τn,n } are bounded above, then for each closed set E ⊂ C \ D there exist positive constants c1 , c2 such that pn (z) (3.6) c1 ≤ n ≤ c2 , z ∈ E, n = 0, 1, . . . . z In particular, (3.6) holds if dνn (z) = ρn (z)|dz| on T and there exist positive constants m1 , m2 such that T ρn (z)|dz| ≤ m1 and ρn (z) ≥ m2 for all z ∈ T, n = 0, 1, . . . . Remark: For the case when νn = ν is independent of n, the condition that the coefﬁcients τn,n = τn are bounded is equivalent to ν being in the Szeg˝o class, for which it is well-known (cf.[6], Theorem 12.1.1) that the sequence {pn (z)/z n } converges to the Szeg˝ o function in E ⊂ C \ D. The above proposition can therefore be regarded as a weak form of that famous result. Next we consider measures that are absolutely continuous with respect to area measure dA over the unit disk D. We begin with the observation that for any radially symmetric density function w(z) = w(|z|) on D, the polynomials αn (w)z n are orthonormal with respect to w(z)dA, where 1 r 2n+1 w(r)dr]−1/2 . αn (w) = [2π 0

For example, if w = w is the characteristic function of the annulus {z : 1 − ≤ |z| ≤ 1}, then n+1 (3.7) αn (w ) = [1 − (1 − )2n+2 ]−1/2 . π Proposition 3.3. Let Qn (z) = κn,n z n + · · · , κn,n > 0, be orthonormal with respect to the measure dµn (z) = ρn (z)dA(z) on D. If there exist positive constants M1 , M2 such that √ (3.8) κn,n ≤ M1 n and ρn (z) ≤ M2 , z ∈ D, then, for any closed set E ⊂ C \ D, there exist positive constants c1 , c2 such that Qn (z) (3.9) c1 ≤ √ n ≤ c2 , z ∈ E, n = 1, 2, . . . . nz In particular, (3.9) holds if there exist positive constants m1 , M2 and such that ρn (z) ≤ M2 , z ∈ D, and ρn (z) ≥ m1 for 1 − ≤ |z| ≤ 1, n = 1, 2, . . . . Proof. Again from (2.5) we have (3.10)

√ Qn (z) √ n < −z n ·n k(·, z), Qn >µn =< −zk(·, z), |Qn |2 >µn , nz

and so (3.9) will follow if we show that √ f n (z) :=< −z n ·n k(·, z), Qn >µn Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

238 6

E. B. SAFF

√ is uniformly bounded on closed subsets of C \ D and f n (∞) = n/κn,n is uniformly bounded below by a positive constant. The latter condition is immediate from (3.8); the former property follows from the fact that on any closed subset E of C \ D, 1 √ |f n (z)|2 ≤ − z n ·n k(·, z)2µn ≤ K1 n |t|2n ρn (t)dA(t) ≤ K2 n r 2n+1 dr ≤ K3 , D

0

where the Ki ’s are constants. Regarding the last statement of the proposition, it suﬃces to note that κn,n 1 =< Qn , αn (w )·n >w dA ≤ Qn w dA ≤ √ , αn (w ) m1 which, in view of (3.7), implies the ﬁrst condition in (3.8). 3.0.2. Ratio asymptotics. A normal families argument can also be used to obtain a weak form of ratio asymptotics for the polynomials Pn (·; µ) by selecting z0 ∈ Sµ and applying (2.5) with Q(z) = (z − z0 )Pn−1 (z). Proposition 3.4. Let Pn (z) = κn z n + · · · , κn > 0, be orthonormal with respect to dµ and let z0 ∈ Sµ . If the ratios {κn /κn−1 } are bounded from above for some subsequence of integers N , then for any closed set E ⊂ C \ Co(Sµ ) there exist positive constants c1 , c2 such that Pn (z) κn−1 (3.11) c1 ≤ ≤ c2 , z ∈ E, n ∈ N . κn (z − z0 )Pn−1 (z) If, in addition, Sµ ⊂ D, then for any closed set E ⊂ C \ D there exist positive constants c3 , c4 such that P (z) n (3.12) c3 ≤ ≤ c4 , z ∈ E, n ∈ N . zPn−1 (z) We remark that for Sµ ⊂ D, the characterization 1 = min pµ , κn p=zn +··· can be used to show that κn /κn−1 ≥ 1, so that (3.12) follows from (3.11). We conclude with the following generalized form of a conjecture of B. Simon concerning possible extensions of Rakhmanov’s theorem (cf. [5]). Conjecture: If dµ(z) = ρ(z)dA, z ∈ D , where ρ(z) > 0 a.e. in an annulus A := {z : 1 − ≤ |z| ≤ 1}, then

in particular, κn /κn−1

Pn (z) → 1, zPn−1 (z) → 1 as n → ∞.

z ∈ C \ D;

4. Acknowledgments The author is grateful to Razvan Teodorescu for bringing to the author’s attention the work of Bender and Ben-Naim. Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

239 RELATIVE ASYMPTOTICS FOR GENERAL ORTHOGONAL POLYNOMIALS REMARKS ON RELATIVE ASYMPTOTICS FOR GENERAL ORTHOGONAL POLYNOMIALS7

References [1] Carl M. Bender and E. Ben-Naim, Nonlinear-integral-equation construction of orthogonal polynomials, J. Nonlinear Math. Phys. 15 (2008), suppl. 3, 73–80. [2] P.J. Davis, Interpolation and Approximation, Blaisdell Pub. Co, 1963; Dover reprint, 1975. [3] D. Khavinson and N. Stylianopoulos, Recurrence relations for orthogonal polynomials and algebraicity of solutions of the Dirichlet problem,in: Topics Around the Research of Vladimir Maz’ya, International Mathematical Series 11-13, (in press) Springer, New York, 2009. [4] M. Putinar and N. Stylianopoulos, Finite-term relations for planar orthogonal polynomials, Complex Anal. Oper. Theory, 1, (2007), 447–456. [5] B. Simon, Orthogonal Polynomials on the Unit Circle, Part 2: Spectral Theory, Amer. Math. Soc. Colloq. Pub. Vol. 54, Amer. Math. Soc., Providence, R.I., 2004. [6] G. Szeg˝ o, Orthogonal Polynomials, 3rd ed., Amer. Math. Soc. Colloq. Pub., Vol. 23. Amer. Math. Soc., Providence, R.I., 1967. Center for Constructive Approximation, Department of Mathematics, Vanderbilt University, Nashville, TN 37240

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http://dx.doi.org/10.1090/conm/507/09963

Contemporary Mathematics Volume 507, 2010

Fine Structure of the Zeros of Orthogonal Polynomials: A Progress Report Barry Simon∗ Abstract. We consider the asymptotics of zeros of OPRL and POPUC as n → ∞, focusing on the structure on a scale in x of order 1/n. We discuss three recent results (on Poisson behavior in the random case, clock on the a.c. case, and β-distribution in between) and open questions and speculations.

1. Introduction In this paper, we will discuss zeros of orthogonal polynomials. This subject goes back to Gauss’ discovery that the best discrete approximations of Riemann integrals involve zeros of Legendre polynomials, and has spawned a huge literature not only among workers on OPs but within the general theoretical and mathematical physics communities who study “eigenvalue statistics.” In particular, much of the work on random matrices is connected to this subject. The general theory has two levels, both going back to a 1940 paper of Erd¨os– Tur´ an [10]. The bulk behavior concerns what fraction of the n zeros of Pn lie in a given subset, S, of R. The ﬁne structure uses a microscope to look on a scale of size O( n1 ) where, as n → ∞, there are typically ﬁnitely many zeros. The two most heavily studied regimes are where the zeros are Poisson distributed, that is, no correlation between nearby zeros, and where there is rigid equal spacing between neighboring zeros. Over the past few years, I have written a series of papers on this subject [38, 39, 40, 21, 2], including one review [42], but it is time for another review of recent progress and open questions. We begin by setting notation and terminology. In much of this paper, we study orthogonal polynomials on the real line (OPRL). We start with a measure, dµ, on R of the form dµ(x) = w(x) dx + dµs (x) (1.1) where dµs is Lebesgue singular. We will only consider cases where dµ is compactly supported. See [46] for background on OPRL. 2000 Mathematics Subject Classiﬁcation. 26C10,65H17,05E35. Key words and phrases. Orthogonal polynomials, clock behavior, Poisson process. Supported in part by NSF grant DMS-0652919. c 0000 c (copyright 2010 Barry holder) Simon

1 241 Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

242 2

B. SIMON

Given µ, Pn (x) are the monic orthogonal polynomials and pn (x) the orthonormal polynomials. Pn (x) has n simple zeros, all on R, indeed on the convex hull of supp(dµ). We will be interested in zeros near a ﬁxed x0 ∈ R, which we label (n) xj (x0 ), with (n)

(n)

(n)

(n)

. . . x−k (x0 ) < · · · < x−1 (x0 ) < x0 ≤ x0 (x0 ) < x1 (x0 ) . . .

(1.2)

µ determines, and in turn is determined by, a set of Jacobi parameters, {an , bn }∞ n=1 , given by the recursion relations for the Pn : xPn (x) = Pn+1 (x) + bn+1 Pn (x) + a2n Pn−1 (x)

(1.3)

Here an > 0, bn ∈ R, and there is a one-one correspondence between uniformly bounded Jacobi parameters and nontrivial probability measures of compact support. We will also consider orthogonal polynomials on the unit circle (OPUC) and paraorthogonal polynomials (POPUC). See [36, 37] for a discussion of OPUC and [4, 43, 53] for POPUC. The OPUC are determined by µ with monic polynomials, Φn (z), and orthonormal, ϕn (z). The recursion relations now have the Szeg˝o form ¯ n Φ∗n (z) Φn+1 (z) = zΦn (z) − α

(1.4)

z) Φ∗n (z) = z n Φn (1/¯

(1.5)

where The αn ∈ D are called Verblunsky coeﬃcients and there is a one-one correspondence ∞ between {αn }∞ and nontrivial probability measures on ∂D. n=0 ∈ D n−1 and αn−1 ∈ ∂D by (1.4). The The POPUC are deﬁned by {αj }n−2 j=0 ∈ D new element is αn−1 ∈ ∂D, not in D. OPUC have all their zeros in D. POPUC have their zeros in ∂D and are simple, and so similar to OPRL. The bulk behavior of the zeros is described by deﬁning a probability measure, dνn , which assigns a weight 1/n to each zero of Pn in the OPRL case or, in the POPUC case, Φn . In the OPUC case, zeros can have multiplicity k > 1 and dνn gives such zeros a weight k/n. We say that the density of zeros (or density of states) exists if dνn has a weak limit, dν. It often happens that dν is the equilibrium measure for supp(dµ) (see [47, 41]) and even more generally that for some L1 (R, dx) function, dν(x) = ρ(x) dx (1.6) When one turns to ﬁne structure, two regimes have received the most attention—indeed, until very recently, all the attention. One is the regime of truly random recursion parameters where the distribution of zeros, after scaling distances by n about a point x0 , is a Poisson process. The key early papers are by Molchanov [32] and Minami [31]. The other extreme is the nice a.c. spectrum region where it is often known that one has clock behavior 1 (n) (n) xj+1 (x0 ) − xj (x0 ) ∼ (1.7) nρ(x0 ) Here the key works are Erd¨os–Tur´ an [10], Freud [12], and Deift et al. [9], and this is the main area I studied in [38, 39, 40, 21]. Freud realized a connection of zeros and the asymptotics of the CD kernel slightly oﬀ diagonal, so we recall Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

FINE STRUCTURE OF THE ZEROS OF OP: A PROGRESS REPORT

243 3

the deﬁnition and some properties of the kernel. The Christoﬀel–Darboux kernel is deﬁned for x, y ∈ R by Kn (x, y) =

n

pj (x)pj (y)

(1.8)

j=0

The connection to zeros depends on the CD formula Kn (x, y) =

an+1 (pn+1 (x)pn (y) − pn+1 (y)pn (x)) x−y

(1.9)

For a review of the CD kernel, see [44]. One aspect of CD kernel asymptotics is “classical.” It involves the notion that the diagonal asymptotics is often given by 1 ρ(x) Kn (x, x) ∼ n w(x)

(1.10)

a notion associated to work of Freud and Nevai, discussed by Nevai in [33]. This Freud–Nevai vision was realized especially in M´ at´e–Nevai–Totik [30] and Totik [49]. A revolution in establishing oﬀ-diagonal asymptotics came from two remarkable papers of Lubinsky [27, 25], part of a series that also includes [22, 23, 24, 26] and several other papers. In particular, his second approach in [25] will play a major role below. Before leaving the subject of CD kernels, we want to recall the second kind polynomials, qn (x), deﬁned as follows. Let p˜n (x) be the OPRL associated to oncestripped Jacobi parameters, that is, to {˜ an , ˜bn }∞ n=1 where a ˜n = an+1

˜bn = bn+1

(1.11)

Then ˜n (x) qn (x) = a−1 1 p

(1.12)

We will need the associated CD kernel Kn(q) (x, y) =

n

qj (x)qj (y)

(1.13)

j=0

Section 2 describes “older” results, including Lubinsky’s ﬁrst approach [27] to the oﬀ-diagonal CD kernel. In particular, it describes in some detail Poisson and clock behavior. Section 3 discusses three recent signiﬁcant results on the Poisson, intermediate, and clock regimes. Section 4 discusses open questions. It is pleasure to thank Paco Marcell´ an and Andrei Martinez-Finkelstein for the invitation to speak at the IWOPA’08 conference and Bill L´opez for the excuse he gave us for the conference (and for his many signal contributions). I would like to thank Jonathan Breuer, Rowan Killip, Nikolai Makarov, Andrei MartinezFinkelshtein, Eric Ryckman, Mihai Stoiciu, and Vilmos Totik for useful discussions. 2. The Past Is Prologue Here we will recall some results that describe the two main types of reasonably well-understood ﬁne structure. We begin with Stoiciu’s results [48] on Poisson behavior for random POPUC (recalling that Molchanov [32] and Minami [31] were the pioneers in Poisson behavior for OPs): Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

244 4

B. SIMON

Theorem 2.1 (Stoiciu [48]). Fix R < 1. Let ∞ Ω = × {z ∈ C | |z| ≤ R} × ∂D n=0

and let dη be the measure on Ω describing independent, identically distributed random variables, uniformly distributed on {z | |z| ≤ R} with ω∞ independent and uniformly distributed on ∂D. For ω ∈ Ω, let Φn (z; ω) be the POPUC with αj = ωj , j = 0, . . . , n − 2; αn−1 = ω∞ . Then for any z0 = eiθ0 ∈ ∂D and any a1 < b1 < · · · < ak < bk ﬁxed, 2πaj 2πbj , θ0 + , lim Prob Φn (z; ω) has exactly mj zeros in θ ∈ θ0 + n→∞ n n (2.1) k (bj − aj )mj −(bj −aj ) e j = 1, . . . , k = mj ! j=1 Remarks. 1. What is critical is rotation invariance of the individual distribution, not the exact form. 2. For related results on the zeros of the OPUC in this case, see Davies–Simon [8]. Two things are critical in this proof: (a) Exponential localization of the eigenstates, which implies asymptotic independence of zeros produced by subboxes. (b) An estimate that assures the probability of two zeros in an interval of size 2πε/n is o(ε). The best results on clock spacing for OPRL with spectrum [−2, 2] is Theorem 2.2 (Lubinsky [27, 22]). Let dµ be a measure supported on [−2, 2] which is regular. Let [a, b] be an interval in [−2, 2] so that on [a, b], dµ(x) = w(x) dx with w continuous and strictly positive. Deﬁne x0 ∈ (a, b), (n)

(n)

(2.2)

(n) xj (x0 )

by (1.2). Then for any

1 ρ(x0 )

(2.3)

n(xj+1 (x0 ) − xj (x0 )) → where ρ(x0 ) is given by (1.6).

Remarks. 1. Regularity here means (a1 . . . an )1/n → 1; see [47, 41]. 2. (2.3) is called clock behavior. 3. It is known [11, 27, 45, 51] that the three conditions, w(x) continuous and nonvanishing and dµs = 0, can be replaced by the three conditions of local Szeg˝o condition, x0 being a Lebesgue point for w, and limk→∞ kµs (x0 − k1 , x0 + k1 ) = 0. Also, supp(dµ) = [−2, 2] can be replaced by an essential support requirement. Lubinsky’s approach is related to proving universality. Let Kn be the CD kernel given by (1.8). Universality at x0 says (uniformly in |a| < A, |b| < A for each A < ∞) lim

n→∞

Kn (x0 + na , x0 + nb ) sin(πρ(x0 )(b − a)) = Kn (x0 , x0 ) πρ(x0 )(b − a)

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(2.4)

FINE STRUCTURE OF THE ZEROS OF OP: A PROGRESS REPORT

245 5

Since sin(π(c − d)) = 0 ⇔ c − d ∈ Z and the CD formula, (1.9), says that if pn (x1 ) = 0, then pn (x2 ) = 0 for x2 = x1 if and only if Kn (x1 , x2 ) = 0, (2.4) should be connected to clock spacing. Indeed, Theorem 2.3 (Freud–Levin Theorem). Universality, (2.4), at x0 implies clock spacing, (2.3). Remarks. 1. This result is implicit in Freud [12] who proved universality for suﬃciently nice measures on [−2, 2]. It was rediscovered by Levin and reported in Levin–Lubinsky [22]. 2. Under some additional growth assumptions on Kn , clock spacing implies universality; see, for example, [23]. The same argument relates two weaker sets of notions. The striking thing about clock behavior is equal spacing, which is independent of the spacing multiplied by n having a limit. We say quasi-clock behavior holds at x0 if for any ﬁxed j ∈ Z as n → ∞, (n) (n) xj+1 (x0 ) − xj (x0 ) →1 (2.5) (n) (n) x1 (x0 ) − x0 (x0 ) Deﬁne 1 ρn (x0 ) = w(x0 )Kn (x0 , x0 ) (2.6) n We say weak universality holds if Kn (x0 +

a nρn

, x0 +

Kn (x0 , x0 )

b nρn )

→

sin(π(b − a)) π(b − a)

(2.7)

Theorem 2.4 (Weak Freud–Levin). Weak universality at x0 implies quasi-clock behavior at x0 . If ρn (x0 ) → ρ(x0 ) (2.8) then weak universality implies universality. This links up ﬁne structure to the Freud–Nevai vision discussed in Section 1. Lubinsky proved Theorem 2.2 by using a model measure for which universality holds, and then used a clever comparison argument. He used Legendre polynomials as his model, but the calculations are more explicit and easier for Chebyshev polynomials of either the ﬁrst or second kind. Lubinsky’s result was extended to general compact subsets, e, of R by Findley [11], Simon [45], and Totik [51]. Findley and Totik use the method of polynomial mappings and approximation [49, 50]. Simon uses approximation and Jost functions for the isospectral torus as a model. One important aspect of the second approach of Lubinsky [25], discussed in the next section, is that it goes beyond the need for a model to compare to. 3. The Present: Three Breakthroughs We want to focus on three recent major results that deal with three diﬀerent regimes of zeros: (1) The result of Combes, Germinet, and Klein [7] on Poisson statistics for multidimensional Schr¨ odinger operators. (2) The results of Killip and Stoiciu [15] on decaying random models. Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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(3) The new approach of Lubinsky [25] and the associated work of Avila, Last, and Simon [2] on clock spacing. We will say very little about (1) since it is peripheral to our concerns here and a lot about (3) since it was the content of my talk at the conference for which this is the proceedings. The bulk of this review concerns OPs which arise from one-dimensional difference equations. Combes–Germinet–Klein [7] study the continuum diﬀerential Schr¨odinger operator, −∆ + V , in higher dimension. While the initial results on Poisson statistics [32] were on continuum models, they were deﬁnitely onedimensional. Minami [31] could handle arbitrary dimensions, but his models were discrete. In part, he had one key result (the Minami lemma) that seemed to be restricted to rank one perturbations. Extending and understanding this (done by Bellissard–Hislop–Stolz [3], Combes–Germinet–Klein [6], and Graf–Vaghi [13]) was a ﬁrst step—but even with it, the full argument in [7] is a subtle piece of work that settles a problem that has been open for more than ﬁfteen years. Killip–Stoiciu [15] discuss POPUC with random decaying Verblunsky coeﬃcients. They require rotation invariance, so to simplify exposition, we will suppose the distributions are over suitable disks. Thus, we suppose An (ω) are iidrv with individual distribution uniform over the disk of radius 12 . c0 , c1 , . . . will be a sequence with 0 < cj ≤ 1 and we take αn (ω) = cn An (ω)

(3.1)

For background, we recall the following from [37]: Theorem 3.1. Let αn obey (3.1) where cn < 1 and for some N0 and n > N0 , 0 < γ < 1, cn = kn−γ Then, with dµω the measure with Verblunsky coeﬃcients αn (ω), (i) If γ > 12 for a.e. ω, dµω is purely absolutely continuous with support ∂D and for Lebesgue a.e. θ, bounded transfer matrix at z = eiθ . (ii) If γ < 12 , for a.e. ω, dµω is pure point with eigenvalues dense in ∂D and eigenvectors decaying as fast as exp(−dn1−2γ ). (iii) For γ = 12 , if k2 > 8, for a.e. ω, dµω is dense pure point with polynomially decaying eigenfunctions. If k2 ≤ 8, dµω is purely singular continuous with constant Hausdorﬀ dimension 1 − 18 k2 . Remarks. 1. |An (ω)|2 = 18 , so the Γ of (12.7.11) of [37] for γ =

1 2

is Γ =

√k . 8

2. Constant Hausdorﬀ dimension, d, means µω (A) = 0 for any set of dimension smaller than d, and µω is supported on a set of dimension d. Killip–Stoiciu have results on the zeros that have a similar three-part breakdown. For POPUC, we need a phase, αn−1 ∈ ∂D, which we take random, uniform on ∂D, and independent of αj (ω): Theorem 3.2 (Killip–Stoiciu [15]). Under the hypotheses of Theorem 3.1, (i) For γ > 12 , for a.e. ω and z0 , the zeros of the POPUC have clock spacing (n) with density 2π/n in the sense that if z0 = eiθ0 and θj (θ0 ) are deﬁned by (n)

(n)

. . . < θ−1 < θ0 ≤ θn near z0 , then

(n)

(n)

< θ−2 < . . . , so eiθj (n)

(θ0 )

are all the zeros of Φ(αn ) (z)

(n)

n(θj+1 − θj ) → 2π Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

(3.2)

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as n → ∞. (ii) For γ < 12 , the zeros of the POPUC are locally Poisson distributed in the sense of Theorem 2.1. (iii) If γ = 12 , the zeros asymptotically have a CβE distribution where 16 (3.3) k2 By a ﬁnite CβE (for circular beta ensemble) distribution, we mean the measure on (∂D)n given by the density for (eiθ1 , . . . , eiθn ) −1 Nβ,n |θj − θk |β (3.4) β=

j 0. (ii) For a.e. x0 ∈ Σ0 , 1 ρ∞ (x0 ) Kn (x0 , x0 ) → >0 n+1 w(x0 ) (q)

(iii) For a.e. x0 ∈ Σ with Kn

sup n

(3.5)

given by (1.13), 1 K (q) (x0 , x0 ) < ∞ n+1 n

(3.6)

Then for a.e. x0 ∈ Σ, universality (and so, quasi-clock behavior) holds at x0 . Remark. ρ∞ is not assumed to be the density of a density of zeros. Rather, it is deﬁned by (3.5), that is, condition (ii) is a statement that the limit exists (and is ﬁnite and nonzero). Furthermore, [2] shows the hypotheses hold in the ergodic case. Let (Ω, dη(ω)) be a probability measure space where Ω is a compact metric. Let T : Ω → Ω be continuous and invertible and ergodic for dη. Let A, B be continuous functions on Ω where A has values in (0, ∞) and B in R. To each ω ∈ Ω, we deﬁne a Jacobi matrix with parameters an (ω) = A(T n−1 ω)

bn (ω) = B(T n−1 ω)

(3.7)

A canonical example is the almost Mathieu equation where λ ∈ (0, ∞) and α ∈ R \ Q are ﬁxed, Ω = ∂D, dη = dθ/2π, T (eiθ ) = ei(θ+πα) , A(eiθ ) ≡ 1, B(eiθ ) = 2λ cos(θ) (so bn (eiθ0 ) = 2λ cos(παn + θ0 )). This model is known to have purely a.c. spectrum if 0 < λ < 1. Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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Theorem 3.4 ([2]). For any ergodic Jacobi matrix with |Σac | > 0, for a.e. ω ∈ Ω and a.e. x0 ∈ Σac , 1 Kn (x0 , x0 ) = ρ∞ (x0 ) (3.8) lim w(x0 ) n→∞ n+1 where ρ∞ (x0 ) is the density of the a.c. part of the density of zeros. Remarks. 1. Ergodicity implies the density of zeros exists. It is a result of Kotani [18] (see Simon [35] for the discrete case) that Σac = {x | γ(x) = 0}

(3.9)

where γ is the Lyapunov exponent, and of Kotani [19] that on this set, the a.c. part of the density of zeros, ρ∞ (x), is exactly the average of weights, w. 3. For regular measures with local Szeg˝ o conditions on the weight (false when Σac is a Cantor set), results like (3.8) are known due to M´at´e–Nevai–Totik [30] and Totik [49]. Here we want to emphasize the ideas of Lubinsky [25] that get the kernel sin(πx)/πx. The following is essentially implicit in his paper: (i) (ii) (iii) (iv) (v)

Theorem 3.5. Let f (z) be an entire function obeying ∞ 2 |f (x)| dx ≤ 1. −∞ f (0) = 1; |f (x)| ≤ 1 on R. For constants C and A, |f (z)| ≤ CeA|z| for all z ∈ C. f is real on R; all zeros of f lie on R. If . . . x−n < · · · < x−1 < 0 < x1 < · · · < xn < . . . are all the zeros of f , then |xj | ≥ |j| − 1

(3.10)

Then

sin πz (3.11) πz Remarks. 1. Lubinsky uses stronger hypotheses. In this form, it appears in [2]. [2] can replace (iii) by a weaker hypothesis if (3.10) is replaced by |xj − xk | ≥ |j − k| − 1. The ALS proof of this result proceeds by using (iii) to get a Hadamard factorization z αz f (z) = e 1− ez/xj (3.12) xj f (z) =

j∈Z j=0

with α real. Thus,

y2 1+ 2 xj j=0

∞ which, by (v) and the Euler formula for sinh(x) = x n=1 (1 + |f (iy)|2 ≤

(3.13) x2 n2 ),

|f (iy)| ≤ Cε e(π+ε)|y|

leads to (3.14)

Phragm´en–Lindel¨of, (3.14), and f bounded on R yields |f (x + iy)| ≤ Cε e(π+ε)|y| Thus, by the Paley–Wiener theorem, fˆ(k), the Fourier transform of f , is supported in [−π, π]. (i), (ii), and the Schwarz inequality imply that fˆ(k) is (2π)−1/2 Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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times the characteristic function of [−π, π]. The Fourier transform of this characteristic function is sin(πz)/πz. 3. Lubinsky appeals to various results on the sin(z)/z kernel, but the proofs of these results depend on Paley–Wiener as above. Lubinsky [25] used this result to prove Theorem 3.6 ([25]). Suppose dµ has a weight, w, and an interval, I, so that (i) inf w(x) > 0

(3.15)

x∈I

(ii) For x0 ∈ I int , we have that x0 is a Lebesgue point of dµ in that limδ↓0 (2δ)−1 |w(x) − w(x0 )| dx = 0 and limδ↓0 (2δ)−1 µs (x0 − δ, x0 + δ) = 0. (iii) For a real, Km (x0 + na , x0 + na ) →1 (3.16) Kn (x0 , x0 ) uniformly in |a| ≤ A for any A > 0. Then weak universality (and so, quasi-clock behavior) holds at x0 . Remark. (3.16) is called the Lubinsky wiggle condition. Lubinsky gets his result by ﬁxing a ∈ R and considering limit points, f (z), of Kn (x0 +

a nρn , x0

+

a+z nρn )

Kn (x0 , x0 ) By Montel’s theorem and the a priori bound, 1 w z lim sup Kn x0 + , x0 + ≤ CeA(|z|+|w|) n n n→∞ n

(3.17)

(3.18)

(which follows from the Schwarz inequality and the case w = z which one gets from (3.15) and the Christoﬀel variational principle [44]) limit points exist, and if all limit points are sin(πx)/πz, then the limit exists. Condition (i) of Theorem 3.5 follows from the use of Kn (x, y)Kn (y, x) dµ(y) = Kn (x, x) (3.19) and the use of Lebesgue points (and ρn scaling). Condition (ii) follows from the Lubinsky wiggle condition (3.16). Condition (iii) follows from (3.18). Condition (iv) follows from properties of the CD kernel [44]. Condition (v) follows from a clever argument of Lubinsky using the Markov–Stieltjes inequalities [44] and the wiggle condition. In his paper, Lubinsky could only prove the wiggle condition in situations where Totik’s methods [49, 50] hold and where Totik already used Lubinsky’s ﬁrst method ([51]). But Lubinsky expressed his belief (vindicated by [2]) that the wiggle condition could be proven in other cases. In proving Theorem 3.3, [2] cannot use (3.15) and a comparison argument to get (3.18) since (3.15) is not assumed. Instead, they use a general perturbation bound that gets exponential bounds from boundedness of the Ces`aro averaged transfer

n 1 2 matrix n+1 T (x j 0 ) (Theorem 3 of [2]). They get the wiggle condition from j=0 the assumed existence (3.5), a use of Egoroﬀ’s theorem, and an equicontinuity result for Kn (x0 + na , x0 + na ) that follows from (3.18). Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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To get Theorem 3.4, [2] prove the a.e. existence of the limit in (3.8) by a use of the ergodic theorem (which is subtle since w(x0 ) is ω-dependent) and separate arguments that prove the limit is ω-independent, and then that it is ρ∞ (x0 ). 4. The Future: Open Questions and Speculations This section has two parts. First, we discuss questions left open in each of the three limit regions we partially understand: Poisson, clock, and β-distributed. Then we speculate about other singular continuous situations. Poisson Questions. With regard to Poisson behavior, one especially interesting question is the following: Question 4.1. Is it true in the Anderson case (OPRL, with an ≡ 1 and bn independent, identically distributed random variables (iidrv) with nice density) that asymptotically as n → ∞, for any x0 = x1 , {number of zeros of pn (x) in [x0 + a b c d n , x0 + n ]} and {number of zeros of pn (x0 ) in [x1 + n , x1 + n ]} are independent? Here a < b and c < d. If x0 = x1 and a < b < c < d, this independence is part of what is known as Poisson behavior. Intuitively, independence of nearby O( n1 ) boxes would seem less likely than distant boxes, and one expects the answer to this question is yes. But it is open. Question 4.2. Prove the analog of the Killip–Stoiciu Poisson result for OPRL. That is, if an ≡ 1 and bn = cn An (ω) where An are iid, say uniformly distributed on [−1, 1], and cn = kn−γ for γ < 12 , then zeros are locally Poisson distributed. We do not expect this to be hard. Question 4.3. What is the ﬁne structure of the eigenvalues of the Anderson model near the edges of the spectrum? This is not here because we necessarily expect the answer to have anything to do with Poisson, but because this subsection is really on situations where one has eigenfunctions decaying at least as fast as exp(−nα ) for some α > 0. This question is asking for reﬁnements of Lifschitz tails (see [16] for a discussion of Lifschitz tails). The basic questions are how big an interval at the top of the spectrum do you need to get O(1) for the expected number of zeros, and what are their statistics. Question 4.4. Poisson behavior is a statement about probabilities. Since probabilities on O( n1 ) are not constant, on that scale there is not almost sure behavior, but there may be almost sure√behavior on larger scales. For example, one might expect almost surely on a (1/ n) scale that the fraction of neighboring pairs of eigenvalues with separations in [0, nc ] is given by the expected number in the Poisson process. What can be said about such almost sure behavior? Clock Questions. The most interesting open question is a conjecture of [2]. Question 4.5. Prove the following conjecture of Avila, Last, and Simon [2]: Consider a probability measure of compact support given by (1.1). For Lebesgue a.e. x in {x | w(x) > 0}, one has quasi-clock behavior. [41] has an example where supp(dµ) = [−2, 2], {x | w(x) > 0} = [−2, 0], dµ has dense mass points in [0, 2], and the density of zeros does not exist. Indeed, both the equilibrium measures for [−2, 0] and [−2, 2] are limit points of the density of zeros. Thus, clock behavior does not hold (and ρn (x) does not have a limit). But weak universality can hold—and I believe it does. I suspect that any proof for this special case will allow a treatment of the general conjecture. Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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Question 4.6. What can be said about edge behavior in the a.c. ergodic case? Consider ﬁrst the deterministic case in a single interval. Suppose a, b ∈ supp(dµ) but (a, b) is disjoint from this support. One cannot have clock behavior at a since (a, b) has at most one eigenvalue. For the edge of measures on [−1, 1] with Jacobi weight asymptotics at an edge, Lubinsky [24, 26] has proven Bessel kernel behavior rather than sin kernel. It is reasonable to guess that unlike bulk behavior which is universal across the a.c. ergodic case, edge behavior is not. In particular, for the almost Mathieu equation at 0 < |λ| < 1, I would guess that if α has good Diophantine properties, the edge behavior is the same as for Chebyshev of the ﬁrst kind (i.e., the whole-line free Jacobi matrix). But if α is a Liouville number, the behavior is diﬀerent—most likely diﬀerent on diﬀerent scales. Question 4.7. Prove the OPRL analog of the Killip–Stoiciu [15] result in the a.c. region, namely if an ≡ 1, bn = cn An (w), where An are as in Question 4.2, with cn = kn−γ and 1 ≥ γ > 12 , then one has clock behavior for a.e. ω. Given prior work on this case ([17]), this should be straightforward. The result for γ > 1 follows from [21]. β-distribution Questions. With regard to β-distributions, we pose the analog of Questions 4.2 and 4.7, namely Question 4.8. Determine the asymptotic local zero distribution of OPRL with an ≡ 1, bn = cn An (w), and cn = kn−1/2 . Now the rate of polynomial decay of eigenfunctions and local Hausdorﬀ dimension [17] is a function of both k and x. Presumably that is true of the β in the β-distribution. We have no good tools for identifying β-distributions directly ([15] use the known Verblunsky coeﬃcients of Haar measure), so this seems diﬃcult. Speculations on Singular Spectrum. Singular spectrum is like Tolstoy’s remark on dysfunctional families—singular spectrum is unusual in many ways, but that’s precisely the point—each is unusual in its own manner. The one example we understand, that of [15], has power decaying eigenfunctions and spectrum (i.e., closed support of the measure) which is an interval. But there are also examples where the support is a closed set of measure zero. Here are two examples: Question 4.9. Let dµ be the classical Cantor measure on the middle third sets. What can be said about the an ’s and about the local structure of the zeros? Question 4.10. Consider the almost Mathieu equation at the critical coupling, λ = 1. It is known (see Last [20]) that for irrational frequencies, the spectrum is singular continuous and of measure zero. What can be said about the global and the local structure of the zeros? Let us expand on these two examples. It is known (see, e.g., Makarov [28, 29]) that the potential theory equilibrium measure for the classical Cantor set lives on a set of Hausdorﬀ dimension strictly smaller than the dimension of the Cantor set itself. Totik (private communication) has informed me that his methods with Stahl [47] allow one to prove that this equilibrium measure is the density of zeros measure for this case. Here is a bold, probably foolhardy, speculation: Perhaps in this case, these zeros are, a.e. with respect to the equilibrium measure, quasi-clock distributed. Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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1 The spacing though, rather than O( n1 ), is O( n1/d ) where d is the dimension of the support of the equilibrium measure. For this case, because of reﬂection symmetry about x = 12 , we have bn ≡ 12 . Motivated by the ﬁnite gap case (see [52, 1, 34, 5]), I would conjecture that the an ’s are almost periodic with frequency module determined by the harmonic measures of the subsets of the Cantor set between two gaps. Motivated by these aspects of the density of zeros for the Cantor measure:

Question 4.11. Does the density of zeros for the critical almost Mathieu model live on a set of smaller local Hausdorﬀ dimension than the spectrum? As discussed in [41], for noncritical coupling, the density of zeros is the equilibrium measure for the spectrum and that likely persists at critical coupling. It seems to be at least possible that the zeros for the critical almost Mathieu model are quasi-clock spaced but with O(n−α ) (α > 1) spacing in the bulk.

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[45] B. Simon, Two extensions of Lubinsky’s universality theorem, J. Anal. Math. 105 (2008), 345–362. [46] B. Simon, Szeg˝ o’s Theorem and Its Descendants: Spectral Theory for L2 Perturbations of Orthogonal Polynomials, in preparation; to be published by Princeton University Press. [47] H. Stahl and V. Totik, General Orthogonal Polynomials, in “Encyclopedia of Mathematics and Its Applications,” 43, Cambridge University Press, Cambridge, 1992. [48] M. Stoiciu, The statistical distribution of the zeros of random paraorthogonal polynomials on the unit circle, J. Approx. Theory 39 (2006), 29–64. [49] V. Totik, Asymptotics for Christoﬀel functions for general measures on the real line, J. Anal. Math. 81 (2000), 283–303. [50] V. Totik, Polynomial inverse images and polynomial inequalities, Acta Math. 187 (2001), 139–160. [51] V. Totik, Universality and ﬁne zero spacing on general sets, in preparation. [52] H. Widom, Extremal polynomials associated with a system of curves in the complex plane, Adv. in Math. 3 (1969), 127–232. [53] M.-W. L. Wong, First and second kind paraorthogonal polynomials and their zeros, J. Approx. Theory 146 (2007), 282–293. Mathematics 253-37, California Institute of Technology, Pasadena, CA 91125, USA E-mail address: [email protected]

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http://dx.doi.org/10.1090/conm/507/09964

Contemporary Mathematics Volume 507, 2010

A Potential-Theoretic Problem Connected with Complex Orthogonality Herbert Stahl Dedicated to Guillermo Lopez Lagomasino on the occasion of his 60th birthday.

Abstract. A potential-theoretic problem is studied that arises in connection with complex orthogonality relations. Let K ⊂ C be a compact set that possesses a symmetry property with respect to the Green function g(·, ∞) in the domain C \ K that is typical for this type of analysis. (The property is equivalent to a principle of minimum capacity). It is shown that for any positive measure µ of mass at most 1 that is diﬀerent from the equilibrium distribution ωK on K, there exists a companion measure ν with a mass less than 1 such that the logarithmic potential of the sum µ + ν assumes its minimum on K at a point with a neighborhood in which the potential of µ + ν also possesses a symmetric behavior with respect to K. The connection of this problem with the asymptotic analysis of orthogonal polynomials satisfying a complex orthogonality relation is discussed in some detail. Thus, for instance, the measure µ typically is an hypothetical asymptotic distribution of zeros of orthogonal polynomials, and the result plays a key role in proving that such an asymptotic distribution has to be the equilibrium distribution ωK .

1. Main Result 1.1. The Background. The study of Pad´e approximants, rational interpolants, or rational best approximants in the complex plane is closely connected with the analysis of orthogonal polynomials pn of deg pn ≤ n that satisfy a relation of the form j z pn (z) f (z)dz = 0 for j = 0, . . . , n − 1, (1.1) C wn (z) where the wn are given functions, which typically are polynomials (cf., [Nut77], [NS77], [Sta86a], [Sta86b], [Nut86], [GR87], [Gon87], [Sta89], [Nut90], [Sta90], [Apt02], [SS08]). In the present paper we are concerned with the special case wn ≡ 1, i.e., the unweighted complex orthogonality z j pn (z)f (z)dz = 0 for j = 0, . . . , n − 1, (1.2) C

1991 Mathematics Subject Classiﬁcation. 31A15, 31A05, 30C85. Key words and phrases. Logarithmic potentials, complex orthogonality, symmetry property. Research has been supported by the Deutsche Forschungsgemeinschaft (AZ: STA 299/13-1). 255 Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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where we assume that the function f is analytic near ∞. The integration path C is chosen in the region of analyticity of f , and therefore there exists a lot of freedom for its choice. For the investigations of the convergence of the diﬀerent types of approximants it is necessary to have knowledge about the asymptotic location of the zeros of polynomials pn that satisfy relation (1.1) or (1.2), and often it is also desirable to know the asymptotic distribution of these zeros. In many situations of interest, almost all zeros cluster asymptotically (n → ∞) on a compact set K ⊂ C that possesses a certain symmetry property, which will be speciﬁed in more detail further below, and in case of relation (1.2) the asymptotic distribution of the zeros typically is the equilibrium measure ωK associated with the set K. Result of the type just mentioned have been proved in [Sta86a] for relation (1.2) and in [GR87] and [SS08] for orthogonality relations with wn not identical 1. In the present paper we are not concerned with all aspects of such an asymptotic analysis, nor do we study the symmetry property of the set K ⊂ C, which is also a fundamental topic for complex orthogonality. Our issue here is a problem that is purely potential-theoretic in nature, but nevertheless very important for the asymptotic analysis of complex orthogonal polynomials. Deﬁnitions and assumptions that are necessary for a formal statement of our main result will be introduced after the next subsection. 1.2. The Aims of the Present Paper. In the main theorem we show that if a compact set K ⊂ C of positive capacity possesses the symmetry property, then for any given positive measure µ of mass at most 1 that is diﬀerent from the equilibrium distribution ωK on K one can ﬁnd a companion measure ν with a mass less than 1 such that the logarithmic potential of the sum µ + ν assumes its minimum on K at a single point z0 ∈ K \ E0 , and in a neighborhood of this point z0 the logarithmic potential of µ + ν possesses as speciﬁc symmetric behavior with respect to K. The compact set E0 appears already in the deﬁnition of the symmetry property as an exceptional set of capacity zero, and it plays also a signiﬁcant role in the asymptotic analysis of complex orthogonal polynomials. In applications the given measure µ is typically a hypothetical asymptotic distribution of zeros of orthogonal polynomials pn that satisfy relation (1.2), and the conclusions in the main theorem are a major tool for proving that the equilibrium measure ωK is the only possibility for such an asymptotic distribution. Concepts and results similar to our main theorem have already been used in [Sta86a], [Sta86b], [GR87], and [Sta89] for the study of convergence of Pad´e approximants and rational best approximants. However, there the development of the concepts has been done in an ad hoc manner, and the proofs are limited to the very speciﬁc situation. We hope that the more systematic treatment in the present paper will shed more light on the potential-theoretic background of the analysis. The diﬀerence between relation (1.1) and (1.2) means in potential-theoretic terms that we have to deal with an external ﬁeld in case of the weighted orthogonality in (1.1), while this is not necessary in case of relation (1.2). It is hoped that the systematic approach to the problem in the present paper will also be helpful for the study of the analogous problem in case of an orthogonality relation of type (1.1). In the next subsection we introduce several deﬁnitions and prove some auxiliary results. Among them properties of a reﬂection function Φ that is closely Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

A POTENTIAL-THEORETIC PROBLEM

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connected with the symmetry property of the set K. The main theorem is then formulated and discussed. It is proved in Section 2. Some auxiliary results of an exclusively potential-theoretic nature that can be formulated without reference to the symmetry property of the set K are proved in Section 3. 1.3. The Symmetry Property. By M(S) we denote the set of all positive measures deﬁned on a closed set S ⊂ C such that the measure is ﬁnite on a compact subset, and by M1 (S) the subset of probability measures. We deﬁne the logarithmic potential p(µ; ·) of a measure µ ∈ M(C) as 1 |x| def (1.3) p(µ; z) = dµ(x) + dµ(x), log log |z − x| |z − x| D CD which is somewhat diﬀerent from the usual deﬁnition. However, it has the advantage that its existence is guaranteed even if the measure µ has high density near ∞. In applications, where the measure µ ∈ M(C) typically is a hypothetical asymptotic distribution of zeros of orthogonal polynomials pn of a complex orthogonality relation of type (1.2), we often have not enough information to exclude that the measure µ has too much mass near ∞ for the usual deﬁnition of the logarithmic potential p(µ; ·). If supp(µ) ⊂ D, then obviously deﬁnition (1.3) coincides with the usual one. We say that a property holds quasi everywhere (in short: qu.e.) on a set S ⊂ C if it holds on S with possible exceptions on a set of (inner) capacity zero. Definition 1. Let K ⊂ C be a compact set of positive capacity. The probability measure ωK ∈ M1 (K) that satisﬁes (1.4)

p(ωK ; z) = c0

for quasi every

z∈K

with c0 ∈ R is called equilibrium distribution or equilibrium measure on K (cf., [Ran95, Theorem 3.3.2]). By gC\K (·, ∞) we denote the Green function in the unbounded component of C \ K (cf., [Ran95, Section 4.4]), for which we have the representation (1.5)

gC\K (·, ∞) = p(ωK ; ·) − c0 .

Because of the somewhat nonstandard deﬁnition of the logarithmic potential in (1.3), the usual expression for the constant c0 in (1.4) and (1.5) has to be modiﬁed. We have 1 (1.6) c0 = log + log |x|dωK (x). cap(K) CD We now come to the important deﬁnition of the symmetry property of a compact set K ⊂ C. Definition 2. A set K ⊂ C is said to possess the symmetry property with respect to the Green function gC\K (·, ∞) and an exceptional set E0 ⊂ K if the following two assertions hold true: (i) The set K is compact and of positive capacity, the set E0 ⊂ K is also compact, but of capacity zero, and we have (1.7) K \ E0 = γi i∈I

with γi , i ∈ I ⊂ N, being disjoint, open, analytic Jordan arcs, and C \ K is connected. Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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(ii) The Green function gC\K (·, ∞) satisﬁes the symmetry relation (1.8)

∂ ∂ gC\K (z, ∞) = g (z, ∞) for z ∈ γi , i ∈ I, ∂n+ ∂n− C\K where ∂/∂n+ and ∂/∂n− denote the normal derivatives to both sides of the arc γi .

Since the set K is assumed to be of positive capacity, there always exists at least one arc γi in K \ E0 , i.e., I = ∅ in (1.7). Lemma 3. If K ⊂ C possess the symmetry property from Deﬁnition 2 with exceptional set E0 , then equality (1.4) in Deﬁnition 1 holds for every z ∈ K \ E0 , i.e., we have gC\K (z, ∞) = 0 for every z ∈ K \ E0 . Proof. It follows from the special structure of K \E0 in (1.7) that all irregular points of the boundary of the domain C \ K belong to E0 . Indeed, if for an arc γi in K \ E0 , ϕi is the Riemann mapping function of C \ γ i onto C \ D with ϕi (∞) = ∞, then log |ϕi | is a barrier for each point z ∈ γi (cf., [Ran95, Deﬁnition 4.1.4]). The assertion of Lemma 3 follows then immediately from the deﬁnition of regular and irregular points with respect to solutions of Dirichlet problems (cf., [Ran95, Deﬁnition 4.4.9]). From Lemma 3 together with the symmetry relation (1.8) it follows by Schwarz reﬂection principle that gC\K (·, ∞) can be harmonically continued across each arc γi in K \ E0 from each side. As a consequence we see that the normal derivatives in (1.8) exist for each z ∈ K \ E0 . The possibility of harmonic continuations of gC\K (·, ∞) as a consequence of (1.8) further shows that it was not really necessary to assume after (1.7) that the Jordan arcs γi , i ∈ I, are analytic, some smoothness would have been good enough since the analyticity then follows automatically. As a consequence of the symmetry property of Deﬁnition 2, there exists a function Φ that maps a neighborhood W0 of K \ E0 onto itself, and in a geometrical sense it is a reﬂection on each arc γi in K \ E0 . Proposition 4. If the set K ⊂ C possesses the symmetry property as introduced in Deﬁnition 2 with an exceptional set E0 ⊂ K, then there exist a neighborhood W0 ⊂ C of K \ E0 and a function (1.9)

Φ : W0 −→ W0

with the following properties: (i) Φ is bijective and W0 is an open set (ii) We have Φ(z) = z for all z ∈ K \ E0 . (iii) Each component Wi of W0 is simply connected, it contains exactly one Jordan arc γi , i ∈ I, from representation (1.7) of K \ E0 , and this arc γi divides the domain Wi into two subdomains W+,i and W−,i , i.e., we have (1.10)

Wi \ γi = W+,i ∪ W−,i

for each

i ∈ I.

(iv) We have Φ(W±,i ) = W∓,i for each i ∈ I. (v) The function Φ is anti-analytic, i.e., Φ is analytic. Definition 5. The function Φ from Proposition 4 is called the reﬂection function associated with the symmetric set K. Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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It has turned out that the reﬂection function Φ is of great importance for the asymptotic analysis of polynomials pn that satisfy an orthogonality relation of type (1.2). In this context the function Φ can be seen as a substitute for the conjugation that is often used in the analysis of orthogonal polynomials deﬁned on real intervals, and by the same token, it can also be seen as a generalization of the reﬂection on the unit circle that plays a fundamental role in the analysis of Szeg¨ o polynomials on the unit circle. Proof of Proposition 4. For each i ∈ I we can choose a simply connected domain Wi such that Wi ∩ K = γi and Wi \ γi = W+,i ∪ W−,i with W+,i and W−,i being two disjoined subdomains of Wi . It has been shown in Lemma 3 that equality (1.4) holds for all z ∈ γi , i ∈ I. With this equality we conclude from Proposition 19 that the normal derivatives in (1.8) are strictly positive for each z ∈ γi , i ∈ I. From representation (1.5) we further conclude that the normal derivatives in (1.8) give us the density function fi of the equilibrium distribution ωK on γi , we have ∂ 1 ∂ gC\K (z, ∞) = gC\K (z, ∞) = fi (z) for z ∈ γi , i ∈ I. (1.11) ∂n+ ∂n− 2 where ∂/∂n+ and ∂/∂n− denote the normal derivatives to both sides of the arc γi . From equality (1.4) and symmetry (1.8) it follows that the function for z ∈ W+,i ∪ γi gC\K (z, ∞) def (1.12) ui (z) = gC\K (z, ∞) for z ∈ W−,i is harmonic throughout Wi for each i ∈ I. Since Wi is simply connected, there an analytic completion gi of ui in Wi that can be chosen in such a way that (1.13)

Im gi = ui

and

1 ωK (γi ) > 0. 2 Indeed, the existence of gi in (1.13) is immediate. From the Cauchy Riemann diﬀerential equations applied to gi together with the positivity of the normal derivatives in (1.8) and (1.11), it follows that the function gi is strictly monotonic on γi . Because of (1.4) the function gi maps γi into R. Hence, gi (γi ) is a real interval, which we can choose to start at the origin x = 0. In order to ensure that the other end point xi of this interval lies on the positive real axis, it may be necessary to interchange the role of the two domains W+,i and W−,i in (1.12), which is the same as to change the orientation of the arc γi . The last equation in (1.14) follows from (1.11). From the observations just made, it further follows that the function gi is univalent in a neighborhood of γi since the real part of gi is strictly monotonic on γi . By taking a subdomain of Wi , if necessary, we can assume that the function gi is univalent throughout Wi , and by the same token, we can also assume that def the image domain Vi = gi (Wi ) is symmetric with respect to R. Further, we have Vi ∩ R = (0, xi ), and without loss of generality we assume that all domains Wi , i ∈ I, are disjoined. I.e., we have (1.14)

(1.15)

gi (γi ) = (0, xi )

with xi =

Vi = Vi , Vi ∩ R = (0, xi ) for i ∈ I and

Wi ∩ Wj = ∅ for i = j.

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With the functions gi , i ∈ I, we deﬁne partial mappings def

Φi (z) = gi−1 gi (z) for z ∈ Wi , i ∈ I,

(1.16)

and observe that these functions Φi possess the properties (i) through (v) stated in Proposition 4 if we there replace Φ by Φi , W0 by Wi , and K0 by γi . Since all Wi , i ∈ I, are disjoined, we deﬁne def W0 = Wi , Φ : z → Φi (z) for z ∈ Wi , i ∈ I, i∈I

and it is immediate that the function Φ together with the open set W0 completes the proof of Proposition 4. In the main theorem we are interested in measures and potentials that are invariant under the reﬂection function Φ at least locally. In the next deﬁnition we introduce a disc-like set that is typically used as a neighborhood, and it is also invariant under Φ. Definition 6. For every z0 ∈ K \ E0 there exists 0 < r0 = r0 (z0 ) such that the set (1.17)

def

∆(z0 , r) = { z ∈ W0 | |gi (z) − gi (z0 )| < r, z0 , z ∈ Wi , i ∈ I },

is well deﬁned for every 0 < r ≤ r0 . The function gi is the mapping introduced in (1.13) with the help of (1.12) and (1.14). The index i ∈ I is selected in such a way that z0 ∈ Wi . The value of r0 (z0 ) can be determined by the condition { v : |v − gi (z0 )| < r(z0 ) } ⊂ Vi with Vi being the image domain introduced before (1.15). The set ∆(z0 , r) is called Φ−disc with centre z0 and radius r. It is easy to see that the set ∆(z0 , r) is Φ−invariant, i.e., we have Φ(∆(z0 , r)) = ∆(z0 , r), and therefore also Φ(∂∆(z0 , r)) = ∂∆(z0 , r). For small values of r > 0 the set ∆(z0 , r) is similar to an open disc. However, r is not the actual radius of ∆(z0 , r), it is distorted by a factor that is approximately equal to the derivative gi (z0 ) of the mapping function gi , and because of (1.11) the factor is connected with the density of the equilibrium distribution ωK at the point z0 . 1.4. The Main Theorem. Theorem 7. Let K ⊂ C be a compact set that possesses the symmetry property in the sense of Deﬁnition 2 with an exceptional set E0 ⊂ K, ωK the equilibrium distribution on K from Deﬁnition 1, and Φ the reﬂection function from Deﬁnition 5 with region of deﬁnition W0 . If a positive measure µ ∈ M(C) of total mass at most 1 is not identical with ωK , then there exist a companion measure ν ∈ M(C), a point z0 ∈ K \ E0 , and a def

neighborhood U0 = ∆(z0 , r) ⊂ W0 , r > 0, of z0 in the form of a Φ−disc such that we have (1.18) (1.19)

ν < 1, p(µ + ν; z0 )

0, a point z0 ∈ KE0 , a Φ−disc U0 = ∆(z0 , r0 ) with r0 > 0, and two auxiliary probability measures η1 ∈ M1 (E0 ) and η2 ∈ M1 (KU0 ) with properties described in Proposition 9 and 10, respectively, such that for the measure def (2.1) µ1 = µ + ε0 (η1 + η2 ) ∈ M(C) we have (2.2) p(µ1 ; z0 ) = min p(µ1 ; v), v∈K

(2.3)

p(µ1 ; z0 ) < min p(µ1 ; v) v∈K\U

for any open neighborhood U ⊂ C of z0 with K \ U = ∅, (2.4) U0 ∩ K is a single subarc of K \ E0 , Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

A POTENTIAL-THEORETIC PROBLEM

263 9

and further for any r1 > 0 with r1 ≤ r0 and any δ0 > 0, there exists r2 > 0 with def r2 < r1 , δ1 > 0 with δ1 ≤ δ0 , the Φ−discs Uj = ∆(z0 , rj ), j = 1, 2, c2 ∈ R, and a measure (2.5) µ2 ∈ M(K ∪ U 2 ) such that (2.6) µ2 ≤ 1 − 4 ε0 , (2.7)

p(µ2 ; z) = p(µ1 ; z) + c2

z ∈ U2 ,

for

(2.8)

µ2 |U2 = µ1 |U2 = µ|U2 ,

(2.9)

p(µ2 ; z0 ) = min p(µ2 ; v),

(2.10) (2.11)

v∈K

p(µ2 ; z) ≤ p(µ1 ; z) + c2 p(µ2 ; z) = δ1 + min p(µ2 ; v) v∈K

for for

z ∈ K \ E0 , z ∈ K \ (E0 ∪ U1 ),

and µ2 is minimal among all measures (2.5) that satisfy (2.6) through (2.11) for the given choice of r2 and δ1 . It may be appropriate to make some remarks about the logical structure of the statements in Proposition 8. In a ﬁrst layer it is stated that the objects ε0 , z0 , U0 , η1 , η2 , and µ1 exist for a given measure µ with the stated speciﬁcations, and these objects possess the immediate properties (2.2), (2.3), and (2.4). In a second layer then the existence of r2 , U1 , U2 , δ1 , c2 , and µ2 is postulated depending on the choice of δ0 , r1 , and the already speciﬁed other objects. For the new objects a list of properties follows in (2.5) through (2.11), and they are mainly concerned with the measure µ2 , which is the main object of interest in Proposition 8. The following properties can be seen as central: (i) the mass µ2 of µ2 is smaller than 1, (ii) the measure µ2 reproduces µ in the neighborhood U2 of z0 , (iii) like p(µ1 ; ·), so also the potential p(µ2 ; ·) has a unique minimum at z0 , (iv) globally, the potential p(µ2 ; ·) can be forced to be as close to p(ωK ; ·) as one wants. Proof. In the proof of Proposition 8 the concept of balayage with mass reduction and levelling that is developed in Proposition 13 through Proposition 17 in Subsection 3.1 and 3.2, further below, plays a key role. In some sense the whole proof is an adaptation of these results to the speciﬁc needs in Proposition 8. Before we come to the technical details we make some general remarks: In the proof of Lemma 3 it has been shown that all irregular points of K are contained in the exceptional set E0 . In the propositions in Subsection 3.1 and 3.2 the set of irregular points in K is denoted by KII ,, and in the boundary ∂G of a domain G by ∂GII . These sets often play the role of exceptional sets. Since in assertion (i) of Deﬁnition 2 we have assumed that cap(E0 ) = 0, and since KII ⊂ E0 , we often substitute KII by E0 in results taken from Proposition 14 through 17. Without loss of generality we can assume that µ = 1 since this is the critical situation in Proposition 8. If µ < 1, then inequality (2.6) becomes trivial, and all the other assertions of the proposition are then also easy to verify. Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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In a ﬁrst step we apply the concept of mass reduction and levelling from Proposition 14 to the measure µ instead of the measure µ1 from (2.1), and get a preliminary approximation of a measure that later will be µ2 . Shortcomings in this approximation will be discussed, and based on this discussion we will explain the basic line of the proof proper. From the lower semicontinuity of p(µ; ·) it follows that there exists a point z0,0 ∈ K \ E0 , at which the essential minimum (2.12)

def

m0,0 = inf{ c ∈ R : cap{ z ∈ K : p(µ; z) ≤ c } > 0 }

is assumed by p(µ; ·). From Proposition 14 we know that for any choice of δ0,0 > 0 and any open neighborhood U0,0 ⊂ C of z0,0 there exists c1,0 ∈ R and a measure µ1,0 ∈ M(K ∪ U 0,0 ) such that we have (2.13)

p(µ1,0 ; z) + c1,0 = min(p(µ; z), m0,0 + δ0,0 ) for z ∈ U0,0 ∪ K \ E0

together with the other conclusions in Proposition 14. From the Propositions 15 and 16, it then further follows that δ0,0 > 0 and a neighborhood U0,0 of z0,0 can be chosen so small that we have (2.14)

µ1,0 < µ = 1.

Indeed, the existence of a measure µ1,0 follows directly from Proposition 14 if we there replace m0 , U , z0 , δ, and µ1 by m0,0 , U0,0 , z0,0 , δ0,0 , and µ1,0 , respectively. Inequality (2.14) is a consequence of the assumption µ = ωK since it follows from the considerations in Proposition 15 and 16 that we have b0,0 < µ in limit (3.23) of Proposition 15. With the monotonicity proved in Proposition 15 and limit (3.23) this then implies that we can choose δ0,0 > 0 and a neighborhood U0,0 so small that µ1,0 is close to the limit value b0,0 , and consequently (2.14) is proved. At ﬁrst sight one could expect that the measure µ1,0 from (2.13) and (2.14) could be modiﬁed with the techniques described in Proposition 17 in such a way that it could become a candidate for the measure µ2 in Proposition 8. However, there are essential shortcomings that force us to choose a more complicated route for the proof. In Proposition 8 it is assumed that the point z0 lies inside of one of the open arcs γi , i ∈ I ⊂ N, that form K \ E0 in (1.7), and this assumption is also essential for applications of Theorem 7. It is obvious that the assertion z0 ∈ K \ E0 can not be guaranteed for the point z0,0 introduced in (2.12). The diﬃculty is overcome by using the measure η1 ∈ M1 (E0 ) from Proposition 9 in the deﬁnition of µ1 in (2.1). Another diﬃculty that has to be overcome is the requirement that the potential should assume its minimum only at a single point in K \E0 , and this should happen in a strong sense as, for instance, described by relation (2.3). In order to meet this requirement the measures η2 ∈ M1 (KU0 ) from Proposition 10 has also been included into the deﬁnition of µ1 in (2.1). Notice that the two measures µ and µ1 are identical in a neighborhood of z0 . A further complication arises from the fact that the new measure µ1 has a total mass µ1 = 1 + 2 ε0 . Thus, it is greater than 1. In order to achieve a reduction of its mass below 1, like this is the case in (2.14), the parameter ε0 > 0 has to be chosen small. In order to show that an appropriate value of ε0 can be found, we consider a whole sequence of constellations. Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

A POTENTIAL-THEORETIC PROBLEM

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We choose a sequence εn > 0, n ∈ N, with limn→∞ εn = 0. It follows from the properties of the measure η1 ∈ M1 (E0 ) in Proposition 9 and the lower semicontinuity of p(µ + εn η1 ; ·) that for each n ∈ N there exists zn ∈ K \ E0 such that (2.15)

p(µ + εn η1 ; zn ) = min p(µ + εn η1 ; z). z∈K

By taking, if necessary, a subsequence of N and possibly also another choice of z0,0 after (2.12), we can assume that lim zn = z0,0 .

(2.16)

n→∞

Notice that contrary to the zn the point z0,0 is not necessarily an element of K \ E0 . From the structure of the set K described in assertion (i) of Deﬁnition 2 and zn ∈ K \ E0 , it follows that for each n ∈ N there exists rn > 0 such that the def

intersection Un ∩ K of K with for the Φ−disc Un = ∆(zn , rn ) consists of a single subarc of one of the arcs γj , j ∈ I, that form K \E0 , and do we assume that rn → 0. Corresponding to (2.1) we deﬁne (2.17)

def

µn = µ + εn (η1 + η2 ) ∈ M(C), n ∈ N,

with η2 ∈ M1 (KUn ) the measure from Proposition 10, where we have taken Un and zn as U and z0 , respectively. Because of Un , the measures η2 depends on n, while η1 in (2.17) is independent of n. It follows from (2.15) together with (2.27) and (2.28) in Proposition 10 that (2.18)

def

p(µn ; zn ) = min p(µn ; z) = mn , z∈K

and further it follows from (2.28) and (2.29) that for any open neighborhood U of z0 there exists δ > 0 such that (2.19)

{ z ∈ K : p(µn ; z) ≤ mn + δ } ⊆ U .

For each n ∈ N we choose δn > 0 such that δn → 0 and (2.20)

{ z ∈ K : p(µn ; z) ≤ mn + δn } ⊆ Un

with Un = ∆(zn , rn ) the Φ−disc that has already been speciﬁed earlier. We now apply Proposition 14 with µn , zn , Un , δn , and µ1,n as µ, z, U , δ, and µ1 , respectively. For the measure µ1,n ∈ M(K ∪ U n ), n ∈ N, from Proposition 14 we then know that µ1,n < µn = 1 + 2εn . In order to prove that we have µ1,n < 1 for n ∈ N suﬃciently large, we use the limits (3.23), (3.24), and (3.25) from Proposition 15, which are proved there for δn → 0, Un → {zn }, and µn ﬁxed. But these limits hold also if we have εn → 0 and µn deﬁned by (2.17) instead of µ. We then have (2.21)

lim µ1,n = b0,0 < µ = 1

n→∞

with b0,0 the same limit value in (3.23) of Proposition 15 that corresponds to the original measure µ, and has already appeared in the discussion after (2.14). Notice that because of (2.16) and rn → 0 we have U n ⊂ U0,0 for n ∈ N suﬃciently large. Since εn → 0, it follows from (2.21) that there exists n1 ∈ N such that (2.22)

µ1,n1 ≤ 1 − 6 εn1 ,

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and with this choice of n1 we deﬁne (2.23)

def

def

def

def

ε0 = εn1 , z0 = zn1 , r0 = rn1 , U0 = ∆(z0 , r0 ).

Next, the values δ0 > 0 and r1 > 0 are chosen freely up to the restrictions r1 ≤ r0 def and δ0 ≤ δn1 . Further, we set U1 = ∆(z0 , r1 ), and in Proposition 17 we substitute the objects U , µ, and µ1 by U1 , µ1 , and µ1,n1 , respectively. From Proposition 17 in combination with Proposition 14 we then deduce that def there exists r2 > 0 with r2 ≤ r1 , U2 = ∆(z0 , r2 ), δ1 > 0 with δ1 ≤ δ0 , a constant c2 ∈ R, and a measure µ2 ∈ M(K ∪ U 1 ) such that the assertions (2.5) through (2.11) in Proposition 8 hold true. Indeed, the measure µ2 in (2.5) of Proposition 8 is taken to be identical to the measure µ2 in Proposition 17. The existence of an appropriate constant c2 ∈ R is obvious from the circumstances. In the formulation of conclusions in Proposition def 17 there appears the geometric disk D(z0 , r) = {|z − z0 | < r} with a certain r > 0 where we need the Φ−disc U2 = ∆(z0 , r2 ). However, from the introduction of Φ−discs in Deﬁnition 6, it is obvious that there always exists r2 > 0 such that (2.24)

U2 = ∆(z0 , r2 ) ⊂ D(z0 , r),

which shows that we can substitute D(z0 , r) by U2 everywhere in Proposition 17. Inequality (2.6) in Proposition 8 is the consequence of equality (3.33) in Proposition 17, the monotonicity proved in Proposition 15, and (2.22). Notice that by our deﬁnition the measure µ1 in Proposition 15 is identical with the measure µ1,n1 in (2.22). Identity (2.7) is identical with (3.34) in Proposition 17 if one takes into account (2.24), the deﬁnitions in (2.23), and the subsequent change of notations. Analogously, the equalities in (2.8) follow from (3.31) in Proposition 17 and the observation that supp(η1 + η2 ) ∩ U2 = ∅. Equality (2.9) is a consequence of (2.2), (2.7), and (2.3). Inequality (2.10) follows from (3.35) in Proposition 17 together with (3.14) in Proposition 14. In an analogous way, identity (2.11) follows from (2.3) together with (3.35) in Proposition 17 and (3.14) in Proposition 14.

In the next two propositions we formulate and prove auxiliary results that have already been used in the proof of Proposition 8. Proposition 9. Let K ⊂ C be a compact set with symmetry property and exceptional set E0 ⊂ K as introduced in Deﬁnition 2. Then there exist a probability measure η1 ∈ M1 (E0 ) such that (2.25)

p(η1 ; z) = ∞

for all

z ∈ E0 .

Proof. The set E0 ⊂ K is compact, and in Deﬁnition 2 it has been assumed that cap(E0 ) = 0, therefore we know from Evans’ Theorem (cf., [Ran95, Theorem 5.5.6]) in potential theory that a probability measure η1 with supp(η1 ) ⊂ E0 and (2.25) exists.

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A POTENTIAL-THEORETIC PROBLEM

267 13

Proposition 10. Let K ⊂ C be a compact set with symmetry property and exceptional set E0 ⊂ K as introduced in Deﬁnition 2. For any z0 ∈ K \ E0 and any Φ−disc U0 = ∆(z0 , r0 ) with r0 > 0 so small that U 0 ∩ E0 = ∅

(2.26)

and

U 0 ∩ K a single Jordan arc,

there exists a probability measure η2 ∈ M(K \ U0 ) such that (2.27)

p(η2 ; z0 ) = p(η2 ; z0 )

0, the point z0 ∈ KE0 , and the measure µ1 be determined as described in Proposition def 8. Then for every r3 > 0 suﬃciently small and Φ−disc U3 = ∆(z0 , r3 ) ⊂ W0 there exists a measure µ3 ∈ M(K ∪ U3 ) and a constant c3 ∈ R such that (2.31)

µ3 ≤ 1 − 3 ε0 ,

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268 14

HERBERT STAHL

(2.32)

p(µ3 ; z0 )

0 be chosen so small that the conclusions of Proposition 11 hold true and that we have r3 ≤ r0 for r0 in Proposition 8. Then def def we set U0 = U3 = ∆(z0 , r3 ) or the Φ−disc U0 in Theorem 7, r = r3 , and deﬁne (2.35)

def

ν = µ3 + ε0 (η1 + η2 )

with the measure µ3 from Proposition 11, where we use the same two auxiliary probability measures η1 and η2 from the Propositions 9 and 10, respectively, that have been used in (2.1) of Proposition 8, and also ε0 > 0 is chosen like that in Proposition 8. As an immediate consequence of (2.35) and (2.1) in Proposition 8 we then have (2.36)

µ + ν = µ1 + µ3 .

From (2.31) in Proposition 11 together with (2.35) it follows that ν = µ3 + 2 ε0 ≤ 1 − ε0 , which proves (1.18) in Theorem 7. Inequality (1.19) in Theorem 7 is an immediate consequence of (2.36) together with (2.3) in Proposition 8 and (2.32) in Proposition 11. Because of (2.36), assertion (1.19) in Theorem 7 is practically identical with (2.33) in Proposition 11. Notice that we have U0 = U3 . Since we have assumed that r3 ≤ r0 , it follows from the deﬁnition of µ1 in (2.1) of Proposition 8 together with the Propositions 9 and 10 that µ1 |U3 = µ|U3 , and in the same way it follows from (2.35) that ν|U3 = µ3 |U3 . Therefore it follows from (2.34) in Proposition 11 that ν|U3 = µ3 |U3 = Φ µ1 |U3 = Φ µ|U3 which proves (1.21) in Theorem 7. At last, assertion (1.22) in Theorem 7 is a direct consequence of property (2.25) in Proposition 9 of the probability measure η1 in the deﬁnition of µ1 in (2.1) of Proposition 8.

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A POTENTIAL-THEORETIC PROBLEM

Proof of Proposition 11. Like Proposition 11 itself, so is also its proof in many respects a continuation of the analysis done for Proposition 8. In the sequel, the objects K, µ, E0 , z0 , ε0 , µ1 , and r0 with U0 = ∆(z0 , r0 ) are the same as those in Proposition 8. In the second part of Proposition 8 it has been shown that r1 > 0 and δ0 > 0 can be chosen within certain bounds so that all assertions after (2.5) in Proposition 8 are satisﬁed. We make now such a choice not only for a pair r1 and δ0 , but for two sequences of values r1,n > 0 and δ0,n > 0, n ∈ N, and assume that r1,n → 0 and δ0,n → 0 as n → ∞.

(2.37)

From Proposition 8 it then follows that for each n ∈ N there exist r2,n = r2 > 0, δ1,n = δ1 > 0, c2,n = c2 ∈ R, and a measure µ2,n = µ2 ∈ M(K ∪ U 2,n ) such that the assertions (2.5) through (2.11) in Proposition 8 hold true for each n ∈ N. In accordance with Proposition 8, we have 0 < r2,n < r1,n , 0 < δ1,n ≤ δ0,n , and deﬁne def

Ujn = ∆(z0 , rjn ) for j = 1, 2, n ∈ N. From (2.11) in Proposition 8 we deduce that for each n ∈ N there exists a constant c0,n ∈ R such that p(µ2,n ; z) = c0,n

(2.38)

for z ∈ K \ (E0 ∪ U1,n ).

In (3.23), (3.24), and (3.25) of Proposition 15 it has been proved under conditions that are satisﬁed here that the limits def

lim µ2,n = b2 ,

(2.39)

n→∞ ∗

µ2,n −→ b2 ωK

(2.40)

as

n → ∞,

lim p(µ2,n ; z) = c0 − b2 gC\K (z, ∞)

(2.41)

n→∞

exist. The last limit holds locally uniformly for z ∈ C \ K, and we have (2.42) lim c0,n = c0 = −b2 log cap(K) − b2 log |v|dωK (v). n→∞

The second equality in (2.42) follows from representation (1.5) and (1.6) of the Green function. Besides of the potentials p(µ2,n ; ·), we consider also a modiﬁed variant, which will be deﬁned next. Motivated by the representation (2.43)

p(ωK∪U 1,n ; ·) = −gC\(K∪U 1,n ) (z, ∞)

− log cap(K ∪ U 1,n ) −

C\D

log |v|dωK∪U 1,n (v)

for the Green function (cf., (1.5) and (1.6)), we deﬁne def

(2.44)

µ3,n = µ2,n + ε0 ωK∪U 1,n

(2.45)

c3,n = c0,n + ε0 log cap(K ∪ U 1,n ) + ε0

and

def

C\D

log |v|dωK∪U 1,n (v).

Since gC\(K∪U 1,n ) (z, ∞) = 0 for z ∈ U 1,n ∪ K \ E0 , we conclude with (2.38) that (2.46)

p(µ3,n ; z) − c3,n = p(µ2,n ; z) − c0,n

for z ∈ U 1,n ∪ K \ E0 .

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It is obvious that the limits (2.39), (2.40), (2.41), and (2.42) hold also true in a modiﬁed form for µ3,n , c3,n , and p(µ3,n ; ·). The potential p(µ3,n ; ·) is the modiﬁed variant of p(µ2,n ; ·), which we were looking for. Both variants are now transformed with the help of the reﬂection function Φ : W0 −→ W0 from Deﬁnition 5. First, we deﬁne (2.47)

def

h2,n (z) = p(µ2,n ; Φ(z)) − c0,n

for z ∈ W0 , n ∈ N,

which is a superharmonic function in W0 for each n since Φ is anti-analytic in W0 . The potential p(µ3,n ; ·) will also be transformed with the help of the reﬂection function Φ, but in its case the transformed function will be extended to the whole complex plane C. There exists a function h3,n , n ∈ N, in C, which is uniquely determined by the following conditions ⎧ = p(µ3,n ; Φ(z)) − c3,n for z ∈ K \ E0 ∪ U 1,n ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ + O(1) as z→∞ ⎨ = ||µ3,n || log |z| (2.48) h3,n (z) ⎪ ⎪ harmonic in C \ (K ∪ U 1,n ) ⎪ ⎪ ⎪ ⎪ continuous in C \ (E0 ∪ U1,n ) ⎪ ⎩ lower semicontinuous in C In order to prove the existence of the function h3,n , we ﬁrst deﬁne an auxiliary function h3,n as the solution of a Dirichlet problem in C \ (K ∪ U 1,n ) with boundary function h3,n (cf., [Ran95, Corollary 4.2.6 and Theorem 4.3.3]), which is deﬁned on K ∪ ∂U1,n by the ﬁrst line in (2.48). In general, the boundary of the domain C \ (K ∪ U 1,n ) contains irregular points. We know from the proof of Lemma 3 that they are all contained in E0 . Because of these irregular boundary points, we consider a generalized solution of a Dirichlet problem. For its existence and uniqueness it is important that the boundary function is bounded and continuous quasi everywhere on the boundary, which is indeed the case since r2,n < r1,n for all n ∈ N. The solution is continuous at every regular boundary point, where the boundary function is also continuous (cf., [Ran95, Theorem 4.3.3 together with Theorem 4.3.4]). This property proves the fourth line in (2.48). With the basic properties of the Green function gC\(K∪U 3,n ) (·, ∞) stated in Deﬁnition 1, we then deduce that the function (2.49) h3,n = h3,n − µ3,n g (·, ∞) C\(K∪U 3,n )

satisﬁes (2.48), and its uniqueness follows from the uniqueness of the solution h3,n of the Dirichlet problem. Our next task is to prove that the function h3,n is superharmonic throughout C for n ∈ N suﬃciently large, which then will imply that h3,n can be represented as a potential of a positive measure plus a constant. The proof of the superharmonicity of h3,n is rather involved, and many of the constructions introduced so far have specially been made for this purpose. The property of symmetry from Deﬁnition 2 is also essential here. The measure in representation (2.58), further below, of the function h3,n is a key piece of the whole proof. It is obvious that for the question of superharmonicity the behavior of h3,n is critical only on the sets ∂U1,n and K \ U1,n since there the new deﬁnitions in (2.48) are pasted together. In U1,n the transformed potential p(µ3,n ; Φ(·)) is superharmonic like p(µ3,n ; ·). Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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A POTENTIAL-THEORETIC PROBLEM

We start the study with some preparations. From (2.47), (2.48), (2.38), (2.44), (2.49), the limits (2.39), (2.41), (2.42), and the symmetry (1.8) of the Green function gC\K (·, ∞) in Deﬁnition 2, it follows that (2.50) (2.51)

def

lim h2,n (z) = −b2 gC\K (z, ∞) = h2∞ (z),

n→∞

def

lim h3,n (z) = −(b2 + ε0 )gC\K (z, ∞) = h3∞ (z).

n→∞

The convergence in (2.50) and (2.51) holds point-wise in W0 \ {z0 }, and locally uniformly at least in W0 \ K. Because of (2.38), (2.47), and (2.49) we have (2.52)

hjn (z) = 0 for z ∈ K \ (E0 ∪ U 1,n ), j = 2, 3. def

For the next estimates we introduce a further Φ−disc U4 = ∆(z0 , r4 ) with r4 > r1,n for all n ∈ N, U4 ⊂ W0 , and the assumption that K ∩ U 4 is a single Jordan arc in K \ E0 . With the help of Proposition 20 we then investigate the convergence of h2,n and h3,n on ∂U4 as n → ∞. With the convergence in (2.50) and (2.51) and the properties of the limit functions h2∞ and h3∞ in (2.50) and (2.51), we see that the assumptions of Proposition 20 are satisﬁed, and from (3.47) and (3.46) in Proposition 20 we deduce that for any choice of ε3 > 0 there exists n3 ∈ N such that h3,n (z) h2,n (z) ≤ 1 + ε3 and ≥ 1 − ε3 for z ∈ ∂U4 \ R, n ≥ n3 . (2.53) h2∞ (z) h3∞ (z) From the deﬁnition of the functions hj∞ , j = 2, 3, in (2.50) and (2.51) it immediately follows that h3∞ (z) ε0 (2.54) =1+ for z ∈ ∂U4 \ R. h2∞ (z) b2 Notice that b2 in (2.37) is positive, or at least we can assume without loss of generality that it is positive. With ε3 satisfying ε0 , (2.55) 0 < ε3 ≤ 2b2 + ε0 or equivalently, ε0 /b2 − ε3 ε0 /b2 − ε3 > ε3 , we deduce from (2.53) and (2.54) that

ε0 ε0 h3,n (z) ε0 h3,n (z) ε0 = 1+ ≥ 1+ (1 − ε3 ) = 1 + − ε3 − ε3 ≥ h2∞ (z) b2 h3∞ (z) b2 b2 b2 h2,n (z) for all z ∈ ∂U4 \ R, n ≥ n3 . (2.56) 1 + ε3 ≥ h2∞ (z) Since h2∞ = −b2 gC\K (·, ∞) < 0 in C \ K, this proves that (2.57)

h3,n (z) ≤ h2,n (z)

for

z ∈ ∂U4 \ R, n ≥ n3 .

From estimate (2.57) we then conclude that h3,n is superharmonic in U4 for n ≥ n3 . Indeed, h2,n is obviously superharmonic in W0 , and therefore also in U4 . Further, we have h3,n (z) = h2,n (z) for z ∈ U 1,n ∪ (K ∩ U 4 ), n ≥ n3 , and h3,n is harmonic in C\(K ∪U 1,n ), which with (2.57) then shows that h3,n is superharmonic in U4 for n ≥ n3 . The superharmonicity of h3,n on K \ (E0 ∪ U 4 ) is an immediate consequence of the fact that h3,n (z) =

sup

h3,n (v)

fore z ∈ K \ (E0 ∪ U 4 ), n ≥ n3 ,

v∈C\U1,n

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which follows from (2.11) of Proposition 8, the ﬁrst line in (2.48), and (2.57). It now only remains to study the behavior of h3,n on E0 . The function h3,n is bounded from below on any compact subset of C, it is lower semicontinuous on E0 because of the last line in (2.48), and we have cap(E0 ) = 0 by the assumptions made in Deﬁnition 2. Using these properties together, it follows that the function h3,n , n ∈ N, is suberharmonic in E0 . The superharmonicity of h3,n is important since with the help of the PoissonJensen Formula (cf., [Ran95, Theorem 4.5.1]) it implies that the function h3,n3 with the behavior near inﬁnity as stated in the second line of (2.48) can be represented by a potential of a positive measure plus a constant, i.e., there exists µ 3 ∈ M(K ∪U 1,n ) and c3 ∈ R such that (2.58)

µ3 ; ·) + c3 h3,n3 = p(

and

µ3 = µ3,n3 .

With representation (2.58) we have made the decisive step to the conclusion of def

the proof. For any choice of r3 > 0 such that r3 ≤ r3,n3 and Φ−disc U3 = ∆(z0 , r3 ), we deﬁne the measure (2.59)

µ3 ∈ M(K ∪ U 3 )

as the result of balayage in the sense of Proposition 12 of the measure µ 3 out of the domain C \ (K ∪ U 3 ) onto K ∪ U 3 . For the measure µ3 the assertions (2.31) through (2.34) of Proposition 11 follow from properties of the corresponding measures µ3,n3 and µ 3 . Indeed, it follows from (2.2) in Proposition 8 that µ2,n ≤ 1 − 4 ε0

for all

n ∈ N,

and with (2.44) we conclude that µ3,n ≤ 1 − 3 ε0

for n ∈ N,

which immediately proves (2.31) in Proposition 11. The same type of assertion as in (2.32) of Proposition 11 has been proved in (2.3) of Proposition 8 for the potential p(µ2,n ; ·), n ∈ N. Since the reﬂection function Φ is the identity on K \ E0 , this property carries over to the functions h2,n deﬁned in (2.47), and further to h3,n because of the ﬁrst line in (2.48) together with (2.46). The subsequent manipulation of h3,n in (2.58) and the balayage that has led to (2.59) does not change the situation on K \ E0 , and consequently assertion (2.32) is proved for p(µ3 ; ·). From (2.7) in Proposition 8 and the deﬁnitions made after (2.37) we know that p(µ1 ; z) = p(µ2,n ; z) + c2,n for z ∈ U2,n and n ∈ N. With (2.46), the ﬁrst line in (2.48), (2.58), and the balayage that has led to (2.59), it then further follows that (2.60)

p(µ1 ; z) = p(µ3 ; Φ(z)) + c3 for z ∈ U3

with a certain constant c3 ∈ R. Since we have Φ(U3 ) = U3 for the Φ− disc U3 , and since Φ ◦ Φ is the identity, we derive from (2.60) that (2.61)

p(µ3 ; z) = p(µ1 ; Φ(z)) − c3 for z ∈ U3 .

Adding up (2.60) and (2.61) proves (2.33) in Proposition 11. Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

A POTENTIAL-THEORETIC PROBLEM

273 19

Identity (2.34) in Proposition 11 will be proved in a very similar way as (2.61). From (2.8) in Proposition 8 and the deﬁnitions made after (2.37) we know that (2.62)

µ1 |U2,n = µ2,n |U2,n for all

n ∈ N.

From (2.44) together with the ﬁrst line of (2.48), representation (2.58), and the balayage that has led to (2.59) it then follows that for n ∈ N and µ3,n3 |U3 = µ3 |U3 . (2.63) µ3,n |U3,n = Φ µ2,n |U3,n The two identities (2.62) and (2.63) together prove (2.34) in Proposition 11. Notice that U2,n3 ⊃ U3,n3 ⊃ U3 . 3. Auxiliary Results from Potential Theory In this last section of the paper we prove results that are purely potentialtheoretic in nature and can be formulated without reference to the symmetry property from Deﬁnition 2. For general reference to potential theory we use [Ran95] and [ST97]. The somewhat non-standard deﬁnition of the logarithmic potential in (1.3) causes only minor changes in the general theory. Necessary adaptations will be mentioned only if they are not immediately obvious. 3.1. Balayage with Mass Reduction. The technique of balayage with mass reduction is perhaps the most interesting tool from potential theory in the present paper. In the next proposition we ﬁrst put together some basic properties of the general technique of balayage in a form that is best suited for our needs. This is then followed by the introduction of the technique of balayage with mass reduction. Proposition 12. Let G ⊂ C be an open set with a compact boundary ∂G ⊂ C of positive capacity. For every measure µ ∈ M(C) there exists a measure µ ∈ M(C \ G) with µ (∂GII ) = 0 and a constant c ∈ R such that (3.1) (3.2)

µ = µ

and

p( µ; z) = p(µ; z) + c for every z ∈ C \ (G ∪ ∂GII ).

By ∂GII we denote the set of all irregular points of the boundary ∂C of each component C of the open set G with respect to the Dirichlet Problem in C. The measure µ is uniquely determined by (3.1), (3.2), and the condition µ (∂GII ) = 0. It is called the balayage measure generated by sweeping (balayage) the measure µ out the open set G (or more precisely out of G ∪ ∂GII ). If µ(G ∪ ∂GII ) > 0, then we have (3.3)

p( µ; z) < p(µ; z) + c

for all

z ∈ G ∪ ∂GII ,

otherwise we have µ = µ and c = 0. In this later case we have equality in (3.3). For the balayage measure µ we have the representation ωv dµ(v), (3.4) µ = µ|C\(G∪∂GII ) + G∪∂GII

where ωz ∈ M(∂G) is the harmonic measure with respect to the point z ∈ G ∪ ∂GII in the component C of the open set G with z ∈ C ∪ ∂CII . The constant c in (3.2) Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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and (3.3) is given by (3.5) c= log |v|d( µ − µ)(v) − C\D

G∞ ∪∂G∞,II

gG∞ (v, ∞)dµ(v)

with G∞ ⊂ G being the unbounded component of G if such a component exists, otherwise we assume the second integral in (3.5) to be 0. For a deﬁnition of the harmonic measure ωz we refer to [Ran95, Section 4.3], and for irregular points to [Ran95, Section 4.1] or [ST97, Section I.4]. The harmonic measure is typically deﬁned in the literature for a domain, but the concept can be extended to an open set G in a natural way by considering individual harmonic measures in each of the components C of the set G. This strategy has been followed in (3.4). Since irregular points are isolated in the ﬁne topology, each z ∈ ∂GII can be attributed in a one-to-one manner to a component C of G, and so it is possible to extend the strategy of deﬁning the harmonic measure by composition also to the points of ∂GII . From Kellogg’s Theorem we have cap(∂GII ) = 0 (cf., [Ran95, Theorem 4.2.5]), it therefore follows that equality holds in (3.2) for quasi every z ∈ ∂G. In (3.5) the ﬁrst term is a consequence of our special deﬁnition of the logarithmic potential in (1.3), the second one is equal to zero if ∞ ∈ / G. It is possible that the two integrals in (3.5) do not exist independently if the measure µ does not thin out fast enough near inﬁnity. Proposition 12 has essentially been proved in [ST97, Theorem II.4.7] for an open set G, and a more special version for case that G is a domain has been proved in [ST97, Theorem II.4.1 and II.4.4]. Some aspects of the balayage technique have been approached in [ST97] in a slightly diﬀerent way from Proposition 12. So, for instance, in [ST97] it has been assumed that p(µ; ·) is bounded on ∂G, instead we assume in Proposition 12 that the measure µ is not only swept out of the open set G, but also out of the set ∂GII of irregular points, which also ensures that the balayage measure µ is unique. Such slight deviations can be proved with the same tools as those applied in [ST97, Section II.4], and we will not go into further details here. In the last integral in (3.5) we have assumed that the Green function gD (·, ·) is deﬁned throughout C × C, which can be done in a natural way. We now come to the main object of the present subsection: the balayage with mass reduction. For a domain G ⊂ C with ∞ ∈ G the balayage technique is combined with a reduction of the mass of µ, i.e., equality (3.1) will be replaced by an inequality, and the reduction will be done in an optimal way. Proposition 13. Let G ⊂ C be a domain with ∞ ∈ G and ∂G of positive capacity. By ∂GII we denote the set of all irregular points in ∂G. For every measure µ ∈ M(C) there exists a measure µ ∈ M(C \ G) with µ (∂GII ) = 0 and a constant c ∈ R such that (3.6)

p( µ; z) = p(µ; z) + c for every z ∈ C \ (G ∪ ∂GII ).

If another measure µ 1 ∈ M(C \ G) with µ 1 (∂GII ) = 0, µ 1 = µ , and c1 ∈ R satisﬁes (3.6) with µ and c replaced by µ 1 and c1 , respectively, then we have (3.7)

µ1 > µ .

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Because of (3.7) the measure µ is unique, and it is called the balayage measure with mass reduction. If µ(G) > 0, then we have µ < µ .

(3.8)

For the measure µ we have the representation (3.9) µ = µ|C\(G∪∂GII ) + ωv dµ(v) + ( µ − µ)ωC\G G∪∂GII

with ωv ∈ M(∂G) the harmonic measure in G with respect to the point v ∈ G, and ωC\G ∈ M(∂G) the equilibrium distribution of the compact set C \ G, which actually is the same as the harmonic measure with respect to ∞. For the constant c in (3.6) we have the representation (3.10) c= log |v|d( µ − µ)(v) − gG (v, ∞)dµ(v). C\D

G∪∂GII

There exists an instructive interpretation for the balayage technique with mass reduction of the last proposition. Unlike the balayage in Proposition 12, here only as much of the mass of µ is swept out of G onto ∂G as is necessary to satisfy (3.6), the remaining part of the mass of the measure µ is swept to ∞, where it has no longer any inﬂuence in the logarithmic potential p( µ; ·). This interpretation is also the basic idea for the proof of Proposition 13. Proof. In the ﬁrst part of the proof we show the existence of a candidate for µ ∈ M(C \ G) with µ (∂GII ) = 0 and c ∈ R satisfying (3.6) with minimal mass. Let here µ 0 and c0 denote the measure µ and the constant c, respectively, from Proposition 12. By ωK we denote the equilibrium distribution of the compact set def K = C\G. From [Ran95, Theorem 4.3.14] we know that ωK = ω∞ , from [Ran95, Theorem 4.3.6] we further know that ωK (∂GII ) = 0, and from Proposition 12 that µ 0 (∂GII ) = 0. With (3.11)

def

0 − b ωK ≥ 0 } b0 = sup{ b ≥ 0 : µ

we deﬁne (3.12)

def

µ = µ 0 − b0 ωK .

Since p(ωK ; ·) is constant on C \ (G ∪ ∂GII ) (cf., [Ran95, Theorem 4.2.4]), identity (3.6) follows from (3.12) and (3.2) in Proposition 12, which concludes the ﬁrst part of the proof. Next, we prove (3.7). From the assumptions made with respect to the measure µ 1 ∈ M(C \ G) it follows from (3.6) that p( µ−µ 1 ; z) = c1 − c

for all

z ∈ C \ (G ∪ ∂GII ).

Taking into account the behavior of p( µ−µ 1 ; ·) in G and especially at inﬁnity, we deduce that p( µ−µ 1 ; ·) + c − c1 = ( µ1 − µ)gG (·, ∞), which implies that µ1 − µ) ωK . µ =µ 1 + ( From the extremality of b0 in (3.12) it follows that µ1 − µ ≥ 0, and in case of µ1 − µ = 0 we would have µ 1 = µ , which proves (3.7). Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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From Harnack’s inequality (cf., [Ran95, Corollary 1.3.3, Theorem 4.3.14] and [Ran95, Corollary 4.3.5]) we deduce that for every compact set V ⊂ G there exists r > 0 such that 1 r ωv ≤ ωw ≤ ωv for v, w ∈ V. r If µ(G) > 0, then it follows from the last relation and from (3.4) in Proposition 12 that we have b0 > 0 in (3.12), which proves (3.8). Representation (3.9) follows from (3.4) in Proposition 12 together with (3.12). Notice that we have b0 = µ − µ. Identity (3.10) follows in the same way as the analogous identity (3.5) in Proposition 12. 3.2. Balayage with Mass Reduction and Levelling. In the proof of Theorem 7 the technique of balayage with mass reduction alone has turned out to be not suﬃcient, it has to be combined with a technique that we have called levelling. In the next four propositions this technique is introduced and studied together with balayage. Levelling means here that a given measure µ is changed in such a way that its potential is levelled oﬀ at a certain level without changing its values below this level. This manipulation is done simultaneously with balayage and mass reduction. The most interesting result in this direction for our applications is contained in Proposition 17. The concept has been used in situations, where the compact set K ⊂ C possesses the symmetry property in the sense of Deﬁnition 2. In order to keep our treatment free from too many technical assumptions and details, we assume in the present section only that K ⊂ C is a compact set of positive capacity that has no inner points and its complement C \ K is connected. For a measure µ ∈ M(C) the essential minimum m0 of p(µ; ·) on K is deﬁned as (3.13)

def

m0 = inf{ c ∈ R : cap{ z ∈ K : p(µ; z) ≤ c } > 0 }.

From the deﬁnition of m0 and the lower semicontinuity of p(µ; ·) it immediately follows that a minimal point z0 ∈ K can be chosen in K so that p(µ; z0 ) = m0 and cap(K ∩ U ) > 0 for every open neighborhood U ⊂ C of z0 . Proposition 14. Let the compact set K ⊂ C, a given measure µ ∈ M(C), and the point z0 ∈ K possess the properties just stated. By KII ⊂ K we denote the set of irregular points in K with respect to the Dirichlet problem in the domain C \ K. Let further U ⊂ C be a simply connected open neighborhood of z0 with ∂U being a smooth Jordan curve, and let δ > 0 be arbitrary. Then there exists µ1 ∈ M(K ∪ U ) with µ1 (KII ) = 0 and c1 ∈ R such that (3.14)

p(µ1 ; z) + c1 = min(p(µ; z), m0 + δ)

for every z ∈ U ∪ K \ KII , µ1 |U1 ≥ µ|U1

(3.15) def

for the set U1 = { z ∈ U : p(µ; z) ≤ m0 + δ }, and for every measure µ2 ∈ M(K ∪ U ) with µ2 (KII ) = 0 and µ2 = µ1 that satisﬁes (3.14) and (3.15) with (µ1 , c1 ) replaced by (µ2 , c2 ), c2 ∈ R, we have (3.16)

µ1 < µ2 .

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A POTENTIAL-THEORETIC PROBLEM

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It is an immediate consequence of the conditions that lead to (3.16) that for every open neighborhood U with the assumed properties and every δ > 0, the measure µ1 is uniquely determined by the two assertions (3.14), (3.15), and the principle of minimal mass expressed by (3.16). In most cases, the measure µ1 is diﬀerent from µ, but there are interesting exceptions that will be studied in Proposition 16, further below. There, it is shown that if µ = b ωK , b > 0, then we have µ1 = µ for every U and every δ > 0. Proof. In the ﬁrst part of the proof a measure µ1 ∈ M(K ∪ U ) that satisﬁes (3.14) and (3.15) is constructed in two steps. After that, in a second part, it is shown that this measure µ1 satisﬁes also (3.16). The two steps of the construction in the ﬁrst part consist of a levelling procedure applied to the potential p(µ; ·) followed by balayage with mass reduction. Since the minimum of two superharmonic functions is again superharmonic, and since this is also true for logarithmic potentials (cf., [Ran95, Theorem 3.4.2]), there exists µ 1 ∈ M(C) and c1 ∈ R with µ1 = µ such that p( µ1 ; ·) + c1 = min(p(µ; ·), m0 + δ).

(3.17) def

The set A1 = { z ∈ C : p(µ; z) > m0 + δ } is open and obviously also bounded in C. It is important for the subsequent investigation that the measure µ 1 in (3.17) can also be seen as the result of balayage of the measure µ out of the open set A1 . The procedure has been described in Proposition 12. Since U1 = U \ A1 , relation (3.18)

µ 1 |U1 ≥ µ|U1

follows from (3.4) in Proposition 12. Notice that each component of A1 is simply connected, and therefore there exist no irregular points, i.e., (∂A1 )II = ∅. From the interpretation of µ 1 as the result of balayage out of the bounded and open set A1 it follows that (3.19)

supp(µ) ⊂ P c (supp( µ1 )) ,

where P c (S) denotes the polynomial convex hull of a compact set S ⊂ C, which is the union of S with all bounded components of C \ S. On the other hand, the construction of the measure µ 1 just described is the result of cutting (levelling oﬀ) the potential p(µ; ·) at a level m0 + δ. In the second step we start from µ 1 and deﬁne the measure µ1 ∈ M(K ∪ U ) by balayage with mass reduction. The procedure has been described in Proposition 13, and there we take µ 1 as µ, µ1 as µ , c1 as c, and C \ (K ∪ U ) as the domain G. From (3.6) in Proposition 13 together with (3.17) we then have equality (3.14) for all z ∈ C \ (G ∪ ∂GII ) = K \ KII ∪ U . The assertion µ1 (KII ) = 0 is identical with the analogous assertion µ (∂GII ) = 0 in Proposition 13. Relation (3.15) is a consequence of (3.18) and representation (3.9) in Proposition 13, since (3.9) implies that (3.20)

µ1 |U = µ 1 |U ,

Indeed, it follows from (3.9) that the measure µ in Proposition 13 diﬀers from µ only outside of C \ G, and C \ G corresponds to U in the present environment. Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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For later use we add that because b0 ≥ 0 of (3.11) in the proof of Proposition 13 we always have µ1 ≤ µ1 = µ .

(3.21)

However, if supp( µ1 ) \ (K ∪ U ) = ∅, then from (3.8) in Proposition 13 the stronger result µ1 < µ1 = µ

(3.22)

follows. The veriﬁcation of (3.16) is practically a uniqueness proof for the measure µ1 . We assume that there exists a measure µ2 ∈ M(K ∪ U ) with µ2 (KII ) = 0 that satisﬁes (3.14) and (3.15) with (µ1 , c1 ) replaced by (µ2 , c2 ), c2 ∈ R. From (3.7) of Proposition 13 it then follows that either µ1 = µ2 or µ1 < µ2 . Since the ﬁrst alternative has been excluded, (3.16) is proved. The measure µ1 ∈ M(K ∪ U ) in Proposition 14 depends on the choice of the neighborhood U of z0 and on δ > 0. In the next proposition we show that this dependence possesses a certain monotonicity. Further, we consider the limit of the measure µ1 and that of the potential p(µ1 ; ·) + c1 in (3.14) of Proposition 14 for δ → 0 and U → {z0 }. We write Uj → {z0 } for a sequence {Uj }j∈N if Uj = {z0 }. j∈N

Proposition 15. Let K, µ, m0 , and z0 be the same objects as in Proposition 14, and let Uj and δj , j = 1, 2, . . ., be two sequences of open neighborhoods and positive values, respectively, that have the same properties as the corresponding individual objects U ⊂ C and δ > 0 in Proposition 14. By µj ∈ M(K ∪ U j ) and cj ∈ R, j = 1, 2, . . ., we denote the analogues of the measure µ1 and the constant c1 from Proposition 14 corresponding to U = Uj and δ = δj . (i) If U1 ⊇ U2 and δ1 ≥ δ2 , then we have µ1 ≥ µ2 . (ii) For any sequence {(Uj , δj )}j∈N with Uj ⊇ Uj+1 , Uj → {z0 }, δj ≥ δj+1 , and δj → 0 as j → ∞, the limit lim µj = b0

(3.23)

j→∞

exists, and we have ∗

µj −→ b0 ωK for j → ∞, and

(3.24) (3.25)

lim [p(µj ; ·) + cj ] = −b0 gC\K (·, ∞) + m0

j→∞

locally uniformly in C \ K. Proof. The monotonicity in part (i) follows rather immediately from the construction of the measure µ1 in two steps in the proof of Proposition 14. The conclusions become more obvious if one starts in Proposition 14 with µ1 as the choice for µ, and if one takes U2 , δ2 as the choice for U and δ. Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

A POTENTIAL-THEORETIC PROBLEM

279 25

The existence of limit (3.23) is an immediate consequence of the monotonicity in part (i). From this monotonicity, it further follows together with (3.14) in Proposition 14 and µj (KII ) = 0 that p(µj ; z) + cj − δj ≤ p(µj+1 ; z) + cj+1 − δj+1 for all z ∈ C, j ∈ N. From the last inequality and Harnack’s monotone convergence (cf., [Ran95, Theorem 1.3.9]) we get the existence of limit (3.25) locally uniformly in C \ K. The concrete form of the limit function on the right-hand side of (3.25) follows then from the observation that this function is equal to m0 quasi everywhere on K, and at inﬁnity it has the form b log |z|+O(1) as z → ∞. Limit (3.24) is then a consequence of (3.25). From (3.21) in the proof of Proposition 14 we know that we always have µ1 ≤ µ for the measure µ1 in Proposition 14. But in most cases, we have the proper inequality µ1 < µ, i.e., an eﬀective mass reduction takes place, as will be shown in the next proposition. The only exception from this rule are multiples of the equilibrium distribution ωK . Proposition 16. We have b0 < 1

(3.26)

in (3.23) of Proposition 15 for any µ ∈ M(C) with µ ≤ 1 and µ = ωK . def

For the measures µ = b ωK with b > 0 the new measure µ1 in Proposition 14 is identical with the original one, i.e., we have µ1 = b ωK = µ for any open neighborhood U with the assumed properties and any δ > 0. Proof. If µ < 1, then we automatically have b0 ≤ µ1 ≤ µ < 1. Hence, without loss of generality we can assume that µ = 1. The two cases supp(µ)\K = ∅ and supp(µ) ⊂ K will be considered separately. If (3.27)

supp(µ) \ K = ∅,

then we also have supp(µ) \ (K ∪ U ) = ∅ for a neighborhood U of z0 suﬃciently small. Since the neighborhoods U have been assumed to be simply connected, it follows from (3.27) together with (3.19) in the proof of Proposition 14 that for all δ > 0 we have supp( µ1 ) \ (K ∪ U ) = ∅ for the measure µ 1 introduced in (3.17) in the proof of Proposition 14. From (3.22) we then know that (3.28)

µ1 < µ = 1,

and with the monotonicity from Proposition 15 it follows that b0 < 1, which proves (3.26) under assumption (3.27). Next we assume that (3.29)

supp(µ) ⊂ K.

If the potential p(µ; ·) is constant quasi everywhere on K, then from the deﬁnition of m0 in (3.12) it follows that p(µ; z) = m0 for quasi every z ∈ K. With the Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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properties of the Green function (cf., Deﬁnition 1 or [Ran95, Section 4.4]) we then conclude that p(µ; ·) − m0 = −gC\K (·, ∞), which implies that µ = ωK . But this last possibility has explicitly been excluded, and so we have proved that p(µ; ·) is not constant quasi everywhere on K. From the last conclusion it follows that there exists δ0 > 0 such that the set def A1 = { z ∈ C : p(µ; z) > m0 + δ } is not empty for all 0 < δ ≤ δ0 . The set A1 has already been considered after (3.17) in the proof of Proposition 14. From A1 = ∅ it follows that the measure µ 1 from (3.17) is diﬀerent from µ, and since K has no inner points, this implies that supp( µ1 ) \ K = ∅

(3.30)

for all

0 < δ ≤ δ0 .

With the same argumentation as used after (3.27) we then deduce that (3.28) holds true for a neighborhood U of z0 suﬃciently small and 0 < δ ≤ δ0 . Hence, (3.26) is also proved under assumption (3.29). def

Set now µ = b1 ωK with 0 < b1 . Since the potential p(ωK ; ·) is constant quasi everywhere on K, it follows from (3.13) that p(µ; ·) = m0 quasi everywhere on K. Consequently, in the proof of Proposition 14, we have µ = µ 1 = µ1 and c1 = c1 = −m0 for all δ > 0, which proves that µ1 = b1 ωK and b0 = b1 in (3.23) of Proposition 15. In general, the point z0 in Proposition 14 and 15 is not an inner point of the set U1 ⊂ U deﬁned after (3.15) in Proposition 14. Therefore, the new measure µ1 ∈ M(K ∪ U ) from Proposition 14 does in general not reproduce the original measure µ in a neighborhood of z0 . In the next proposition we will show that the measure µ1 from Proposition 14 can be modiﬁed in a neighborhood of z0 in such a way that the new measure µ2 is an exact copy of the original measure µ in this neighborhood. Proposition 17. Let the objects K, µ, U , z0 , µ1 , and c1 be the same as in Proposition 14. Then there exists a measure µ2 ∈ M(K ∪ U ), a constant c2 ∈ R, and r > 0 with D(z0 , r) ⊂ U such that (3.31)

µ2 |D(z0 ,r) = µ|D(z0 ,r) ,

(3.32)

µ2 |(K∪U )\U = µ1 |(K∪U )\U ,

(3.33)

µ2 = µ1 ,

(3.34) (3.35)

p(µ2 ; z) + c2 = p(µ; z) for |z − z0 | ≤ r, = p(µ1 ; z) + c1 for z ∈C\U p(µ2 ; z) + c2 ≥ p(µ1 ; z) + c1 for z ∈ U.

In (3.31) and (3.32) it is shown that the new measure µ2 is identical with the original measure µ in a small neighborhood D(z0 , r) of z0 , and on the other hand, it coincides with the measure µ1 from Proposition 14 outside of the larger neighborhood U . Nothing explicit is said here about the measure µ2 in the set U \ D(z0 , r), but the second line in (3.35) gives some relevant information about the behavior of the potential in U \ D(z0 , r). Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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Because of (3.33), minimality (3.16) of the measure µ1 in Proposition 14 holds also for the new measure µ2 , and obviously the same is true for the assertions of Proposition 15 and 16. These observations are formulated in the next corollary. Like the measure µ1 from Proposition 14, so also the new measure µ2 from Proposition 17 depends on the choice of the neighborhood U of z0 and on δ > 0. For the next corollary the ﬁrst line in (3.35) is decisive since it shows that outside of the open set U the two potentials p(µ1 ; z) and p(µ2 ; z) are identical up to a constant. Corollary 18. The conclusions of the Propositions 15 and 14 remain true if the sequences of measures and constants considered there are replaced by analogous objects from Proposition 17. Especially, one can exchange the measure µ1 from Proposition 14 against measure µ2 from Proposition 17. Proof of Proposition 17. In the proofs of Proposition 14 and 16 we have already used the set A1 = { z ∈ C : p(µ; z) > m0 + δ } at several places. As a consequence of the superharmonicity of p(µ; z), the set A1 is open. It further follows from the superharmonicity that each component C of A1 is simply connected. For each r > 0 with D(z0 , r) ⊂ U , let A0 (r) denote the union of all components C of A1 with C ∩ D(z0 , r) = ∅. There exists r0 > 0 such that (3.36)

def

A0 = A0 (r0 ) satisﬁes

A0 ⊂ U.

Indeed, if A0 (r) ∩ ∂U = ∅ for all r > 0, then there would exist a component C1 of A1 with z0 ∈ C 1 and C 1 ∩ ∂U = ∅. From the principle of domination (cf., [Ran95, Theorem 3.6.9] or [ST97, Theorem II.3.2]) and the properties of the Green function (cf., [Ran95, Section 4.4]) it then follows that p(µ; z) − (m0 + δ) ≥ − µ gC\C 1 (z, ∞)

for z ∈ C,

and therefore we have p(µ; z0 ) = m0 +δ since C\C 1 is a regular domain with respect to the Dirichlet problem (cf., [Ran95, Theorem 4.2.1]). But this contradicts (3.13), and (3.36) is veriﬁed. The measure µ2 is now deﬁned by (3.37)

def

µ2 |A0 = µ|A0

and

def

µ2 |C\A0 = µ1 |C\A0 .

Identity (3.32) follows immediately from this deﬁnition and A0 ⊂ U . For the proof of (3.31) we need some preparations. def

Beside of the set U1 = U \ A1 we also consider U0 = U \ A1 , which is an open set, and U0 ⊂ U1 . As a sharpening of (3.15) in Proposition 14 we show that (3.38)

µ1 |U0 = µ|U0 .

Indeed, in the proof of Proposition 14 it has been shown after (3.17) that the measure µ 1 , which has been introduced in (3.17), can be seen as the result of balayage out of the open set A1 . From representation (3.4) in Proposition 12 of the balayage measure together with the deﬁnition of the set U0 we therefore have µ 1 |U0 = µ|U0 . Identity (3.38) then follows from the last identity and (3.20) in the proof of Proposition 14. We have A0 ⊂ A1 and A0 ∩ D(z0 , r0 ) = A1 ∩ D(z0 , r0 ). Hence, we also have U0 ∩ D(z0 , r0 ) = D(z0 , r0 ) \ A0 , and with (3.38) this proves that µ1 |D(z0 ,r0 )\A0 = µ|D(z0 ,r0 )\A0 . Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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With deﬁnition (3.37) we further conclude that µ2 |D(z0 ,r0 ) = µ|D(z0 ,r0 )∩A0 + µ1 |D(z0 ,r0 )\A0 = µ|D(z0 ,r0 ) , which proves (3.31). Now, we can interpret the measure µ1 as the result of balayage of the new measure µ2 out of the open set A0 . Indeed, by a closer look on the proof of Proposition 14 we see that for each component C1 of A1 that is fully contained in U , the measure µ1 |C 1 result from balayage of µ|C 1 out of C1 . Notice that the second step in the construction of µ1 lets everything invariant in U . Our claim then follows from µ|C 1 = µ2 |C 1 , which is a consequence of (3.37), and the fact that balayage out of an open set can be broken down into the balayage out of each of its components (cf., [ST97, Theorem II.4.7]). From the description of balayage in Proposition 12 we know that µ2 = µ1 , which is equation (3.33), and further that there exists c2 ∈ R with = p(µ1 ; z) + c1 for z ∈ C \ A0 (3.39) p(µ2 ; z) + c2 ≥ p(µ1 ; z) + c1 for z ∈ A0 . The last relation is a slightly more precise version of (3.35). Identity (3.34) is a consequence of (3.31) and the ﬁrst line of (3.39). 3.3. Normal Derivatives and Convergence on Half-Circles. The results of the next two propositions have played a role in the proof of Proposition 11. In both propositions we start our considerations with real-valued functions def that are harmonic in the upper half-disc D+ = { z ∈ D : Im(z) > 0 } or the upper def

half-ring D+ (r1 , r2 ) = r2 D+ \ r1 D with 0 < r1 < r2 ≤ 1. Proposition 19. Let u be a positive and harmonic function in the upper halfdisc D+ with boundary-values u(x) = 0 for x ∈ (−1, 1). Under this assumption the normal derivatives def ∂ u(x + i 0), x ∈ (−1, 1), (3.40) g(x) = ∂y exist, the function g is real analytic in (−1, 1), and we have (3.41)

g(x) > 0

for all

x ∈ (−1, 1).

Proof. We extend the function u into D by reﬂection on (−1, 1), i.e., we set def

u(z) = −u(z) for z ∈ D+ . It is an immediate consequence of the mean-value property that the extended function is harmonic in D, and consequently, it can be completed to an analytic function f in D that satisﬁes Im f = u and f (0) = 0. Elementary calculations show that ∂ u(z) = Re f (z) ∂y

for z = x + i y ∈ D.

Hence, the function g in (3.40) is the restriction of an harmonic function, which implies that it is real-analytic. Considering the development of f in D yields π dt ∂ u(0) = Re f (0) = f (0) = >0 u(reit ) sin(t) (3.42) ∂y rπ 0 Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

A POTENTIAL-THEORETIC PROBLEM

283 29

for 0 < r < 1, and the inequality in (3.42) therefore is a consequence of the assumption u > 0 in D+ . Inequality (3.41) follows from (3.42) for all x ∈ (−1, 1) by using a Moebius transform for mapping x to 0 in the usual way. Proposition 20. Let u be a positive and harmonic function in the upper halfring D+ (r, 1) = D+ \ rD with 0 < r < 1, and assume that it is continuous in D+ (r, 1). Let further un , n ∈ N, be a sequence of functions that are harmonic in D+ (r, 1) and continuous in D+ (r, 1). We assume that (3.43)

for x ∈ [−1 − r] ∪ [r, 1] and n ∈ N,

u(x) = un (x) = 0

(3.44)

lim un (z) = u(z)

n→∞

point-wise for z ∈ D+ (r, 1),

and un is bounded on D+ (r, 1) uniformly for all n ∈ N. Under these assumptions we have lim u − un D+ (r ,r ) = 0

(3.45)

n→∞

for r < r < r < 1

with ·S denoting the uniform norm on S. With def

qn =

(3.46)

inf

z∈D+ (r

,r )

un (z) , u(z)

def

Qn =

un (z) z∈D+ (r ,r ) u(z) sup

we further have lim qn = lim Qn = 1.

(3.47)

n→∞

n→∞

Proof. Like in Proposition 19, we can extend the functions u and un harmonically into the ring D \ rD and continuously into the closed ring D \ rD by reﬂection on R because of (3.43). The extended functions are again denoted by u and un . Analytic completions f and fn of u and un , respectively, exist throughout the ring D \ rD. We assume that 1+r 1+r ) = Re fn (i ) = 0, n ∈ N. 2 2 The last two equalities in (3.48) assure the uniqueness of f and fn . The existence of the analytic completions f and fn in the ring domain D \ rD may deserve some remarks since D\rD is not simply connected. We ﬁrst look for analytic completions in the simply connected upper ring D+ \rD, which obviously exist, and extend them then into the whole ring D \ rD by (3.48) Im f = u, Im fn = un , and Re f (i

f (z) = f (z), fn (z) = fn (z),

(3.49)

z ∈ D+ \ rD.

Using for the harmonic functions u and un a representation by a Poisson integral shows that it follows from the assumed point-wise convergence (3.44) and the uniform boundedness of {un } that we have (3.50)

lim fn = f and

n→∞

lim f n→∞ n

= f locally uniformly in D \ rD

The uniform convergence (3.45) follows immediately from the ﬁrst limit in (3.50) since we have D+ (r , r ) ⊂ D+ (r, 1) for r < r < r < 1. In the proof of (3.47) we consider two separate parts of D+ (r , r ). One of these parts consists of a small strip along [−1 − r] ∪ [r, 1]. We start with the small strip. Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

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HERBERT STAHL

From (3.41) in Proposition 19 it follows that there exist ε > 0 and δ > 0 such that we have def V = { z ∈ D+ (r , r ) : Im z ≤ ε } ⊂ D+ (r, 1) and ∂ u(z) ≥ δ ∂y

(3.51)

∂ We have ∂y u = Re f and limit in (3.50) that

∂ ∂y un

for

z = x + i y ∈ V.

= Re fn , and therefore it follows from the second

∂ ∂ un (z) = u(z) uniformly for n→∞ ∂y ∂y lim

z = x + i y ∈ D+ (r , r ),

and together with (3.51), we then have (3.52)

lim

n→∞

un (z) = 1 uniformly for u(z)

z ∈ D+ (r , r ) ∩ V.

From (3.45) together with the assumption that u(z) > 0 for z ∈ D+ (r, 1), it follows that un (z) = 1 uniformly for z ∈ D+ (r , r ) \ V. (3.53) lim n→∞ u(z) Limit (3.47) then follows from (3.52) and (3.53) together with (3.46).

References [Apt02] A. I. Aptekarev. Sharp constants for rational approximations of analytic functions. Mat. Sb., 193:1–72, 2002. [Gon87] A. A. Gonchar. Rational approximation of analytic functions. Proc. Int. Congr. Math. Berkely/Calif. 1986, Vol. 1:739–748, 1987. [GR87] A. A. Gonchar and E. A. Rakhmanov. Equilibrium distributions and the degree of rational approximation of analytic functions. Matem. Sbornik, 134(176)(3):306–352, 1987. English transl. in Math. USSR Sbornik 62(2):305–348, 1989. [NS77] J. Nuttall and S. R. Singh. Orthogonal polynomials and Pad´e approximants associated with a system of arcs. J. Approx. Theory, 21:1–42, 1977. [Nut77] J. Nuttall. The convergence of pad´e approximants to functions with branch points. In Pad´ e and Rational Approximation, (E. Saﬀ and R. S. Varga, eds), New York, 1977. Academic Press. [Nut86] J. Nuttall. Asymptotics of generalized jacobi polynomials. Constr. Approx., 2:59–77, 1986. [Nut90] J. Nuttall. Pad´e polynomial asymptotics from a singular integral equation. Constr. Approx., 6:157–166, 1990. [Ran95] T. Ransford. Potential Theory in the Complex Plane, volume 28 of London Math. Soc. Students Texts. Cambridge University Press, Cambridge, 1995. [SS08] H. Stahl and T. Schmelzer. An extension of the 1/9 −Problem. submitted to J. Comput. Appl. Math., 2008. [ST97] E. B. Saﬀ and V. Totik. Logarithmic Potentials with External Fields, volume 316 of Grundlehren der Math. Wissenschaften. Springer-Verlag, Berlin, 1997. [Sta86a] H. Stahl. Orthogonal polynomials with a complex weight function I. Constr. Approx., 2:225–240, 1986. [Sta86b] H. Stahl. Orthogonal polynomials with a complex weight function II. Constr. Approx., 2:241–251, 1986. [Sta89] H. Stahl. On the convergence of generalized Pad´e approximants. Constr. Approx., 5:221– 240, 1989. [Sta90] H. Stahl. General convergence results for rational approximants. In Approximation Theory VI, Chui, Schumaker, and Ward eds, pages 605–634, New York, NY, USA, 1990. Acadenic Press.

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A POTENTIAL-THEORETIC PROBLEM

285 31

Technische Fachhochschule Berlin, FB II Mathematik, Luxemburgerstrasse 10, D13353 Berlin, GERMANY E-mail address: [email protected]

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http://dx.doi.org/10.1090/conm/507/09965

Contemporary Mathematics Volume 507, 2010

Orthogonal Polynomials and Approximation Theory: some open problems Walter Van Assche Dedicated to Guillermo L´ opez Lagomasino at the occassion of his 60th birthday

Abstract. During the International Workshop on Orthogonal Polynomials and Approximation Theory there was an open problem session on Friday, September 12, 2008. Some of the open problems, with a brief description of their background and references to relevant literature, are given in this paper.

1. Open problems posed by Barry Simon 1.1. Orthogonal polynomials on the disk. Orthonormal polynomials on the disk B1 (0) = {z ∈ C : |z| ≤ 1} are deﬁned as polynomials {pn , n = 0, 1, 2, . . .} for which pn (z)pm (z) dµ(z) = δm,n ,

z = reiθ ,

B1 (0)

where µ is a positive measure on the disk. If we take the Lebesgue measure (area measure) on the disk, then the orthogonality is 1 2π 1 pn (reiθ )pm (reiθ ) rdr dθ = δm,n π 0 0 and the orthonormal polynomials are given by √ (1.1) pn (z) = n + 1 z n . Observe that these polynomials have the same zeros as the orthonormal polynomials {ϕn , n = 0, 1, . . .} on the unit circle T = {z ∈ C : |z| = 1} with respect to Lebesgue measure (arc length measure) on the circle, which are ϕn (z) = z n , the only diﬀerence is in the leading coeﬃcient, or in the norm of the monic polynomial. Rakhmanov’s theorem [Rakh] [Sim2, Ch. 9] gives information about the ratio of leading coeﬃcients of orthogonal polynomials on the unit circle for measures which are comparable to the Lebesgue measure on the circle: 1991 Mathematics Subject Classiﬁcation. 30E10, 41A21, 42C05, 47B36. Key words and phrases. Orthogonal polynomials, approximation theory. 1

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2 288

WALTER VAN ASSCHE

Theorem 1.1 (Rakhmanov). Suppose µ is a measure on the unit circle T = {z ∈ C : |z| = 1} with µ > 0 almost everywhere on T. If ϕn (z) = κn z n + · · · are the orthonormal polynomials for the measure µ, then (1.2)

lim κn /κn+1 = 1

n→∞

and pn+1 (z) = 1, n→∞ zpn (z) lim

uniformly for |z| ≥ 1. Problem 1.1. Is there a Rakhmanov theorem for the disk? Put diﬀerently, is there a common property of all measures on the disk larger than a measure equivalent to area measure on the disk? The example (1.1) shows that the limit of the ratio κn /κn+1 does not give a distinction between the circle and the disk. If we consider orthonormal polynomials on the annulus {z ∈ C : R ≤ |z| ≤ 1} (0 < R < 1) with respect to Lebesgue measure (area measure) 2π 1 1 pn (reiθ )pm (reiθ ) rdr dθ = δm,n , π(1 − R2 ) 0 R then (1.3)

pn (z) =

√

n+1

1 − R2 zn , 1 − R2n+2

and hence also for these polynomials the ratio κn /κn+1 converges to 1, as is the case for the orthonormal polynomials on the disk and on the unit circle. This example suggests that this phenomenon occurs for more than just the circle and the disk. References [Sim2] B. Simon: Orthogonal Polynomials on the Unit Circle, Part 2: Spectral Theory, Amer. Math. Soc. Colloq. Publ. 54, Amer. Math. Soc., Providence RI, 2005. [Rakh] E.A. Rakhmanov: On the asymptotics of the ratio of orthogonal polynomials, II, Mat. Sb. 118 (160) (1982), 104–117 (in Russian); Math. USSR Sb. 46 (1983), 105–117.

1.2. Isospectral torus. During the workshop, Saﬀ talked about orthogonal polynomials on an Archipelago1 [GPSS1, GPSS2]. Typically the regions for these orthogonal polynomials are disconnected. For orthogonal polynomials on several intervals (of the real line) [CSZ, SoYu, PeYu] or on several arcs (of the unit circle) [Sim2, Ch. 11] there are cases when the recurrence coeﬃcients are periodic or almost periodic, and the set of recurrence coeﬃcients which give rise to the same spectrum (the same support of the orthogonality measure) turns out to be a manifold which is called the isospectral torus. Unfortunately, there is no ﬁnite order recursion formula for the Bergman polynomials on an Archipelago, so it is not obvious to ﬁnd an operator such as the Jacobi operator (for orthogonal polynomials on the real line) or the CMV matrix (for orthogonal polynomials on the unit circle). 1 an

extensive group of islands

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ORTHOGONAL POLYNOMIALS AND APPROXIMATION THEORY: SOME OPEN PROBLEMS 3 289 ORTHOGONAL POLYNOMIALS AND APPROXIMATION THEORY

One can always use the multiplication operator M by expanding zpn (z) into an orthogonal series consisting of the Bergman polynomials zpn (z) =

n+1

ak,n pk (z),

k=0

and then the Hessenberg matrix M = (ak,j )k,j=0,1,2,... has the property that its principal submatrices ⎛ a0,0 a0,1 ⎜a1,0 a1,1 ⎜ ⎜ a2,1 Mn = (ak,j )k,j=0,1,...,n = ⎜ 0 ⎜ .. .. ⎝ . . 0

···

a0,2 a1,2 a2,2 .. . 0

··· ··· ··· ··· an,n−1

⎞ a0,n a1,n ⎟ ⎟ a2,n ⎟ ⎟ .. ⎟ . ⎠ an,n

have (left) eigenvalues which coincide with the zeros of pn+1 . Problem 1.2. Is there an isospectral torus for the regions talked about in Ed Saﬀ ’s talk? References J.S. Christiansen, B. Simon, M. Zinchenko: Finite gap Jacobi matrices, I. The isospectral torus, Constr. Approx. doi:10.1007/s00365-009-9057-z [GPSS1] B. Gustafsson, M. Putinar, E.B. Saﬀ, N. Stylianopoulos: Les polynˆ omes orthogonaux de Bergman sur un archipel, C.R. Acad. Sci. Paris, Ser. I 346 (2008), 499–502. [GPSS2] B. Gustafsson, M. Putinar, E.B. Saﬀ, N. Stylianopoulos: Bergman polynomials on an Archipelago: estimates, zeros and shape reconstruction, Adv. Math. doi:10.1016/j.aim.2009.06.016 [PeYu] F. Peherstorfer, P. Yuditskii: Asymptotic behavior of polynomials orthonormal on a homogeneous set, J. Anal. Math. 89 (2003), 113–154. [Sim2] B. Simon: Orthogonal Polynomials on the Unit Circle, Part 2: Spectral Theory, Amer. Math. Soc. Colloq. Publ. 54, Amer. Math. Soc., Providence RI, 2005. [SoYu] M. Sodin, P. Yuditskii: Almost periodic Jacobi matrices with homogeneous spectrum, inﬁnite dimensional Jacobi inversion, and Hardy spaces of character-automorphic functions, J. Geom. Anal. 7 (1997), no. 3, 387–435. [CSZ]

1.3. Quasi-clock behavior of zeros. Let {pn , n = 0, 1, 2, . . .} be orthogonal polynomials on [−1, 1] and suppose that the zeros are asymptotically distributed according to a measure with density ρ on [−1, 1]: b 1 lim #{zeros of pn ∈ [a, b]} = ρ(x) dx. n→∞ n a Fix a point x0 ∈ (−1, 1) and denote the zeros of pn in the neighborhood of x0 by x−k,n (x0 ) < · · · < x−1,n (x0 ) < x0 ≤ x0,n (x0 ) < x1,n (x0 ) < · · · . The zeros are said to have clock behavior at x0 if lim n (xj+1,n (x0 ) − xj,n (x0 )) =

n→∞

1 ρ(x0 )

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4 290

WALTER VAN ASSCHE

for all ﬁxed j ∈ Z. See, e.g., [LSim], [Sim2]. This notion requires the existence of the density ρ for the zeros. A weaker notion is quasi-clock behavior at x0 , for which xj+1,n (x0 ) − xj,n (x0 ) lim =1 n→∞ x1,n (x0 ) − x0,n (x0 ) for any ﬁxed j ∈ Z. Problem 1.3. In Simon’s equilibrium measure paper [Sim1, Example 5.8] there is an example of a measure on [−1, 1] for which the density of zeros does not have a limit. Prove or disprove that there is quasi-clock behavior of the zeros for this example. See also [ALS] for a more general conjecture. References [ALS] A. Avila, Y. Last, B. Simon: Bulk universality and clock spacing of zeros for ergodic matrices with a.c. spectrum, arXiv:0810.3277 [LSim] Y. Last, B. Simon: Fine structure of the zeros of orthogonal polynomials IV: A priori bounds and clock behavior, Comm. Pure Appl. Math. 61 (2008), 486–538. [Sim1] B. Simon: Equilibrium measures and capacities in spectral theory, Inverse Problems and Imaging 1 no 4 (2007), 713–772. [Sim2] B. Simon: Fine structure of the zeros of orthogonal polynomials: a progress report, this volume.

1.4. Singular measures. The orthogonal polynomials for the classical Cantor measure are not known explicitly, but there are techniques for computing the orthogonal polynomials and the recurrence coeﬃcients [Man]. Much more is known for orthogonal polynomials on the Julia set of T (z) = z 2 − c, which for c > 2 is a real set of measure zero [BGH, BMM]. In this case the orthogonal polynomial of degree 2n is the n-th iterate of the polynomial T : p2n (z) = T ◦n (z), where T ◦n (z) = T ◦(n−1) (T (z)) and T ◦1 (z) = T (z). They could be used to see what may happen for zeros of orthogonal polynomials for singular measures of Cantor type. Problem 1.4. What can be said about the ﬁne structure of the zeros of orthogonal polynomials for singular measures? For example how about the classical Cantor measure? References [BGH] M.F. Barnsley, J.S. Geronimo, A.N. Harrington: Almost periodic Jacobi matrices associated with Julia sets for polynomials, Comm. Math. Phys. 99 (1985), no. 3, 303–317. [BMM] D. Bessis, M.L. Mehta, P. Moussa: Orthogonal polynomials on a family of Cantor sets and the problem of iterations of quadratic mappings, Lett. Math. Phys. 6 (1982), no. 2, 123–140. [Man] G. Mantica: A stable Stieltjes technique for computing orthogonal polynomials and Jacobi matrices associated with a class of singular measures, Constr. Approx. 12 (1996), no. 4, 509–530.

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ORTHOGONAL POLYNOMIALS AND APPROXIMATION THEORY: SOME OPEN PROBLEMS 5 291 ORTHOGONAL POLYNOMIALS AND APPROXIMATION THEORY

1.5. Finite gap sets. Let J be a Jacobi matrix ⎛ ⎞ b1 a1 0 0 ··· ⎜a1 b2 a2 0 · · ·⎟ ⎜ ⎟ J = ⎜ 0 a2 b3 a3 · · ·⎟ ⎝ ⎠ .. .. .. .. . . . . with aj > 0 and bj ∈ R, and let pn be the polynomials generated by xpn (x) = an+1 pn+1 (x) + bn+1 pn (x) + an pn−1 (x), with p0 = 1 and p−1 = 0. Then these polynomials are orthogonal on the real line with respect to some probability measure µ, which is the spectral measure of J. Killip and Simon [KS] have proved some results relating the behavior of the recurrence coeﬃcients (the entries of the Jacobi matrix J) and properties of the measure µ. One of their main results is Theorem 1.2 (Killip-Simon). Let J be a Jacobi matrix and µ its corresponding spectral measure. Then (1.4)

2

∞

(an − 1)2 +

n=1

∞

b2n < ∞

n=1

if and only if the following four properties hold − 1. The support of µ is [−2, 2] ∪ {x+ j , j = 1, . . . , N+ } ∪ {xj , j = 1, . . . , N− }, + − where N± are zero, ﬁnite or inﬁnite, and x+ 1 > x2 > · · · > 2 and x1 < − ± x2 < · · · < −2, and if N± = ∞ then limj→∞ xj = ±2. 2. If µac is the absolutely continuous part of µ, then 2

log µac (x) 4 − x2 dx > −∞. −2

3. The following bound holds N+

3/2 |x+ + j − 2|

j=1

N−

3/2 |x− < ∞. j + 2|

j=1

4. µ is a probability measure. If the interval [−2, 2] is replaced by a collection of intervals on the real line, then the recurrence coeﬃcients no longer converge, but they may be (asymptotically) periodic or almost periodic. Problem 1.5. Prove an analog of the Killip-Simon theorem for general ﬁnite gap sets. See also [DKS] where the result is proved for asymptotically periodic recurrence coeﬃcients. References [DKS] D. Damanik, R. Killip, B. Simon: Perturbations of orthogonal polynomials with periodic recursion coeﬃcients, Ann. of Math. (to appear). [KS] R. Killip, B. Simon: Sum rules for Jacobi matrices and their applications to spectral theory, Ann. Math. 158 (2) (2003) 253–321.

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1.6. Subexponential behavior. The Nevai class for [−1, 1] consists of all families of orthogonal polynomials (all orthogonality measures µ on the real line) with a three term recurrence relation xpn (x) = an+1 pn+1 (x) + bn pn (x) + an pn−1 (x), for which 1 , lim bn = 0. n→∞ 2 Nevai [Nev] proved that for orthogonal polynomials in the Nevai class for [−1, 1] one has lim an =

n→∞

(1.5)

p2 (x) lim n n 2 =0 n→∞ k=0 pk (x)

for every x ∈ [−1, 1], and this holds uniformly on every compact set K ⊂ (−1, 1). The result in fact holds uniformly on [−1, 1] [BLS, NTZ, Szw]. Breuer, Last and Simon [BLS, Conjecture 1.4] have conjectured that for any orthogonality measure µ with compact support, the asymptotic formula (1.5) holds µ-almost everywhere on the support of µ. Problem 1.6. Prove or ﬁnd a counterexample of the Breuer-Last-Simon conjecture. References [BLS] J. Breuer, Y. Last, B. Simon: The Nevai condition, Constr. Approx. doi:10.1007/500365-009-9055-1 [Nev] P. Nevai: Orthogonal Polynomials, Memoirs Amer. Math. Soc. 18 (1979), number 213, Amer. Math. Soc., Providence RI, 185 pp. [NTZ] P. Nevai, V. Totik, J. Zhang: Orthogonal polynomials: their growth relative to their sums, J. Approx. Theory 67 (1991), 215–234. [Szw] R. Szwarc: Uniform subexponential growth of orthogonal polynomials, J. Approx. Theory 81 (1995), 296–302.

1.7. The Schr¨ odinger conjecture. Let H : 2 (N) → 2 (N) be a discrete one-dimensional Schr¨odinger operator: (Hy)n = yn−1 + yn+1 + V (n)yn ,

n ∈ N,

with y(0) = 0. Let B be the set of all λ for which all solutions y of yn−1 + yn+1 + V (n)yn = λyn are bounded as functions of n ∈ N. The Schr¨odinger conjecture2 says that the absolutely continuous part ρac of the spectral measure of H is equivalent to χB (λ) dλ, where χB is the characteristic function (indicator function) of the set B. There was an earlier version that conjectured it for the entire spectrum, but Jitomirskaya [Jit] found a counterexample to that. A variation of this conjecture, applied to Jacobi operators and orthogonal polynomials, is in terms of the three-term recurrence relation xpn (x) = an+1 pn+1 (x) + bn pn (x) + an pn−1 (x), 2 this formulation is taken from Christian Remling: http://www.math.ou.edu/∼cremling/misc/mill/schrodinger.htm

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ORTHOGONAL POLYNOMIALS AND APPROXIMATION THEORY: SOME OPEN PROBLEMS 7 293 ORTHOGONAL POLYNOMIALS AND APPROXIMATION THEORY

Problem 1.7. Prove or ﬁnd a counterexample of the following: orthogonal polynomials are almost everywhere bounded on the absolutely continuous spectrum of the Jacobi matrix (the support of the absolutely continuous part µac of the orthogonality measure). In 1921, Steklov [Stek] had a much stronger conjecture: Conjecture 1.3 (Steklov). Suppose pn are orthonormal polynomials on [−1, 1] with weight function w(x) ≥ δ > 0 for all x ∈ [−1, 1]. Then |pn (x)| ≤ C for every x ∈ (−1, 1). This conjecture is certainly not true at the endpoints ±1, which is already n + 12 . In clear by looking at Legendre polynomials, for which |pn (±1)| = 1979, Rakhmanov [Rakh] gave a counterexample where |pn (0)| is unbounded. The Schr¨odinger conjecture only claims that the polynomials are bounded almost everywhere on [−1, 1] for such polynomials. References S. Jitomirskaya: Singular spectral properties of a one-dimensional Schr¨ odinger operator with almost periodic potential, in “Dynamical Systems and Statistical Mechanics” (Moscow, 1991), Adv. Soviet Math. 3, Amer. Math. Soc. Providence RI, 1991, pp. 215– 254. [Stek] V.A. Steklov: Une m´ ethode de la solution du probl` eme de d´ eveloppement des fonctions en s´ eries de polynomes de Tch´ ebychef ind´ ependante de la th´ eorie de fermeture, I, II, Izv. Ross. Akad. Nauk; Bull. Acad. Sci. Russie (6), 15 (1921), 281–302, 303–326. [Rakh] E.A. Rakhmanov: On Steklov’s conjecture in the theory of orthogonal polynomials, Mat. Sb. (N.S.) 108 (150) (1979), no. 4, 581–608 (in Russian); Math. USSR Sb. 32 (1980), 549–575.

[Jit]

2. Open problems posed by Paul Nevai 2.1. Tur´ an determinants. Tur´ an’s inequality says that Pn2 (x) − Pn−1 (x)Pn+1 (x) ≥ 0,

−1 ≤ x ≤ 1

for the Legendre polynomials {Pn , n ≥ 0}. This inequality inspired Karlin and Szeg˝ o [KS] to investigate a general theory dealing with inequalities of this type for classical orthogonal polynomials. Later such quadratic forms were also studied for more general classes of orthogonal polynomials and higher order analogues were also considered (M´at´e, Nevai and Totik [MNT], Geronimo and Van Assche [GVA], Szwarc [Szw], Osilenker [Os], Berg and Szwarc [BSz]). Problem 2.1. Study the positivity of Tur´ an-type determinants such as p2n (x) − pn−1 (x)pn+1 (x) (for orthonormal polynomials) or Pn2 (x) − Pn−1 (x)Pn+1 (x) (for monic orthogonal polynomials), or their variations and their higher order analogues, for general classes of measures.

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References [BSz] C. Berg, R. Szwarc: Bounds on Tur´ an determinants, J. Approx. Theory doi:10.1016/j.jat.2008.08.010 [KS] S. Karlin, G. Szeg˝ o: On certain determinants whose elements are orthogonal polynomials, J. Analyse Math. 8 (1960/1961), 1–157. [MNT] A. M´ at´e, P. Nevai, V. Totik: Strong and weak convergence of orthogonal polynomials, Amer. J. Math. 109 (1987), 239–282. [Os] B.P. Osilenker: The Tur´ an determinant for orthogonal polynomials with asymptotically periodic recurrence coeﬃcients, Dokl. Akad. Nauk 361 (1998), no. 3, 318–320. [Szw] R. Szwarc: Positivity of Tur´ an determinants for orthogonal polynomials, in “Harmonic Analysis and Hypergroups” (Delhi, 1995), Birkh¨ auser, Boston MA, 1998, pp. 165–182. [GVA] J.S. Geronimo, W. Van Assche: Approximating the weight function for orthogonal polynomials on several intervals, J. Approx. Theory 65 (1991), no. 3, 341–371.

2.2. Christoﬀel functions. Let {pn , n ≥ 0} be orthonormal polynomials on the real line with orthogonality measure α. The Christoﬀel function is deﬁned as 1 λn (x) = λn (α; x) = n−1 . 2 k=0 pk (x) It used to be the case that conditions for the existence of lim nλn (x) > 0

n→∞

and conditions such that lim inf nλn (x) > 0, n→∞

a.e.

were diﬀerent, the latter being considerably weaker except that they are naturally local since Christoﬀel functions are monotone with respect to their measures. Now it appears that the gap has been closed, and nowadays both are proved under essentially the same suﬃcient conditions that are, clearly, far from being necessary. Problem 2.2. Find conditions for lim inf n→∞ nλn (x) > 0 (almost everywhere) that are substantially weaker than the currently known suﬃcient conditions for the existence of the limit. Regarding Problem 2.2, the following conjecture by Totik and Nevai (and probably also conjectured by many others) is relevant: Conjecture 2.1. Let α > 0 on an interval ∆ ⊂ supp(α). Then lim sup n→∞

n−1 1 2 pk (x) < ∞ n k=0

for almost every x ∈ ∆. The (much) weaker lim inf statement has been known for almost 30 years. If the above conjecture holds, then the orthogonal polynomials are (C, 1) bounded which would be a nice addition to Steklov’s conjecture that was proved to be false by Rakhmanov. 2.3. Comparative results for orthogonal polynomials. Based on the well-developed comparative theory of orthogonal polynomials, one can prove that (2.1)

lim

n→∞

λn (gdα, x) = g(x) λn (dα, x)

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ORTHOGONAL POLYNOMIALS AND APPROXIMATION THEORY: SOME OPEN PROBLEMS 9 295 ORTHOGONAL POLYNOMIALS AND APPROXIMATION THEORY

for large classes of measures α and functions g that satisfy some reasonably weak conditions, cfr. Bello and L´opez [BL, Lop], Peherstorfer and Steinbauer [PS], M´ at´e, Nevai and Totik [Nev, MNT]. The problem is that all these conditions seem to be far from being necessary as well. Problem 2.3. Let g be a nice function, say, positive and continuous (or more), and let α be in a given class of measures, say, supported on [−1, 1]. Find conditions for α that are both necessary and suﬃcient for (2.1) to hold. References M. Bello Hern´ andez, G. L´ opez Lagomasino: Ratio and relative asymptotics of polynomials orthogonal on an arc of the unit circle, J. Approx. Theory 92 (1998), no. 2, 216–244. [Nev] P. Nevai: Extensions of Szeg˝ o’s theory of orthogonal polynomials, in “Polynˆ omes Orthogonaux et Applications” (C. Brezinski et al., Eds.), Lecture Notes in Mathematics 1171, Springer-Verlag, Berlin, 1985, pp. 230–238. [MNT] A. M´ at´e, P. Nevai, V. Totik: Extensions of Szeg˝ o’s theory of orthogonal polynomials, II, III, Constr. Approx. 3 (1987), 51–72, 73–96. [PS] F. Peherstorfer, R. Steinbauer: Comparative asymptotics for perturbed orthogonal polynomials, Trans. Amer. Math. Soc. 348 (1996), no. 4, 1459–1486. [Lop] G. L´ opez Lagomasino: Relative asymptotics for polynomials orthogonal on the real axis, Mat. Sb. (N.S.) 137 (179) (1988), no. 4, 500–525 (in Russian); Math. USSR Sbornik 65 (1990), no. 2, 505–529. [BL]

3. An open problem posed by A.I. Aptekarev and A.A. Gonchar 3.1. Convergence of diagonal Pad´ e approximants. Suppose f is a given function, analytic in a neighborhood of 0. Let fn = Pn /Qn be the sequence of diagonal Pad´e approximants (near 0), i.e., Pn and Qn are polynomials of degree ≤ n for which f (z) − fn (z) = An z 2n+1 + O(z 2n+2 ),

z → 0.

Let D be a domain in C such that 0 ∈ D. Problem 3.1. Prove (or give a counterexample): If all fn are holomorphic in D (i.e., if the Pad´e approximants have no poles in D), then f is holomorphic in D. This result was proved by Gonchar when D is a disk or a domain close to a disk [Gon]. If fn avoids three points {0, 1, ∞}, then Montel’s theorem implies this conjecture. If fn avoids {0, ∞} then Jentzsch’s theorem essentially implies this conjecture. This problem (conjecture) corresponds to fn only avoiding {∞}. References [Gon] A.A. Gonchar: On uniform convergence of diagonal Pad´ e approximants, Mat. Sb., N. Ser. 118(160) (1982), 535–556 (1982) (in Russian); Math. USSR Sb. 46 (1983), 539–559.

4. An open problem posed by J.S. Geronimo 4.1. Orthogonal matrix polynomials. Consider an (m+1)×(m+1) matrix measure dµ with support on the real line. Associated with this measure there are orthogonal matrix polynomials Pn such that Pn (x) dµ(x) PkT (x) = Im+1 δn,k .

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They satisfy a three-term recurrence relation xPn (x) = An+1 Pn+1 (x) + Bn Pn (x) + ATn Pn−1 (x), with P0 = I and P−1 = 0. Suppose the matrix measure is a Hankel matrix ⎛ ⎞ µ0 µ1 ··· µm ⎜ µ1 µ2 · · · µm+1 ⎟ ⎜ ⎟ µ=⎜ . .. ⎟ . ⎝ .. . ⎠ µm

µm+1

···

µ2m

Such measures arise from two variable measures [GW]. Problem 4.1. How is the Hankel structure manifest in the recurrence coeﬃcients An and Bn ? References [GW] J.S. Geronimo, H. Woerdeman: Two variable orthogonal polynomials on the bicircle and structured matrices, SIAM J. Matrix Anal. Appl. 29, no. 3 (2007), 796–825.

5. An open problem posed by E.A. Rakhmanov and A. Mart´ınez-Finkelshtein 5.1. Critical and reﬂectionless measures. In various applications of potential theory in approximation, the notion of (positive) critical measure plays an important role. It can be introduced in diﬀerent ways. Probably the most general one is the following (we restrict ourselves here to the case without external ﬁeld). For a measure µ supported on C we can deﬁne its energy by 1 E(µ) = log dµ(x) dµ(y) . |x − y| Given a compact set A of zero capacity (i.e., a polar set), any function h ∈ C 1 (C)

such that h A ≡ 0 deﬁnes a local variation of the plane by z → z t = z + t h(z), t ∈ C. This transformation induces also a variation of sets e → et = {z t : z ∈ e}, and measures: µ → µt , deﬁned by µt (et ) = µ(e). In diﬀerential form, the pullback measure µt can be written as dµt (xt ) = dµ(x). A measure µ on C is A-critical if for any h as above d E(µt ) = 0 . dt If A is a ﬁnite set, then it is known that any A-critical measure µ is supported on a ﬁnite union of analytic arcs, Γ = Γ1 ∪ · · · ∪ Γk , and the corresponding logarithmic potential 1 V µ (z) = log dµ(x) |x − z| is constant on each component Γj of Γ. Furthermore, ∂V µ (z) ∂V µ (z) = , ∂n+ ∂n−

z ∈ Γ◦ ,

where n± are normals to Γ pointing in opposite directions. Another feature of such a µ is that its Cauchy transform 1 µ C (z) = dµ(t) t−z Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

ORTHOGONAL POLYNOMIALS AND APPROXIMATION THEORY: SOME OPEN PROBLEMS 11 297 ORTHOGONAL POLYNOMIALS AND APPROXIMATION THEORY

satisﬁes for z ∈ C \ Γ: 2 (C µ (z)) =

C(z) , A(z)

A(z) =

(z − a),

a∈A

and deg C = deg A − 2. On the other hand, reﬂectionless measures, that appear in spectral theory and inverse scattering, can be deﬁned in terms of the Cauchy transform. A measure µ on C is reﬂectionless if C µ (z) = 0 µ-a.e. on C, where the integral is understood in terms of principal value. The properties mentioned above show that any critical measure is reﬂectionless. It is also immediate to show that there are reﬂectionless measures with inﬁnite energy. So, it is natural to pose the following question [MFR, Conjecture 5.2]: Problem 5.1. Is it true that any reﬂectionless measure with a ﬁnite energy is A-critical, for a certain polar set A on C? A weaker statement is the following. Assume that µ is a reﬂectionless measure supported on a ﬁnite set of analytic arcs Γ = Γ1 ∪ · · · ∪ Γk of C. Assume that µ is absolutely continuous with respect to the arclength of these arcs. Is it true that this measure is A-critical, for a suitable ﬁnite set A? References [MFR] A. Mart´ınez-Finkelshtein, E. A. Rakhmanov: Critical measures, quadratic diﬀerentials, and weak limits of zeros of Stieltjes polynomials, arXiv:0902.0193

6. Guillermo’s open problem 6.1. Nikishin systems. Suppose (f1 , f2 , . . . , fr ) is a system of Markov functions, i.e., dσk,r (x) , fk (z) = ∆r z − x where ∆r is an interval of the real line. For Hermite-Pad´e approximation one wants to approximate these r functions simultaneously by rational functions with interpolation conditions at inﬁnity. Let n = (n1 , n2 , . . . , nr ) ∈ Nr be a multiindex. There are two types of Hermite-Pad´e approximation: for type I HermitePad´e approximation one wants to ﬁnd polynomials An,k (k = 1, . . . , r) with degree An,k ≤ nk − 1, and a polynomial Bn such that r

An,k (z)fk (z) − Bn (z) = O(1/z |n| ),

z → ∞,

k=1

where | n| = n1 + n2 + · · · + nr . For type II Hermite-Pad´e approximation one wants to ﬁnd rational approximants for each fk , but with a common denominator, i.e., a polynomial Pn of degree ≤ | n| and polynomials Qn,k such that Pn (z)fk (z) − Qn,k (z) = O(1/z nk +1 ),

z → ∞,

for k = 1, 2, . . . , r. We say that a multi-index n is weakly normal is Pn is determined uniquely. A multi-index n is normal if any non-trivial solution Pn has degree | n|. If Pn has exactly | n| simple zeros in ∆r , then the index is strongly normal. When all the multi-indices are weakly normal, normal or strongly normal, then the system (f1 , . . . , fr ) is said to be weakly perfect, perfect or strongly perfect. Licensed to National Taiwan University. Prepared on Sat Jun 3 01:22:02 EDT 2023for download from IP 140.112.243.46.

12 298

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Nikishin introduced an interesting system of functions in [Nik]. These Nikishin systems are deﬁned by induction. A system of order r = 2 is a Nikishin system if f1 and f2 are Markov functions with measures σ1,2 and σ2,2 supported on an interval ∆2 , and if σ2,2 is absolutely continuous with respect to σ1,2 with dσ1,1 (t) dσ2,2 , (x) = dσ1,2 ∆1 x − t where σ1,1 is a measure of constant sign on an interval ∆1 , and ∆1 ∩ ∆2 = ∅. In general, the system (f1 , . . . , fr ) is a Nikishin system of order r when all the measures σk,r (k = 1, . . . , r) are supported on an interval ∆r , and each σk,r is absolutely continuous with respect to σ1,r , with dσk−1,r−1 (t) dσk,r , k = 2, . . . , r, (x) = dσ1,r x−t ∆r−1 where the measures (σ1,r−1 , . . . , σr−1,r−1 ) are such that their Markov functions are a Nikishin system of order r − 1 on an interval ∆r−1 , with ∆r−1 ∩ ∆r = ∅. For Nikishin systems all multi-indices with the property that nk ≤ nj + 1 whenever 1 ≤ j < k ≤ r are known to be strongly normal (Driver and Stahl [DS]). Guillermo L´opez Lagomasino and Ulises Fidalgo Prieto [FPLL] showed that any Nikishin system of order r = 3 is strongly perfect. Problem 6.1. Which multi-indices are normal for a Nikishin system of order r > 3? In an e-mail (dated January 9, 2009), Guillermo L´ opez Lagomasino announced that he and Ulises Fidalgo Prieto deﬁnitely proved that Nikishin systems are perfect. Naturally, if their proof is indeed valid, then this problem is no longer open. References K. Driver, H. Stahl: Normality in Nikishin systems, Indag. Math. 5 (1994), no. 2, 161– 187. [FPLL] U. Fidalgo Prieto, G. L´ opez Lagomasino: On perfect Nikishin systems, Computational Methods and Functions Theory 2 (2002), no. 2, 415–426. [Nik] E.M. Nikishin: On simultaneous Pad´ e approximants, Mat. Sb., N. Ser. 113(155), 499–519 (in Russian); Math. USSR Sb. 41 (1982), 409–425. [DS]

Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B box 2400, Belgium E-mail address: [email protected]

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Titles in This Series 509 Primitivo B. Acosta-Hum´ anez and Francisco Marcell´ an, Editors, Diﬀerential algebra, complex analysis and orthogonal polynomials, 2010 508 Martin Berz and Khodr Shamseddine, Editors, Advances in p-Adic and non-archimedean analysis, 2010 507 Jorge Arves´ u, Francisco Marcell´ an, and Andrei Mart´ınez-Finkelshtein, Editors, Recent trends in orthogonal polynomials and approximation theory, 2010 506 Yun Gao, Naihuan Jing, Michael Lau, and Kailash C. Misra, Editors, Quantum aﬃne algebras, extended aﬃne Lie algebras, and their applications, 2010 505 Patricio Cifuentes, Jos´ e Garc´ıa-Cuerva, Gustavo Garrig´ os, Eugenio Hern´ andez, Jos´ e Mar´ıa Martell, Javier Parcet, Alberto Ruiz, Fern´ ando Soria, Jos´ e Luis Torrea, and Ana Vargas, Editors, Harmonic analysis and partial diﬀerential equations, 2010 504 Christian Ausoni, Kathryn Hess, and J´ erˆ ome Scherer, Editors, Alpine perspectives on algebraic topology, 2009 503 Marcel de Jeu, Sergei Silvestrov, Christian Skau, and Jun Tomiyama, Editors, Operator structures and dynamical systems, 2009 502 Viviana Ene and Ezra Miller, Editors, Combinatorial Aspects of Commutative Algebra, 2009 501 Karel Dekimpe, Paul Igodt, and Alain Valette, Editors, Discrete groups and geometric structures, 2009 500 Philippe Briet, Fran¸ cois Germinet, and Georgi Raikov, Editors, Spectral and scattering theory for quantum magnetic systems, 2009 499 Antonio Giambruno, C´ esar Polcino Milies, and Sudarshan K. Sehgal, Editors, Groups, rings and group rings, 2009 498 Nicolau C. Saldanha, Lawrence Conlon, R´ emi Langevin, Takashi Tsuboi, and Pawel Walczak, Editors, Foliations, geometry and topology, 2009 497 Maarten Bergvelt, Gaywalee Yamskulna, and Wenhua Zhao, Editors, Vertex operator algebras and related areas, 2009 496 Daniel J. Bates, GianMario Besana, Sandra Di Rocco, and Charles W. Wampler, Editors, Interactions of classical and numerical algebraic geometry, 2009 495 G. L. Litvinov and S. N. Sergeev, Editors, Tropical and idempotent mathematics, 2009 494 Habib Ammari and Hyeonbae Kang, Editors, Imaging microstructures: Mathematical and computational challenges, 2009 493 Ricardo Baeza, Wai Kiu Chan, Detlev W. Hoﬀmann, and Rainer Schulze-Pillot, Editors, Quadratic Forms—Algebra, Arithmetic, and Geometry, 2009 492 Fernando Gir´ aldez and Miguel A. Herrero, Editors, Mathematics, Developmental Biology and Tumour Growth, 2009 491 Carolyn S. Gordon, Juan Tirao, Jorge A. Vargas, and Joseph A. Wolf, Editors, New developments in Lie theory and geometry, 2009 490 Donald Babbitt, Vyjayanthi Chari, and Rita Fioresi, Editors, Symmetry in mathematics and physics, 2009 489 David Ginzburg, Erez Lapid, and David Soudry, Editors, Automorphic Forms and L-functions II. Local aspects, 2009 488 David Ginzburg, Erez Lapid, and David Soudry, Editors, Automorphic forms and L-functions I. Global aspects, 2009 487 Gilles Lachaud, Christophe Ritzenthaler, and Michael A. Tsfasman, Editors, Arithmetic, geometry, cryptography and coding theory, 2009 486 Fr´ ed´ eric Mynard and Elliott Pearl, Editors, Beyond topology, 2009 485 Idris Assani, Editor, Ergodic theory, 2009 484 Motoko Kotani, Hisashi Naito, and Tatsuya Tate, Editors, Spectral analysis in geometry and number theory, 2009

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TITLES IN THIS SERIES

483 Vyacheslav Futorny, Victor Kac, Iryna Kashuba, and Eﬁm Zelmanov, Editors, Algebras, representations and applications, 2009 482 Kazem Mahdavi and Deborah Koslover, Editors, Advances in quantum computation, 2009 481 Aydın Aytuna, Reinhold Meise, Tosun Terzio˘ glu, and Dietmar Vogt, Editors, Functional analysis and complex analysis, 2009 480 Nguyen Viet Dung, Franco Guerriero, Lakhdar Hammoudi, and Pramod Kanwar, Editors, Rings, modules and representations, 2008 479 Timothy Y. Chow and Daniel C. Isaksen, Editors, Communicating mathematics, 2008 478 Zongzhu Lin and Jianpan Wang, Editors, Representation theory, 2008 477 Ignacio Luengo, Editor, Recent Trends in Cryptography, 2008 476 Carlos Villegas-Blas, Editor, Fourth summer school in analysis and mathematical physics: Topics in spectral theory and quantum mechanics, 2008 475 Jean-Paul Brasselet, Jos´ e Luis Cisneros-Molina, David Massey, Jos´ e Seade, and Bernard Teissier, Editors, Singularities II: Geometric and topological aspects, 2008 474 Jean-Paul Brasselet, Jos´ e Luis Cisneros-Molina, David Massey, Jos´ e Seade, and Bernard Teissier, Editors, Singularities I: Algebraic and analytic aspects, 2008 473 Alberto Farina and Jean-Claude Saut, Editors, Stationary and time dependent Gross-Pitaevskii equations, 2008 472 James Arthur, Wilfried Schmid, and Peter E. Trapa, Editors, Representation Theory of Real Reductive Lie Groups, 2008 471 Diego Dominici and Robert S. Maier, Editors, Special functions and orthogonal polynomials, 2008 470 Luise-Charlotte Kappe, Arturo Magidin, and Robert Fitzgerald Morse, Editors, Computational group theory and the theory of groups, 2008 469 Keith Burns, Dmitry Dolgopyat, and Yakov Pesin, Editors, Geometric and probabilistic structures in dynamics, 2008 468 Bruce Gilligan and Guy J. Roos, Editors, Symmetries in complex analysis, 2008 467 Alfred G. No¨ el, Donald R. King, Gaston M. N’Gu´ er´ ekata, and Edray H. Goins, Editors, Council for African American researchers in the mathematical sciences: Volume V, 2008 466 Boo Cheong Khoo, Zhilin Li, and Ping Lin, Editors, Moving interface problems and applications in ﬂuid dynamics, 2008 465 Valery Alexeev, Arnaud Beauville, C. Herbert Clemens, and Elham Izadi, Editors, Curves and Abelian varieties, 2008 ´ 464 Gestur Olafsson, Eric L. Grinberg, David Larson, Palle E. T. Jorgensen, Peter R. Massopust, Eric Todd Quinto, and Boris Rubin, Editors, Radon transforms, geometry, and wavelets, 2008 463 Kristin E. Lauter and Kenneth A. Ribet, Editors, Computational arithmetic geometry, 2008 462 Giuseppe Dito, Hugo Garc´ıa-Compe´ an, Ernesto Lupercio, and Francisco J. Turrubiates, Editors, Non-commutative geometry in mathematics and physics, 2008 461 Gary L. Mullen, Daniel Panario, and Igor Shparlinski, Editors, Finite ﬁelds and applications, 2008 460 Megumi Harada, Yael Karshon, Mikiya Masuda, and Taras Panov, Editors, Toric topology, 2008 459 Marcelo J. Saia and Jos´ e Seade, Editors, Real and complex singularities, 2008

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This volume contains invited lectures and selected contributions from the International Workshop on Orthogonal Polynomials and Approximation Theory, held at Universidad Carlos III de Madrid on September 8–12, 2008, and which honored Guillermo López Lagomasino on his 60th birthday. This book presents the state of the art in the theory of Orthogonal Polynomials and Rational Approximation with a special emphasis on their applications in random matrices, integrable systems, and numerical quadrature. New results and methods are presented in the papers as well as a careful choice of open problems, which can foster interest in research in these mathematical areas. This volume also includes a brief account of the scientific contributions by Guillermo López Lagomasino.

CONM/507

AMS on the Web www.ams.org

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