Extended Abstracts Fall 2019: Spaces of Analytic Functions: Approximation, Interpolation, Sampling (Research Perspectives CRM Barcelona) 3030744167, 9783030744168

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Trends in Mathematics Research Perspectives CRM Barcelona Vol. 12

Evgeny Abakumov, Anton Baranov, Alexander Borichev, Konstantin Fedorovskiy, Joaquim Ortega-Cerdà, Editors

Extended Abstracts Fall 2019 Spaces of Analytic Functions: Approximation, Interpolation, Sampling

Trends in Mathematics

Research Perspectives CRM Barcelona Volume 12

Managing Editor David Romero i Sànchez, Centre de Rercerca Matemàtica, Barcelona, Spain

Since 1984 the Centre de Recerca Matemàtica (CRM) has been organizing scientific events such as conferences or workshops which span a wide range of cutting-edge topics in mathematics and present outstanding new results. In the fall of 2012, the CRM decided to publish extended conference abstracts originating from scientific events hosted at the center. The aim of this initiative is to quickly communicate new achievements, contribute to a fluent update of the state of the art, and enhance the scientific benefit of the CRM meetings. The extended abstracts are published in the subseries Research Perspectives CRM Barcelona within the Trends in Mathematics series. Volumes in the subseries will include a collection of revised written versions of the communications, grouped by events. Contributing authors to this extended abstracts series remain free to use their own material as in these publications for other purposes (for example a revised and enlarged paper) without prior consent from the publisher, provided it is not identical in form and content with the original publication and provided the original source is appropriately credited.

More information about this subseries at https://link.springer.com/bookseries/13332

Evgeny Abakumov · Anton Baranov · Alexander Borichev · Konstantin Fedorovskiy · Joaquim Ortega-Cerdà Editors

Extended Abstracts Fall 2019 Spaces of Analytic Functions: Approximation, Interpolation, Sampling

Editors Evgeny Abakumov Laboratoire d’Analyse et de Mathématiques Appliquées Université Gustave Eiffel Marne-la-Vallée, France Alexander Borichev Institut de Mathématiques de Marseille Aix Marseille Université Marseille, France

Anton Baranov Department of Mathematics and Mechanics St. Petersburg State University St. Petersburg, Russia Konstantin Fedorovskiy Faculty of Mechanics and Mathematics Lomonosov Moscow State University Moscow, Russia

Joaquim Ortega-Cerdà Departament de Matemàtiques i Informàtica Universitat de Barcelona Barcelona, Spain

ISSN 2297-0215 ISSN 2297-024X (electronic) Trends in Mathematics ISSN 2509-7407 ISSN 2509-7415 (electronic) Research Perspectives CRM Barcelona ISBN 978-3-030-74416-8 ISBN 978-3-030-74417-5 (eBook) https://doi.org/10.1007/978-3-030-74417-5 Mathematics Subject Classification: 30-06, 41-06, 42-06, 46-06 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

The Intensive Research Program “Spaces of Analytic Functions: Approximation, Interpolation, Sampling” (SAFAIS 2019 program) was held at the Centre de Recerca Matemàtica (Barcelona) in October–December, 2019. This program consisted of several activities. Among them, there was the conference (with the same title), the Workshop “Emergent Trends in Complex Function Theory”, and three Advanced Courses weeks. A research seminar was functioning throughout the whole duration of the program. The present volume contains the extended abstracts of main talks and lectures delivered during the SAFAIS 2019 program. Marne-la-Vallée, France St. Petersburg, Russia Marseille, France Moscow, Russia Barcelona, Spain

Evgeny Abakumov Anton Baranov Alexander Borichev Konstantin Fedorovskiy Joaquim Ortega-Cerdà

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Contents

Intensive Research Program ‘Spaces of Analytic Functions: Approximation, Interpolation, Sampling’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evgeny Abakumov, Anton Baranov, Alexander Borichev, Konstantin Fedorovskiy, and Joaquim Ortega-Cerdà Comparison of Clark Measures in Several Complex Variables . . . . . . . . . Aleksei B. Aleksandrov and Evgueni Doubtsov

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On Spectrum of a Class of Jacobi Matrices on Graph-Trees and Multiple Orthogonal Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alexander I. Aptekarev, Sergey A. Denisov, and Maxim L. Yattselev

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Geometric Properties of Reproducing Kernels in Hilbert Spaces of Entire Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yurii Belov

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A New Life of the Classical Szeg˝o Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . Roman Bessonov and Sergey Denisov

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De Branges Canonical Systems with Finite Logarithmic Integral . . . . . . . Roman V. Bessonov and Sergey A. Denisov

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Rate of Convergence of Critical Interfaces to SLE Curves . . . . . . . . . . . . . Ilia Binder

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Toeplitz and Hankel Operators on Bergman Spaces . . . . . . . . . . . . . . . . . . . M. Bourass, O. El-Fallah, I. Marrhich, and H. Naqos

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Bounds for Zeta and Primes via Fourier Analysis . . . . . . . . . . . . . . . . . . . . . Emanuel Carneiro

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On Zeros of Solutions of a Linear Differential Equation . . . . . . . . . . . . . . . Igor Chyzhykov and Jianren Long

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Contents

On Riesz Bases of Exponentials for Convex Polytopes with Symmetric Faces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alberto Debernardi and Nir Lev

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Remez-Type Inequalities and Their Applications . . . . . . . . . . . . . . . . . . . . . Omer Friedland

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Shift-Invariant Spaces of Entire Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . Karlheinz Gröchenig

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Describing Blaschke Products by Their Critical Points . . . . . . . . . . . . . . . . Oleg Ivrii

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Two Problems on Homogenization in Geometry . . . . . . . . . . . . . . . . . . . . . . Oleg Ivrii and Vladimir Markovi´c

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Toeplitz Operators Between Distinct Abstract Hardy Spaces . . . . . . . . . . . 105 Alexei Karlovich Polynomial Hermite Padé m-System and Reconstruction of the Values of Algebraic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Aleksandr Komlov Quantitative Szeg˝o Minimum Problem for Some non-Szeg˝o Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Anna Kononova Hausdorff Dimension Exceptional Set Estimates for Projections, Sections and Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Pertti Mattila Generic Boundary Behaviour of Taylor Series in Banach Spaces of Holomorphic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Jürgen Müller Szegö-Type ASD for “Multiplicative Toeplitz” Operators . . . . . . . . . . . . . . 145 Nikolai Nikolski Around Uncertainty Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 A. M. Olevskii Inner Functions, Completeness and Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Alexei Poltoratski Schmidt Subspaces of Hankel Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Patrick Gérard and Alexander Pushnitski Maximum Principle and Comparison of Singular Numbers for Composition Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 Hervé Queffélec

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Canonical Systems in Classes of Compact Operators . . . . . . . . . . . . . . . . . . 207 Roman Romanov and Harald Woracek S-Contours and Convergent Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Maxim L. Yattselev Special Conformal Mappings and Extremal Problems . . . . . . . . . . . . . . . . . 219 P. Yuditskii

Intensive Research Program ‘Spaces of Analytic Functions: Approximation, Interpolation, Sampling’ Evgeny Abakumov, Anton Baranov, Alexander Borichev, Konstantin Fedorovskiy, and Joaquim Ortega-Cerdà

Abstract We describe the Intensive Research Program “Spaces of Analytic Functions: Approximation, Interpolation, Sampling” that was held at the Centre de Recerca Matemàtica (Barcelona) in October–December, 2019.

1 Background, Objectives and Perspectives of the Program The main idea of the intensive research program “Spaces of analytic functions: approximation, interpolation, sampling” was to bring together leading experts in topical domains in Complex Analysis, Functional Analysis, and Approximation Theory working on approximation, interpolation and sampling problems in spaces of analytic functions and on their applications to spectral theory, Gabor analysis and random analytic functions. This field is an example of a fruitful interplay of complex analysis, abstract functional analysis and operator theory. During the last three decades it has grown extensively and has seen a tremendous number of important advances and breakthroughs. E. Abakumov LAMA, Univ Gustave Eiffel, Univ Paris Est Creteil, CNRS, 77454 Marne-la-Vallée, France e-mail: [email protected] A. Baranov · K. Fedorovskiy St. Petersburg State University, St. Petersburg, Russia e-mail: [email protected] A. Borichev Aix–Marseille University, CNRS, Centrale Marseille, I2M Marseille, France e-mail: [email protected] K. Fedorovskiy (B) Lomonosov Moscow State University, GSP 1, Moscow 119991, Russia e-mail: [email protected] J. Ortega-Cerdà Universitat de Barcelona, Barcelona, Spain e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. Abakumov et al. (eds.), Extended Abstracts Fall 2019, Trends in Mathematics 12, https://doi.org/10.1007/978-3-030-74417-5_1

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The spaces of analytic functions in question include weighted Bergman and Dirichlet spaces, spaces of entire functions (Paley–Wiener and general de Branges spaces, Fock-type spaces) and many other examples. Among the main topics of the program one ought to emphasize questions about completeness and approximation properties of various systems of functions (such as systems of polynomials, rational functions, analytic functions with prescribed singularities, systems of reproducing kernels etc.) in spaces of continuous, smooth and integrable functions on closed subsets of the complex plane, in the Hardy and the Bergman spaces, in model subspaces of Hardy spaces, in de Branges and Fock-type spaces. During several recent decades the problems of approximation of functions on compact subsets of Euclidean spaces by solutions of homogeneous elliptic partial differential equations (in particular, by polynomial, rational, entire and meromorphic solutions) in norms of various spaces of analytic functions are intensively studied by analysts in many countries (such as Canada, Ireland, Russia, Spain, USA). The main results in this area are related to problems of uniform and smooth approximation by holomorphic, harmonic, and polyanalytic functions and polynomials, as well as closely related problems concerning the properties of corresponding capacities. One can mention here important works by A. O’Farrell, S. Gardiner (Ireland), J. Carmona, M. Melnikov, X. Tolsa, J. Verdera (Spain), A. Boivin, P. Gauthier (Canada), K. Fedorovskiy, M. Mazalov, P. Paramonov (Russia). Interpolation and sampling are among the basic problems in the theory of spaces of analytic functions which have many important applications (e.g., in signal processing, time-frequency analysis, control theory). Their study started with the classical results of L. Carleson, D. Newman, H. Shapiro, A. Shields about interpolation in the Hardy space and those of A. Beurling about sampling and interpolation in spaces of entire functions of exponential type. The work of B. Pavlov, S. Khrushchev and N. Nikolski culminated in the description of complete interpolating sequences for the Paley–Wiener space (and in some model spaces). At the beginning of the 1990s breakthrough results about sampling and interpolation in the Bargmann–Fock space were obtained by K. Seip and R. Wallstén. A (definitely incomplete) list of more recent advances includes: the theory of weighted Paley–Wiener spaces by Yu. Lyubarskii and K. Seip, the description of Fourier frames obtained by J. Ortega-Cerdà and K. Seip, the Toeplitz operator approach to completeness problems by N. Makarov and A. Poltoratski, and a construction of Riesz bases of exponentials on a union of intervals by G. Kozma and S. Nitzan. Other important developments in the area are due to A. Olevskii and N. Lev (Israel), A. Ulanovskii, E. Malinnikova (Norway/USA), A. Nicolau (Spain), A. Borichev, P. Thomas (France), A. Baranov, Yu. Belov (Russia). One of the main goals of the program was to strengthen the contacts between researchers working in two vast areas of complex analysis—theory of approximation of functions by solutions of elliptic equations, rational approximation, singular integrals and capacities on one side and theory of spaces of analytic functions and their operators on the other side. The Intensive Research Program “Spaces of Analytic Functions: Approximation, Interpolation, Sampling” consisted of several activities. These activities included

Intensive Research Program …

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the Conference (with the same title), the Workshop “Emergent Trends in Complex Function Theory” and three Advanced Courses weeks. A research seminar was functioning throughout the whole duration of the program. The Intensive Research Program “Spaces of Analytic Functions: Approximation, Interpolation, Sampling” was supported by the LabEx Bezout, the Norwegian Research Council, in the framework of the Project COMAN (298772), the Norwegian University of Science and Technology, the Centre International de Mathematiques Pures et Appliquees (CIMPA) and the Université Paris-Est Marne-la-Vallée. The Workshop “Emergent Trends in Complex Function Theory” was supported by the Clay Mathematical Institute. We are grateful to these institutions and foundations for their support, which has significantly contributed to the success of the program.

2 Conference The conference “Spaces of Analytic Functions: Approximation, Interpolation, Sampling” took place from November 25 to November 29, 2019. This was the main event of the program which brought together about 60 specialists in complex and harmonic analysis from many countries. The conference was focused on topical problems in operator related function theory, reproducing kernel Hilbert spaces, sampling and interpolation problems, approximation theory, and their various applications. The list of speakers included Alexandru Aleman (Lund University), Alexander Aptekarev (Keldysh Institute of Applied Mathematics), Nicola Arcozzi (Università di Bologna, Frédéric Bayart (Université Clermont Auvergne), Yurii Belov (Saint Petersburg State University), Roman Bessonov (Saint Petersburg State University), Andrii Bondarenko (Norwegian University of Science and Technology), Alexander Bufetov (Aix-Marseille Université), Igor Chyzhykov (University of Warmia and Mazury in Olsztyn), Evgeny Dubtsov (Steklov Mathematical Institute, St. Petersburg), Omar El-Fallah (Mohammed V University of Rabat), Eva Gallardo-Gutiérrez (Universidad Complutense Madrid), Karlheinz Gröchenig (Universität Wien), Oleg Ivrii (Tel Aviv University), Alexei Karlovich (Universidade NOVA de Lisboa), Dmitry Khavinson (University of South Florida), Alexander Komlov (Steklov Mathematical Institute, Moscow), Anna Kononova (St. Petersburg State University), Gady Kozma (Weizmann Institute of Science), Arno Kuijlaars (Katholieke Universiteit Leuven), Nir Lev (Bar-Ilan University), Jürgen Müller (Universität Trier), Artur Nicolau (Universitat Autònoma de Barcelona), Shahaf Nitzan (Georgia Institute of Technology), Hervé Queffélec (Université de Lille), Kristian Seip (Norwegian University of Science and Technology), Xavier Tolsa (ICREA and Universitat Autònoma de Barcelona), Joan Verdera (Universitat Autònoma de Barcelona), Maxim Yattselev (Indiana University–Purdue University Indianapolis), Peter Yuditskii (Johannes Kepler University, Linz). Many conference talks dealt with Banach spaces of analytic functions and operators acting in such spaces. In the talk of A. Aleman computation of the spectrum of the Hilbert matrix on Banach spaces of analytic functions was discussed. N. Arcozzi

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gave a survey of some recent results on analysis of the holomorphic Dirichlet space on the bidisc and on the related bi-parameter potential theory. The talk of F. Bayart was devoted to interpolation in Hardy spaces of Dirichlet series—an object of high current interest due to its relation with number theory. Volterra operators on Hardy spaces of Dirichlet series were considered in the talk of K. Seip. New estimates for the Riemann zeta function on the critical line were presented by A. Bondarenko. In the talks of O. El-Fallah and H. Queffélec singular numbers of operators in Banach spaces of analytic functions were considered. Toeplitz operators between distinct abstract Hardy spaces were discussed in the talk by A. Karlovich, while E. GallardoGutiérrez spoke about Bishop operators and their generalizations. Yu. Belov presented a solution of the spectral synthesis problem in the Bargmann– Segal–Fock space posed by D. Newman and H. Shapiro in the 1960s. In the talk of J. Müller generic boundary behavior of Taylor series in Banach spaces of holomorphic functions was discussed. O. Ivrii considered the problem of reconstructing a Blaschke product from the set of its critical points. I. Chyzhykov discussed the zeros of solutions of certain second order linear differential equations with analytic coefficients. E. Dubtsov spoke about a generalization of Clark measures to the several variables setting. X. Tolsa spoke about the recent solution of the so called ε2 -conjecture of Carleson and the characterization of the tangent points of a Jordan curve in the plane in terms of the finiteness of the associated Carleson’s ε2 -function. In the talk of J. Verdera differentiability of potentials of finite measures was studied. The talk of A. Nicolau was devoted to applications of dyadic martingale technique to the study of the behavior of the divided differences of functions in the Hölder class. N. Lev presented a recent result that for any convex polytope  in Rd which is centrally symmetric and whose faces of all dimensions are also centrally symmetric, the space L 2 () has a Riesz basis of exponential functions. K. Gröchenig considered in his talk interpolation and sampling problems in the shift-invariant space generated by the Gaussian. Several talks revealed close relations between modern complex and harmonic analysis and probability. G. Kozma gave a talk about irreducibility of random polynomials. S. Nitzan presented a series of results on trigonometric polynomials and Gaussian stationary processes, while the talk by A. Bufetov was devoted to relations between determinantal point processes and reproducing kernel Hilbert spaces. One more area covered in the conference was the theory of orthogonal polynomials and its applications. A. Kuijlaars spoke about matrix valued orthogonality in a random tiling problem. In the talk of R. Bessonov zero sets, entropy, and pointwise asymptotics of orthogonal polynomials were discussed. A. Kononova presented results about asymptotics of the norms of orthogonal polynomials for a class of measures on the unit circle which do not satisfy the Szegö condition (i.e., with divergent logarithmic integral). A. Aptekarev spoke about spectra of selfadjoint discrete Schrödinger operators defined on graph-trees whose potential is defined in terms of multiple orthogonal polynomials. Various aspects of Padé-type rational approximation were presented in the talks by A. Komlov and M. Yattselev. The extremal functions in Chebyshev’s problems and their description in terms of some special

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conformal mappings were discussed in the talk by P. Yuditskii. D. Khavinson discussed in his talk several problems about approximation by analytic functions in L 1 -norms.

3 Workshop The workshop “Spaces of Analytic Functions: Approximation, Interpolation, Sampling” took place from October 28 to October 31. The goal of this research workshop was to explore new advances in complex analysis and investigate further the interplay between complex function theory and harmonic analysis, operator theory, mathematical physics, to promote the exchange of new ideas and recent trends in the subject and to formulate some new and ambitious problems. Special emphasis was put on attracting young researchers and Ph.D. students. The workshop was supported by the Clay Mathematics Institute. The workshop included three short courses: • Nikolai Makarov (California Institute of Technology), Études for the Inverse Spectral Problem • Stefanie Petermichl (University of Würzburg), A Probabilistic View on Calderón– Zygmund Operators • Alexei Poltoratski (Texas A&M University), Inner Functions, Spectra and Scattering In addition to the short courses there were five plenary lectures by Ilia Binder (University of Toronto), Sergey Denisov (University of Wisconsin–Madison), Håkan Hedenmalm (KTH—Royal Institute of Technology), Pertti Mattila (University of Helsinki), Sergei Treil (Brown University), as well as 18 short talks mainly by young researchers.

4 Advanced Courses and Research Seminar Three advanced lecture courses were among the main components of the program. They took place from October 21 to 25 (Advanced Course 1), November 4 to 8 (Advanced Course 2) and December 2 to 5 (Advanced Course 3). Below we give a short description of the lecture courses. Note that detailed lecture notes of the courses by Yu. Belov, N. Nikolski and A. Pushnitski are included in the present volume. The Advanced Course 1 included two short courses: • Alexander Olevskii (Tel Aviv University), Around Uncertainty Principle • Alexander Pushnitski (King’s College, London), Schmidt Subspaces of Hankel Operators

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The Uncertainty Principle plays a fundamental role in quantum mechanics. In mathematical language it means that a function and its Fourier transform cannot be both sharply concentrated. The course by Prof. Olevskii discussed some specific forms of this fact and presented a new construction, joint with F. Nazarov, of functions having both “small” support and spectrum. Spectral theory of compact Hankel operator acting on the Hardy space in the unit disc was the subject of the course of A. Pushnitski. One of the main results of the course was that, for a compact Hankel operator , its Schmidt subspaces (i.e., the eigenspaces of  ∗ ) were described as the images of model spaces under the action of isometric multipliers (the so-called nearly invariant subspaces). The Advanced Course 2 consisted of two short courses: • Omer Friedland (Sorbonne University, Paris), Remez-type Inequalities and Their Applications • Kristian Seip (Norwegian University of Science and Technology), Value Distribution of Dirichlet Series The aim of the course of O. Friedland was to give a survey of the classical Remez inequality and of its numerous extensions including the Turán–Nazarov inequality and also his recent results obtained with E. Abakumov and Y. Yomdin. Applications to the uncertainty principle (Logvinenko–Sereda type theorems) were also discussed. The course given by K. Seip dealt with the function theory problems associated to the Riemann zeta function and related questions on the zeros, value distribution and universality. Some important aspects of analytic number theory, such as for example the dual relation between the distribution of the nontrivial zeros of the zeta function and the prime numbers were also considered. The Advanced Course 3 consisted of three short courses: • Yurii Belov (St. Petersburg State University), Geometric Properties of Reproducing Kernel Systems • Alexander Bufetov (Aix-Marseille Université and Steklov Mathematical Institute, Moscow), Determinantal point processes and classical analysis • Nikolai Nikolski (University of Bordeaux), Spectral asymptotics of Toeplitz matrices The spectral synthesis problem is concerned with the possibility of reconstruction of a vector by means of certain “spectrum-preserving” approximations. The course of Yu. Belov discussed the recent solutions of the long standing spectral synthesis problem for exponential systems on an interval, as well as of more general problems for systems of reproducing kernels in Hilbert spaces of entire functions including the Paley–Wiener, de Branges and Fock spaces. The course by A. Bufetov was devoted to the intriguing relations between complex analysis and probability, namely between determinantal point processes, random analytic functions, and interpolation on their zero sets. The course given by N. Nikolski dealt with the asymptotic spectral distributions for large Toeplitz-like matrices when their size goes to infinity. The multiplicative

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Toeplitz matrices were also considered, and the links to the dilation completeness problem were discussed. The lecture notes of this short course are included in the present volume. During the weeks free from other program activities a research seminar was running. Among the speakers of the seminar were Emanuel Carneiro (ICTP Trieste), Roman Romanov (St. Petersburg State University), and Konstantin Dyakonov (ICREA and Universitat de Barcelona).

Comparison of Clark Measures in Several Complex Variables Aleksei B. Aleksandrov and Evgueni Doubtsov

Abstract Let D denote the unit disc of C and let  denote the unit ball Bn of Cn or the unit polydisc Dn , n ≥ 2. Given a non-constant holomorphic function b :  → D, we study the corresponding family σα [b], α ∈ ∂D, of Clark measures on ∂. For  = Bn and an inner function I : Bn → D, we show that the property σ1 [I ]  σ1 [b] is directly related to the membership of an appropriate function in the de Branges– Rovnyak space H(b).

1 Introduction Let D denote the unit disc of C and T = ∂D. Let  denote the unit ball Bn of Cn or the unit polydisc Dn , n ≥ 2, and let ∂ denote the unit sphere Sn = ∂ Bn or the unit torus Tn , respectively. Let C(z, ζ) = C (z, ζ) denote the Cauchy kernel for . Recall that 1 , z ∈ Bn , ζ ∈ Sn , (1 − z, ζ)n n  1 CDn (z, ζ) = , z ∈ Dn , ζ ∈ Tn . 1 − z ζ j j j=1 C Bn (z, ζ) =

The corresponding Poisson kernel is given by the formula P(z, ζ) =

C(z, ζ)C(ζ, z) , z ∈ , ζ ∈ ∂. C(z, z)

A. B. Aleksandrov · E. Doubtsov (B) St. Petersburg Department of Steklov Mathematical Institute, Fontanka 27, St. Petersburg 191023, Russia e-mail: [email protected] A. B. Aleksandrov e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. Abakumov et al. (eds.), Extended Abstracts Fall 2019, Trends in Mathematics 12, https://doi.org/10.1007/978-3-030-74417-5_2

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Let M(∂) denote the space of complex Borel measures on ∂. For μ ∈ M(∂), the Cauchy transform μ+ is defined as  μ+ (z) =

Tn

C(z, ξ) dμ(ξ), z ∈ .

1.1 Clark Measures Given an α ∈ T and a holomorphic function b :  → D, the quotient 1 − |b(z)|2 = Re |α − b(z)|2



 α + b(z) , z ∈ , α − b(z)

is positive and pluriharmonic. Therefore, there exists a unique positive measure σα = σα [b] ∈ M(∂) such that  P[σα ](z) = Re

 α + b(z) , z ∈ . α − b(z)

After the seminal paper of Clark [3], various properties and applications of the measures σα on the unit circle T have been obtained; see, for example, reviews [6–8] for further references.

1.2 Model Spaces and de Branges–Rovnyak Spaces Let  denote the normalized Lebesgue measure on ∂. Definition 1 A holomorphic function I :  → D is called inner if |I (ζ)| = 1 for -a.e. ζ ∈ ∂. In the above definition, I (ζ) stands, as usual, for limr →1− I (r ζ). Recall that the corresponding limit exists -a.e. Also, by the above definition, every inner function is non-constant. Given an inner function I in , we have P[σα ](ζ) =

1 − |I (ζ)|2 = 0 -a.e., |α − I (ζ)|2

therefore, σα = σα [I ] is a singular measure. Here and in what follows, this means that σα and  are mutually singular; in brief, σα ⊥. Let Hol() denote the space of holomorphic functions in . For 0 < p < ∞, the classical Hardy space H p = H p () consists of those f ∈ Hol() for which

Comparison of Clark Measures in Several Complex Variables

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 p

f H p = sup

0 1, unlike in one-dimensional case, we can not prescribe {an, j } and {bn, j } arbitrarily. In fact, coefficients in (7) satisfy the so-called “consistency conditions” which is a system of nonlinear difference equations. This discrete integrable system and associated Lax pair were studied before.

2 Angelesco Systems and Main Results 2.1 Angelesco Systems and Ray’s Limits of NNRR Coefficients We recall that μ is an Angelesco system of measures if supp μ j =  j := [α j , β j ] :

i ∩  j = ∅, i = j, i, j = 1, . . . , d ,

(9)

i.e. supports {i } is the system of d closed segments separated by d − 1 nonempty open intervals. The Angelesco systems are important general class of the perfect systems.1 MOPs with respect to this system were studied by Angelesco in 1919. Perfectness of μ guaranties that the corresponding MOPS satisfy to NNRRs (7). It is not so difficult to see2 that the corresponding NNRRs coefficients P := {an,i , bn,i }n∈Zd≥0 , i=1,...,d satisfy conditions (2) (detailed proof see in [1]). Thus P generated by Angelesco systems can be used as potentials for general class of bounded and self-adjoint on 2 (V) operators Jκ defined by (3). Moreover, asymptotic behavior of the recurrence coefficients {an, j , bn, j } for the ray’s sequences regime, namely

 n i = ci | n | + o n , i ∈ {1, . . . , d},

Nc = { n} :

| c | :=

d

ci = 1 ,

i=1

(10) was studied in [1] for c = (c1 , . . . , cd ) ∈ (0, 1)d . We have Theorem 1 ([1, Theorem 3.5]) Let μ be Angelesco system (9), such that for each i ∈ {1, . . . , d} the measure μi is absolutely continuous with respect to the Lebesgue measure on i and that the density wi := dμi (x)/dx extends to a holomorphic and non-vanishing function in some neighborhood of i .  Then the ray’s limits (10) of coefficients an,i , bn,i from (7) exist for any c ∈ (0, 1)d . lim an,i = Ac,i and lim bn,i = Bc,i , i ∈ {1, . . . , d}, (11) Nc

1 2

Nc

Perfectness of Angelesco system easily follows directly from (5). In fact the condition 0 < an,i for all n ∈ Nd , i ∈ {1, . . . , d} follows directly from (8).

On Spectrum of a Class of Jacobi Matrices on Graph-Trees …

21

We remark that expressions for limits Ac,i , Bc,i were obtained in [1] as well and we recall them in Sect. 2. The validity of Theorem 1 was deduced in [1] from the obtained there results on the strong asymptotics of the Angelesco MOPs for the ray’s regimes with c ∈ (0, 1)d .

2.2 Main Results Here we restrict ourselves by the case d = 2. For this case in [2] we extend the results of [1] on the strong asymptotics of the Angelesco MOPs for the ray’s regimes with c ∈ [0, 1]2 . From the obtained in [2] results on the strong asymptotics of the Angelesco MOPs we deduce for the ray’s regimes with c ∈ [0, 1]2 the following extension of Theorem 1. Theorem 2 Let μ be Angelesco system (9) for d = 2, satisfying conditions of Theorem 1. Then the ray’s limits lim an,i = Ac,i and lim bn,i = Bc,i Nc

Nc

(12)

exist for any c ∈ [0, 1] and i ∈ {1, 2}, where Nc is any subsequence of Z2≥0 such that n | → c as | n | → ∞ along this subsequence. n 1 /| Of course this very technical study in [2] of the Angelesco MOP’s asymptotics in the boundary layers of multi-indices n ∈ Z2≥0 should have a serious motivation. Indeed, it is a spectral theory of Jacobi-matrix operators (3) defined on the graph-tree from Fig. 1. Now, let P := {an,i , bn,i }n∈Z2≥0 , i=1,2 be a collection satisfying (2) for d = 2 and the constants {Ac,1 , Ac,2 , Bc,1 , Bc,2 }c∈[0,1] are the limits of the NNRRs coefficients for an Angelesco MOPs defined on intervals (1 , 2 ). We say that P ∈ P Ang (1 , 2 ) if P satisfies (12). In accordance with Theorem 2 class P Ang (1 , 2 ) is not empty. For the bounded and self-adjoint on 2 (V) operators Jκ defined by (3) with potentials from this class we have the following characterization of the essential spectrum. Theorem 3 Let Jκ be the Jacobi operator defined by (3) corresponding to a collection of parameters P ∈ P Ang (1 , 2 ), then σess (Jκ ) = 1 ∪ 2 .

2.3 Expressions for the Ray’s Limits In this subsection we define values of the limits standing at the right-hand sides in (12). For c ∈ (0, 1) these limits were obtained and proven in [1, Theorem 3.5]. To define these limits for c ∈ {0, 1} is rather easy because on the marginal rays stand usual orthogonal polynomials, see (5). However to prove these limits when one

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A. I. Aptekarev et al.

approaches to the marginal ray from the inside of the lattice is the main technical subject of this paper. Let 1 = [α1 , β1 ] and 2 = [α2 , β2 ] be two intervals on the real line such that β1 < α2 . Denote by ω1 and ω2 the arcsine distributions on 1 and 2 , respectively. Then it is known that E(ωi , ωi ) ≤ E(ν, ν), E(μ, ν) := − log |x − y| dμ(x) dν(y), for any probability Borel ν measure on i . The logarithmic potentials of these measures satisfy i − V ωi ≡ 0 on i , for some constants 1 and 2 , where V ν (z) := − log |z − x| dν(x). Now, given c ∈ (0, 1), define   Mc := (ν1 , ν2 ) : supp(νi ) ⊆ i , ν1  = c, ν2  = 1 − c .

(13)

Then, as it was proven by A. Gonchar and E. Rakhmanov, there exists the unique pair of measures (ωc,1 , ωc,2 ) ∈ Mc such that I (ωc,1 , ωc,2 ) ≤ I (ν1 , ν2 ),

I (ν1 , ν2 ) := 2E(ν1 , ν1 ) + 2E(ν2 , ν2 ) + E(ν1 , ν2 ) + E(ν2 , ν1 ),

(14) for all pairs (ν1 , ν2 ) ∈ Mc . It is also known that there exist constants c,i , i ∈ {1, 2}, such that  c,1 − V 2ωc,1 +ωc,2 ≡ 0 on supp(ωc,1 ), (15) c,2 − V ωc,1 +2ωc,2 ≡ 0 on supp(ωc,2 ). It is further known from A. Gonchar and E. Rakhmanov that supp(ωc,1 ) = [α1 , βc,1 ] =: c,1 and supp(ωc,2 ) = [αc,2 , β2 ] =: c,2 . Thus the intervals c,i dependent on the parameter c. Let Rc , c ∈ (0, 1), be a 3-sheeted Riemann surface realized as follows: cut a copy of C along c,1 ∪ c,2 , which henceforth is denoted by R(0) c , the second copy of C is cut along c,1 and is denoted by R(1) , while the third copy is cut along c,2 and c . These copies are then glued to each other crosswise along the is denoted by R(2) c corresponding cuts, see Fig. 2. It can be easily verified that thus constructed Riemann surface has genus 0. We denote by π the natural projection from Rc to C and employ the notation z for a generic point on Rc with π(z) = z as well as z (i) for a point (i) on R(i) c with π(z ) = z. Since Rc has genus zero, one can arbitrarily prescribe zero/pole divisors of rational functions on Rc as long as the degree of the divisor is zero. Clearly, a rational function with a given divisor is unique up to multiplication by a constant.

On Spectrum of a Class of Jacobi Matrices on Graph-Trees … Fig. 2 Surface Rc when βc,1 = β1 and αc,2 = α2

23 (0)

Rc α1

β1 α2

β2 (1)

Rc

(2)

Rc

Proposition 1 Let Rc , c ∈ (0, 1), be as above and χc (z) be the conformal map of Rc onto C such that



 χc z (0) = z + O z −1 as z → ∞. Further, let numbers Ac,1 , Ac,2 , Bc,1 , Bc,2 , c ∈ (0, 1), be defined by



 χc z (i) =: Bc,i + Ac,i z −1 + O z −2 as z → ∞, i ∈ {1, 2}.

(16)

√ Finally, let wi (z) := (z − αi )(z − βi ) be the branch of the corresponding algebraic function holomorphic outside of i and normalized so that wi (z)/z → 1 as z → ∞; in that case   βi + αi 1 z− + wi (z) (17) ϕi (z) := 2 2 is the conformal map of C \ i onto the complement of the disk of radius (βi − αi )/4 satisfying ϕi (z) = z + O(1) as z → ∞. Then it holds that ⎧ ⎪ A ⎪ ⎪ c,2 ⎪ ⎪ ⎨ Bc,2 lim c→0 ⎪ ⎪ Ac,1 ⎪ ⎪ ⎪ ⎩ Bc,1

 2 = (β2 − α2 )/4 =: A0,2 , = (β2 + α2 )/2

=: B0,2 ,

=0

=: A0,1 ,

(18)

= B0,2 + ϕ2 (α1 ) =: B0,1 ,

and analogous limits hold when c → 1. Moreover, all the constants Ac,1 , Ac,2 , Bc,1 , Bc,2 are continuous functions of the parameter c ∈ [0, 1]. It is worth to notice that the constants Ac,1 and Ac,2 are always positive. Indeed, denote by α1 , β c,1 , αc,2 , β 2 the ramification points of Rc with natural projections α1 , βc,1 , α2 , βc,2 , respectively. Then the symmetries of Rc and χc (z) yield that χc (z) is real and changes from −∞ to ∞ when z moves along the cycle ∞(0) → α1 → ∞(1) → β c,1 → αc,2 → ∞(2) → β 2 → ∞(1)

24

A. I. Aptekarev et al.

whose natural projection is the extended real line. Thus, χc (z) is increasing when it moves past ∞(1) and ∞(2) , which yields the claim (this argument also shows that Bc,1 < Bc,2 ). Acknowledgements The research of Aptekarev and Denisov was carried out with support from a grant of the Russian Science Foundation (project RScF-19-71-30004). The research of Yattselev was supported in part by a grant from the Simons Foundation, CGM-354538.

References 1. A.I. Aptekarev, S.A. Denisov, M.L. Yattselev, Self-adjoint Jacobi matrices on trees and multiple orthogonal polynomials. Trans. Am. Math. Soc. 373(2), 875–917 (2020) 2. A.I. Aptekarev, S.A. Denisov, M.L. Yattselev, On essential spectrum of certain Jacobi operators on a 2-homogeneous rooted tree (in preparation )

Geometric Properties of Reproducing Kernels in Hilbert Spaces of Entire Functions Yurii Belov

Abstract We give a survey of recent results about properties of systems of reproducing kernels in Paley–Wiener spaces, de Branges spaces and Fock spaces. In particular, we consider completeness of biorthogonal systems and spectral synthesis property for such systems.

1 Introduction Hilbert spaces of entire functions H are an important object in complex and harmonic analysis. Usually, they appear as an image of some Hilbert space X under some unitary spectral transform F (i.e. Fourier transform, Krein-de Branges transform, Bargmann transform etc.). The essence of the theory of Hilbert spaces of entire functions are their reproducing kernels. Typically reproducing kernels {kλ }λ∈C are images of some system of special functions in X (exponentials functions, Gaussian functions, solutions of some differential equations etc.). In this paper we survey some classical and recent results about reconstruction of an arbitrary  element f ∈ H via some sequence of samples f (λ) = ( f, kλ )H or via some series λ∈ aλ kλ (z), where  is a discrete subset of C. Such problems have many connections with other parts of analysis (operator theory, function theory) and mathematical physics, see e.g. [1, 9].

1.1 System of Vectors in Abstract Hilbert Space We start with a system of vectors {xn }n∈N in an abstract separable Hilbert space H. We will always assume that {xn }n∈N is a complete and minimal system, that is, {xn }n∈N Y. Belov (B) Department of Mathematics and Computer Science, St. Petersburg State University, St. Petersburg, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. Abakumov et al. (eds.), Extended Abstracts Fall 2019, Trends in Mathematics 12, https://doi.org/10.1007/978-3-030-74417-5_4

25

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Y. Belov

is a complete system but the system {xn }n=m is not complete for any m ∈ N . Given a complete and minimal system there exists a unique biorthogonal system {x˜n }n∈N which satisfies the relation (xm , x˜n ) = δmn . Hence, to every element x ∈ H we can associate its (formal) Fourier series x∼



(x, x˜n )xn .

(1)

n∈N

The first natural requirement is uniqueness of the formal Fourier series ((x, x˜n ) = 0 for any n =⇒ x = 0) or, which is the same, completeness of the biorthogonal system {x˜n }. Such system is said to be a Markushevich basis (or M-basis). A very natural property is the possibility to recover any vector x ∈ H from its Fourier series by some linear method (depending on x): x ∈ Span{(x, x˜n )xn }n∈N . If this holds, then the system {xn }n∈N is called a strong Markushevich basis (strong M-basis), or hereditarily complete system, or one says that the system {xn }n∈N admits the spectral synthesis. The last term is motivated by operator theory, see e.g. [1]. The next property is the existence of uniform linear summation method (not depending on x) for series (1). If the series (1) converges unconditionally for any x ∈ H we will say that {xn }n∈N is a Riesz basis. So, we have a scale of properties for system of vectors {xn }n∈N : i. system {xn }n∈N is complete and minimal; ii. biorthogonal system {x˜n }n∈N is complete (Markushevich basis); iii. every element x belongs to closed linear span of members of its Fourier series (strong Markushevich basis, hereditarily complete system); iv. the formal Fourier series (1) admits a linear summation method; v. system {xn }n∈N is a Riesz basis for H; vi. system {xn }n∈N is an orthogonal basis for H. In general, all these properties are different from each other. But if we restrict ourselves to systems of reproducing kernels in some concrete Hilbert space of entire functions, then some of classes (i)–(vi) may coincide. Another important question is to find a description of systems {kλ }λ∈ satisfying one or another of the conditions (i)–(vi). We consider such questions for Paley–Wiener spaces, de Branges spaces and Fock space of entire functions.

Geometric Properties of Reproducing Kernels in Hilbert Spaces …

27

2 Exponential Systems on an Interval, Paley–Wiener Spaces and de Branges Spaces 2.1 Paley–Wiener Spaces Let {eiλt }λ∈ be an exponential system in the space L 2 (−π, π). We are interested in the properties (i)–(vi) for such systems. Applying the Fourier transform F, Fϕ(z) =

1 π



π

−π

ϕ(t)eit z dt,

one reduces the problem for exponential systems in L 2 (−π, π) to the same problem for systems of reproducing kernels in the Paley–Wiener space P Wπ := F L 2 (−π, π). Recall that the reproducing kernel kλ (z) is of the form kλ (z) =

¯ sin(π(z − λ)) , ¯ π(z − λ)

f (λ) = ( f, kλ ).

In 1981 Young [17] have proved that property (ii) follows from (i) for exponential systems. Theorem 2.1 (Young [17]) Let {eiλt }λ∈ be a complete and minimal system in the space L 2 (−π, π). Then its biorthogonal system is also complete. On the other hand, it is clear that {eiλt }λ∈ is an orthogonal basis if and only if  = Z + δ for some δ ∈ [0, 1). It follows from the completeness and minimality of {eiλt }λ∈ (see, e.g. [13, Lecture 17]) that there exists the generating function, 

G(z) = lim

R→∞

λ∈,|λ| −∞. Finally, Minkin [14] got rid of all assumptions on . The description of linear summation bases of exponentials is not known. In 2016 Belov and Lyubarskii [7] were able to prove that the Muckenhoupt condition (A2 ) is sufficient for existence of linear summation method. Theorem 2.3 (Belov and Lyubarskii [7]) Let the generating function G be of exponential type π in both half-planes C± and also satisfy the Muckenhoupt condition (A2 ) on the real line. Then {eiλt }λ∈ admits a linear summation method in L 2 (−π, π). The question whether spectral synthesis (property (iii)) follows from completeness and minimality (property (i)) was open since 1960s. It was solved in 2013 by Baranov, Belov and Borichev [4]. Theorem 2.4 There exists a system of exponentials {eiλn t }n∈Z , which is complete and minimal in L 2 (−π, π) but is not hereditarily complete. On the other hand, the hereditary completeness holds up to a possible onedimensional defect, see [4, Theorem 1.2]. The description of all hereditarily complete systems from exponentials is not known.

2.2 De Branges Spaces An entire function E is said to be in the Hermite–Biehler class if |E(z)| > |E ∗ (z)|, z ∈ C+ , where E ∗ (z) = E(¯z ). With any such function we associate the de Branges space H(E) which consists of all entire functions F such that F/E and F ∗ /E restricted to C+ belong to the Hardy space H 2 = H 2 (C+ ). The inner product in H(E) is given by  F(t)G(t) (F, G) E = dt. 2 R |E(t)| De Branges spaces naturally appear in theory of canonical systems. The orthogonal bases of reproducing kernels were described by de Branges, see [9]. The Riesz bases of reproducing kernels in de Branges spaces were described only for some special classes of spaces, see e.g. [8].

Geometric Properties of Reproducing Kernels in Hilbert Spaces …

29

Completeness of the system biorthogonal to a system of reproducing kernels in de Branges spaces was studied in [2, 10]. It turns out that for some de Branges spaces property (ii) follows from (i) and for some spaces this is not true. Spectral synthesis property for systems of reproducing kernels in de Branges spaces was studied in [3]. The following theorem describes all de Branges spaces such that property (iii) follows from (ii). The description is given in terms of the spectral measure μ (for details about the spectral measure see [3, Sect. 1.2]). Theorem 2.5 (Baranov, Belov  and Borichev [3]) Let H(E) be a de Branges space with the spectral data μ = n μn δtn . Then H(E) has the strong M-basis property (i.e. (iii) follows from (ii)) if and only if one of the following conditions holds: •



μn < ∞;

n

• The sequence {tn } is lacunary and, for some C > 0 and any n, 

μk + tn2

|tk |≤|tn |

 μk ≤ Cμn . t2 |t |>|t | k k

n

3 Fock Space 

Let F = {F is entire and

|F(z)|2 e−π|z| dm(z) < ∞}; 2

C

here dm denotes the planar Lebesgue measure. Recall that the reproducing kernel kw (z) for F is of the form ¯ . kw (z) = eπwz It is well known that the Bargmann transform is a unitary map between L 2 (R) and the Fock space F. Moreover, the time-frequency shift of the Gaussian is mapped to the normalized reproducing kernel, see e.g. [11] for the details. Since kw (z) = 0, there is no orthogonal bases from reproducing kernels in F. Moreover, Seip have proved [16] that there is no Riesz bases from reproducing kernels in F. On the other hand, there are a lot of complete and minimal systems in F, e.g. {kw : w = m + in, m, n ∈ Z, m 2 + n 2 = 0}. In 2015 Belov [6] have proved the following result which is analogous to Young’s theorem. Theorem 3.1 (Belov [6]) Let {kw }w∈ be a complete and minimal system in the space F. Then its biorthogonal system is also complete.

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On the other hand, property (iii) does not automatically follow from (i), see [5]. Theorem 3.2 (Baranov, Belov and Borichev [5]) There exists a complete and minimal system of reproducing kernels in F which is not hereditarily complete. At the same time, the hereditary completeness holds up to a possible onedimensional defect, see [5, Theorem 1.2]. Acknowledgements The work is supported by Russian Science Foundation grant 17-11-01064.

References 1. A. Baranov, Spectral theory of rank one perturbations of normal compact operators. Algebra i Analiz 30, 5, 1–56 (2018); English transl. in St. Petersburg Math. J. 30, 5, 761–802 (2019) 2. A. Baranov, Y. Belov, Systems of reproducing kernels and their biorthogonal: completeness or incompleteness? Int. Math. Res. Not. 22, 5076–5108 (2011) 3. A. Baranov, Y. Belov, A. Borichev, Spectral synthesis in de Branges spaces. Geom. Funct. Anal. 25(2), 417–452 (2015) 4. A. Baranov, Y. Belov, A. Borichev, Hereditary completeness for systems of exponentials and reproducing kernels. Adv. Math. 235, 525–554 (2013) 5. A. Baranov, Y. Belov, A. Borichev, Summability properties of Gabor expansions. J. Funct. Anal. 274(9), 2532–2552 (2018) 6. Y. Belov, Uniqueness of Gabor series. Appl. Comput. Harmon. Anal. 39, 545–551 (2015) 7. Y. Belov, Y. Lyubarskii, On summation of non-harmonic Fourier series. Constr. Approx. 43(2), 291–309 (2016) 8. Y. Belov, T. Mengestie, K. Seip, Discrete Hilbert transforms on sparse sequences. Proc. Lond. Math. Soc. 103(3), 73–105 (2011) 9. L. de Branges, Hilbert Spaces of Entire Functions (Prentice-Hall, Englewood Cliffs, 1968) 10. E. Fricain, Complétude des noyaux reproduisants dans les espaces modèles. Ann. Inst. Fourier (Grenoble) 52(2), 661–686 (2002) 11. K. Gröchenig, Foundations of Time-Frequency Analysis (Birkhauser, Boston, 2001) 12. S.V. Hruscev, N.K. Nikolskii, B.S. Pavlov, Unconditional bases of exponentials and of reproducing kernels, in Complex Analysis and Spectral Theory (Leningrad, 1979/1980). Lecture Notes in Mathematics, vol. 864 (Springer, Berlin, 1981), pp. 214–335, 13. B.Y. Levin, Lectures on Entire Functions. Translations of Mathematical Monographs, vol. 150 (AMS, Providence, 1996) 14. A.M. Minkin, Reflection of exponents and unconditional bases of exponentials. St. Petersburg Math. J. 3, 1043–1068 (1992) 15. B.S. Pavlov, The basis property of a system of exponentials and the condition of Muckenhoupt. (Russian) Dokl. Akad. Nauk SSSR 247, 37–40 (1979). English transl. in Soviet Math. Dokl. 20 (1979) 16. K. Seip, Density theorems for sampling and interpolation in the Bargmann- Fock space. I. J. Reine Angew. Math. 429, 91–106 (1992) 17. R. Young, On complete biorthogonal system. Proc. Am. Math. Soc. 83(3), 537–540 (1981)

A New Life of the Classical Szeg˝o Formula Roman Bessonov and Sergey Denisov

Abstract We collect several applications of a new version of the classical Szeg˝o formula for orthogonal polynomials. The applications are concerned with the spectral theory of Krein strings, scattering theory for Dirac systems, and triangular factorizations of positive Wiener–Hopf operators.

1 Introduction Let μ be a probability measure on the unit circle T of the complex plane C, and let {ϕn } be the family of orthonormal polynomials generated by μ. It is well-known that ϕn satisfy recurrence relations  1 − |an |2 · ϕ∗n+1 = ϕ∗n − zan ϕn ,

ϕ0 = ϕ∗0 = 1,

n ≥ 0,

(1)

where ϕ∗n (z) = z n ϕn (1/¯z ) and {an } is a sequence of numbers in the unit disk D = {z ∈ C : |z| < 1} depending only on μ. Assuming μ has the form μ = w dm + μs for some density w with respect to the Lebesgue measure m on T and a singular part μs , let us introduce the function K(μ, z) = log P(μ, z) − P(log w, z),

z ∈ D.

(2)

R. Bessonov St. Petersburg State University, Universitetskaya nab. 7-9, 199034 St. Petersburg, Russia e-mail: [email protected] St. Petersburg Department of Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, 191023 St. Petersburg, Russia S. Denisov (B) Department of Mathematics, University of Wisconsin–Madison, 480 Lincoln Dr., Madison, WI 53706, USA e-mail: [email protected] Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya pl. 4, 125047 Moscow, Russia © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. Abakumov et al. (eds.), Extended Abstracts Fall 2019, Trends in Mathematics 12, https://doi.org/10.1007/978-3-030-74417-5_5

31

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R. Bessonov and S. Denisov

As usual, we denote by P the operator of harmonic extension to D:  P(μ, z) =

T

1 − |z|2 dμ(ξ). ¯ 2 |1 − ξz|

We also set P(v, z) = P(v dm, z) for v ∈ L 1 (T). The classical Szeg˝o formula can be written in the following way:  K(μ, 0) = −

T

log w dm = − log



(1 − |an |2 ).

(3)

n≥0

It turns out that one can find expression for K(μ, z) for all z ∈ D: K(μ, z) = log

∞  1 − |z f k (z)|2 n=1

1 − | f k (z)|2

,

where f k are the Schur functions of μ, see [4]. More generally, the formula can be extended to the case where μ is a spectral measure of a self-adjoint differential operator and K(μ, z) is defined for z in the upper half-plane C+ . This approach turns out to be very fruitful: below we discuss several long standing problems that were solved by using the new version of formula (3).

2 Krein Strings The Krein string equation has the form −y  (t, λ) = λρ(t)y(t, λ),

t ∈ [0, L),

λ ∈ C.

Here L > 0 is the length of the string, and ρ denotes its density which is supposed to be an arbitrary σ-finite measure on the positive half-axis, R+ = [0, +∞). We do not exclude “wild” cases where the absolutely continuous part ρac of ρ is zero. So, the piece of string [0, x) has mass M(x) = ρ([0, x)) and M can be arbitrary positive non-decreasing left-continuous function on R+ . The string is called long if L + lim M(x) = ∞. x→L

It is possible to associate a differential operator with each string [M, L]. Its main spectral measure, defined in terms of Weyl function, is called the spectral measure of a string. It turns out that the string is completely determined by its spectral measure. This makes interesting the problem of translating various properties of the spectral measure into the properties of the mass function M. The problem is usually very hard.

A New Life of the Classical Szeg˝o Formula

33

It has strong connection to the problem of determining properties of the sequence of recurrence coefficients {an } in (1) from properties of the orthogonality measure μ on T. Note that (3) implies that log w ∈ L 1 (T) if and only if



|an |2 < ∞.

n≥0

This explains the reason to expect a possibility of describing Krein strings whose spectral measures σ = v d x + σs satisfy the following Szeg˝o-type condition in the domain C \ R+ :  ∞ log v(x) (4) √ d x > −∞. (1 + x) x 0 Note, however, that the point 0 is an inner point of the open unit disk D, while it is a boundary point for the domain C \ R+ . In particular, we cannot hope to use a Szeg˝o formula at 0 for measures on the boundary of C \ R+ . Instead, we need to generalize (3) to other points of D (therefore avoiding usage of properties of orthogonal polynomials/solutions of Krein system related to the special value λ = 0 of the spectral parameter) and then find its analogue for C \ R+ . In fact, the quantity in (4) is nothing but K(σ, −1) for a properly defined entropy function K for C \ R+ and a normalized measure σ. The idea to prove a formula for K(μ, z) for orthogonal polynomials on T and then search for its variant for Krein strings looks very natural, but the real history is somewhat opposite: such a formula was first established for Krein strings and then translated to the unit disk (where its proof is significantly easier). Here is the result for Krein strings [1]. Theorem 1 Let [M, L] be a long string and let σ = v d x + σs be the spectral mea√ / L 1 (R+ ) and sure of [M, L]. Then σ satisfies (4) if and only if ρac ∈ +∞  

 L n Mn − 4 < ∞, L n = tn+2 − tn , Mn = M(tn+2 ) − M(tn ),

(5)

n=0

t √ 

where tn = min x ≥ 0 : n = 0 ρac (t) dt . It is interesting to note that quantities L n , Mn have physical meaning: L n is the length of a piece of the string which is covered by a traveling wave during the period [n, n + 2) of time, Mn is the mass of this piece. We have Mn L n = 4 for any homogeneous string.

3 Scattering Theory for Dirac Operators The one-dimensional Dirac operator on R+ is defined by

34

R. Bessonov and S. Denisov

D Q : X →

 0 −1 1 0

X  + Q X,

Q=

 q1

q2 q2 −q1



(6)

on a dense subset of Lebesgue space L 2 (R+ , C2 ) of squared summable functions on R+ with values in C2 . This is one of the simplest self-adjoint differential operators, and many of analytic tools can be applied to investigation of its properties. As an example, let us consider the problem of the existence of wave operators W± (D Q , D0 ) = lim eitD Q e−itD0 . t→∓∞

(7)

Wave operators are basic objects of the scattering theory, their existence (that is, existence of the limits above in the strong operator topology) and completeness (that is, unitarity of W± as operators between the absolutely continuous subspaces of D0 , D Q ) are the main questions of interest. It is known that the limit in (7) for potentials Q with entries in L p (R+ ), 1 ≤ p ≤ 2, can be expressed in terms of the Szeg˝o function of the spectral measure μ of D Q . The Szeg˝o function of a Poisson summable measure μ = wd x + μs on the real line R such that 

  1 dμ(x) log w(x) 1 − d x < +∞ (8) K(μ, i) = log π R 1 + x2 π R 1 + x2 is the outer function in C+ satisfying |Dμ |2 = w on R. Let χ E is the indicator function of a set of full Lebesgue measure on R such that μs (E) = 0. Denote byF Q , F0 the Fourier transforms generated by generalized eigensolutions of D Q , D0 , correspondingly. We have W− (D Q , D0 ) = γF Q−1 χ E Dμ−1 F0 ,

W+ (D Q , D0 ) = γF ¯ Q−1 χ E Dμ−1 F0

(9)

for every Q with entries in L p (R+ ), 1 ≤ p ≤ 2. This formula, well-known for specialists in scattering theory, immediately rises two questions: (a) does it hold for any Dirac operator with spectral measure such that K(μ, i) < ∞? (b) how to describe potentials Q that generate measures μ such that K(μ, i) < ∞? Both questions were open for a long time and got their answer only recently. To simplify the presentation, we assume q1 = 0 in theorems below, while the general case can be covered as well. Theorem 2 Let q bea real-valued function on R+ such that q ∈ L 1 [0, r ] for every  0q r > 0, and let Q = q 0 . Assume that the spectral measure μ of D Q satisfy (8). Then the wave operators W± (D Q , D0 ) exist, complete, and are given by (9).   t Theorem 3 For n ≥ 0, define the functions gn (t) = exp 2 n q(s) ds , t ∈ [n, n + 2). The spectral measure of D Q satisfies (8) if and only if

A New Life of the Classical Szeg˝o Formula

  n≥0

n+2

35

 gn (t) dt ·

n

n

n+2

 dt − 4 < ∞. gn (t)

(10)

Both theorems were obtained [3] by systematically using the entropy function K(μ, z). In fact, we get a family of such functions K(μr , z), r ≥ 0 if we denote by μr the spectral measure of the operator D Q on [r, +∞). We would like to mention that K(μr , z) satisfies a nonlinear differential equation as a function in r ∈ R+ .

4 Triangular Factorization of Wiener–Hopf Operators Another classical problem that can be solved by using the entropy function K(μ, z) concerns triangular factorization of Wiener–Hopf operators. In the abstract setting, given a bounded invertible positive operator T on a separable Hilbert space H , one may ask about existence of a bounded invertible operator A that is upper triangular with respect to a given chain of subspaces L and factorizes T into the product T = A∗ A. The words “upper triangular with respect to L” simply mean that AE ⊂ E for every E ∈ L. The theory of triangular factorization of positive operators was developed by Gohberg and Krein in 60’s. It was fairly nontrivial problem if there are nonfactorable bounded invertible positive operators. The affirmative answer to this question was given by Larson in 1985. He showed that every uncountable chain L gives rise to a nonfactorable operator. The approach of Larson is highly nonconstructive, and it is desirable to find a concrete example of a nonfactorable operator to understood better this phenomenon. The simplest uncountable chain one can imagine is the chain of subspaces {L 2 [0, r ]}r >0 in L 2 (R+ ). It turns out that the problem of triangular factorization of positive bounded Wiener–Hopf operators  Wψ : f →

R+

ψ(t − s) f (s) ds,

f ∈ L 2 (R+ )

with respect to this chain can be reformulated in terms of the spectral theory of Krein strings and more general objects—canonical Hamiltonian systems. This fact was observed by L. Sakhnovich who posed this problem for Wiener–Hopf operators in 1994. More precisely, a real distribution ψ generates factorable with respect to the chain {L 2 [0, r ]}r >0 bounded positive operator Wψ if and only if the Krein string ˇ √x) d x on R has absolutely continuous corresponding to the spectral measure σ = ψ( mass distribution M. Here ψˇ is the inverse Fourier transform of ψ. Since Wψ is ˇ bounded, positive and invertible, we have c1 ≤ ψ(x) ≤ c2 for almost all x ∈ R and some constants c1 , c2 . So, the spectral measure σ has a density separated from zero and infinity. Asymptotic analysis of the entropy function K(σ, −x) as x → +∞ then gives us information about quantities

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(Mn,ε L n,ε − 4ε2 ),

n≥0

t √ 

where, as before, ts = min x ≥ 0 : s = 0 ρac (t) dt , and we set L n,ε = t(n+2)ε − tnε , Mn,ε = M(t(n+2)ε ) − M(tnε ). Analysing these quantities when ε tends to zero, we get the following conclusion [2]. Theorem 4 Every bounded invertible positive Wiener–Hopf operator Wψ on L 2 (R+ ) admits triangular factorization with respect to the chain {L 2 [0, r ]}r >0 . The reader can find more information in papers listed below. Acknowledgements The work is supported by grant RScF 19-11-00058.

References 1. R.V. Bessonov, S.A. Denisov, A spectral Szeg˝o theorem on the real line. Adv. Math. 359, 1–41 (2020) 2. R.V. Bessonov, Wiener-Hopf operators admit triangular factorization. J. Oper. Theory 82(1), 237–249 (2019) 3. R.V. Bessonov, Szeg˝o condition and scattering for one-dimensional Dirac operators. Constr. Approx. 51, 273–302 (2020) 4. R.V. Bessonov, S.A. Denisov, Zero sets, entropy, and pointwise asymptotics of orthogonal polynomials. Arxiv preprint arXiv:1911.11280

De Branges Canonical Systems with Finite Logarithmic Integral Roman V. Bessonov and Sergey A. Denisov

Abstract Krein–de Branges spectral theory provides a correspondence between canonical Hamiltonian systems and measures on the real line with finite Poisson integral. We revisit this area by giving a description of canonical Hamiltonian systems whose spectral measures have logarithmic integral converging over the real line. Our result can be viewed as a spectral version of the classical Szeg˝o theorem in the theory of polynomials orthogonal on the unit circle. It extends Krein–Wiener completeness theorem, a key fact in the prediction of stationary Gaussian processes.

1 Main Results In this note, which is based on two publications [1, 2] and contains the formulation of two main results from aforementioned papers, we revisit the spectral theory of de Branges’ canonical system, which is defined by the system of differential equations of the form   10 d M(t, z) = zH(t)M(t, z), M(0, z) = I 0 −1 J dt 2×2 = 0 1 , J = 1 0 , t ≥ 0, z ∈ C .

(1) The 2 × 2 matrix-function H on R+ = [0, +∞) is called the Hamiltonian of canonical system (1). Our assumptions of H are: (a) H(t) ≥ 0 and trace H(t) > 0 for Lebesgue almost every t ∈ R+ , R. V. Bessonov St. Petersburg State University, Universitetskaya nab. 7-9, 199034 St. Petersburg, Russia e-mail: [email protected] St. Petersburg Department of Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, 191023 St. Petersburg, Russia S. A. Denisov (B) Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Dr., Madison, WI 53706, USA e-mail: [email protected] Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya pl. 4, 125047 Moscow, Russia © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. Abakumov et al. (eds.), Extended Abstracts Fall 2019, Trends in Mathematics 12, https://doi.org/10.1007/978-3-030-74417-5_6

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(b) the entries of H are real measurable functions absolutely summable on compact subsets of R+ . In 1960s, L. de Branges developed his theory of Hilbert spaces of entire functions. One result in this area is the theorem that establishes a bijection between Hamiltonians H in (1) and nonconstant analytic functions in C+ = {z ∈ C : Im z > 0} with nonnegative imaginary part. Every such function is generated by a nonnegative measure on the real line. Below, we make a another step in de Branges’ theory by identifying Hamiltonians that correspond to measures in the Szeg˝o class, i.e., the measures whose logarithmic integral converges over R. We start with some definitions. A Hamiltonian H on R+ is called singular if 

+∞

trace H(t) dt = +∞.

0

Two Hamiltonians H1 , H2 on R+ are called equivalent if there exists an increasing absolutely continuous function η defined on R+ such that η(0) = 0, limt→+∞ η(t) = +∞, and H2 (t) = η  (t)H1 (η(t)) for Lebesgue almost every t ∈ R+ . Clearly, η(t) rescales the variable t. We say that Hamiltonian H is trivial if there is a non-negative matrix A with rank A = 1, such that H is equivalent to A, i.e., H(t) = η  (t)A for a.e. t ∈ R+ , where η is an increasing absolutely continuous function on R+ , which satisfies η(0) = 0 and limt→+∞ η(t) = +∞. If Hamiltonian is not trivial, it is called nontrivial. We recall that function m belongs to the Herglotz–Nevanlinna class N (C+ ) if it is analytic in C+ and Im m(z) ≥ 0 for z ∈ C+ . It is well-known, that m ∈ N (C+ ) if and only if it admits the following integral representation 1 m(z) = π

  R

1 x − 2 x−z x +1

 dμ(x) + bz + a, z ∈ C+ ,

(2)

where b ≥ 0, a ∈ R, and μ is a Radon measure on R, which satisfies  R

dμ < ∞. 1 + x2

(3)

Those measures on R that satisfy (3) are called Poisson-finite. The class N (C+ ) appears naturally in the theory of canonical Hamiltonian systems. Let H be a nontrivial and singular Hamiltonian. Given condition (b) on H, there exists unique matrix-valued function M that solves (1). Denote by ± , ± its entries so that   +  (t, z) + (t, z) . M(t, z) = ((t, z), (t, z)) = − (t, z) − (t, z)

(4)

Fix a parameter ω ∈ R ∪ {∞}. The Titchmarsh–Weyl function of H is defined by

De Branges Canonical Systems with Finite Logarithmic Integral

ω+ (t, z) + − (t, z) , t→+∞ ω+ (t, z) + − (t, z)

m(z) = lim

39

z ∈ C+ ,

(5)

1 +c2 for non-zero numbers c1 , c3 is interpreted as c1 /c3 . In where the fraction ∞c ∞c3 +c4 Titchmarsh–Weyl’s theory for canonical systems, it is proved that the expression under the limit in (5) is well-defined for large t > 0 (i.e., the denominator is nonzero) for every given singular nontrivial Hamiltonian H. Moreover, the limit m(z) exists, does not depend on ω, m is analytic in z ∈ C+ and has non-negative imaginary part, i.e., m ∈ N (C+ ). In particular, m admits representation (2). The measure μ in (2) is called the spectral measure for the Hamiltonian H. It is obvious that equivalent Hamiltonians have equal Titchmarsh–Weyl functions. Now we can state the result of de Branges that establishes a bijection between Hamiltonians and Herglotz–Nevanlinna functions.

Theorem 1 (de Branges) For every nonconstant function m ∈ N (C+ ), there exists a singular nontrivial Hamiltonian H on R+ such that m is the Titchmarsh–Weyl function (5) for H. Moreover, any two singular nontrivial Hamiltonians H1 , H2 on R+ generated by m are equivalent. For trivial Hamiltonians, function m is a real constant. In fact, in that case, one can solve (1) explicitly   and this calculation shows that m(z) = const ∈ R ∪ ∞. For example, H = 01 00 gives + = 1, − = −zt, + = 0, − = 1,

(6)

  so m = 0. Similarly, if H = 00 01 , then + = 1, − = 0, + = zt, − = 0 and we let m = ∞. Given a Poisson-finite measure μ on R, we will denote by w the density of μ with respect to the Lebesgue measure d x on R, and by μs the singular part of μ, so that μ = w d x + μs . Our goal is to characterize singular nontrivial Hamiltonians whose spectral measures have finite logarithmic integral, i.e., the integral  R

log w(x) dx 1 + x2

converges. The trivial bound log w ≤ w shows that logarithmic integral of a Poissonfinite measure can diverge only to −∞. It will be convenient to call the set of all measures with finite logarithmic integral the Szeg˝o class Sz(R), i.e.,

 dμ(x)  | log w(x)| + d x < +∞ . Sz(R) = μ : 2 1 + x2 R 1+x R If m ∈ N (C+ ) and measure μ in (2) is in Szeg˝o class, we can define Km = log Im m(i) −

     log w(x) dμ log w(x) 1 1 1 d x = log b + dx . − π R 1 + x2 π R 1 + x2 π R 1 + x2

(7)

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R. V. Bessonov and S. A. Denisov

One can use b ≥ 0 and Jensen’s inequality to show that Km ≥ 0. Notice that Km = 0 if and only if m is a constant with positive imaginary part. Next, we introduce the that characterizes measures in Szeg˝o √ class of Hamiltonians class. If H is such that det H ∈ / L 1 (R+ ), define  K(H) =

∞ 

 ηn+2 det

n=0

ηn

  t

H(t)dt − 4 , ηn = min t ≥ 0 : det H(s) ds = n . (8) 0

√ Since the entries of H are locally summable functions, the function t → det H(t) is also locally summable on R+ and {ηn } make sense. One can check the following bound:  ηn+2 2  ηn+2 det H(t) dt ≥ det H(t) dt = 4, n ≥ 0 . ηn

ηn

This inequality shows that the series in (8) contains only non-negative terms and  hence its sum K(H) ∈ R+ ∪ {+∞} is well-defined but could be +∞, in general.  Actually, K(H) can be rewritten in the form reminiscent of matrix A2 Muckenhoupt  condition. Roughly speaking, K(H) measures how fast the entries of H oscillate. In  fact, we have K(H) = 0 if and only if the Hamiltonian H is equivalent to a constant positive matrix. Notice that if the Hamiltonian is trivial then its determinant is zero  is undefined. Define the class H of Hamiltonians by and K

√  / L 1 (R+ ), K(H) < +∞ . H = singular nontrivial H : det H ∈ Here is our main result: Theorem 2 The spectral measure of a singular nontrivial Hamiltonian H on R+ belongs to the Szeg˝o class Sz(R) if and only if H ∈ H. Moreover, we have  ≤ c2 Km ec2 Km , c1 Km ≤ K(H)

(9)

for some absolute positive constants c1 , c2 . The bound (9) is essentially sharp up to numerical values of c1 and c2 . For H such   that K(H) ≤ 1, (9) gives Km ∼ K(H). Moreover, we can present two examples  for both of which K(H) > 1. In the first example, we have Km ∼ log(1 + L) and  K(H) ∼ L, where L is arbitrarily large parameter. This shows that the exponent in the right hand side of (9) can not be dropped. In the second example, we have  ∼ L, where L is again arbitrarily large parameter. Thus, the left Km ∼ L and K(H) bound in (9) can not be improved. Diagonal canonical Hamiltonian systems are related to the equation of a vibrating string:  d d y(t, z) = zy(t, z), t ∈ [0, L), z ∈ C. (10) − d M(t) dt

De Branges Canonical Systems with Finite Logarithmic Integral

41

Here 0 < L ≤ +∞ is the length of the string, M : (−∞, L) → R+ is an arbitrary non-decreasing and right-continuous function (mass distribution) that satisfies M(t) = 0 for t < 0. If M is smooth and strictly increasing on R+ , then Eq. (10) takes the form −y  = z M  y. We consider those L and M that satisfy the following conditions: L + lim M(t) = ∞ and t→L

lim M(t) > 0 ,

t→L

(11)

where the last bound means that M is not identically equal to zero. If (11) holds, we say that M and L form [M, L] pair. To every [M, L] pair one can relate a string and Weyl-Titchmarsh function q with spectral measure σ supported on the positive half-axis R+ . Theorem 2 can be applied to Krein strings as follows. Theorem 3 Let [M, L] satisfy (11) and σ = v d x + σs be the spectral measure ∞ log v(x) √ d x > −∞ if and only if of the corresponding string. Then, we have 0 (1+x) x √ M ∈ / L 1 (R+ ) and  K[M, L] =

+∞   (tn+2 − tn )(M(tn+2 ) − M(tn )) − 4 < ∞,

(12)

n=0

t √   where tn = min t ≥ 0 : n = 0 M  (s) ds . Condition (11) ensures that the string [M, L] has a unique spectral measure and it does the generality of Theorem 3: if (11) is violated, then either M = 0  ∞notlogrestrict v(x) √ d x = −∞. and 0 (1+x) x Our main result has applications to scattering theory of Dirac and wave equations. It provides the necessary framework to prove existence of wave/modified wave operators under optimal assumptions on the decay of coefficients. See, e.g., [3]. Acknowledgements S. A. Denisov was supported by grant RScF-19-71-30004.

References 1. R. Bessonov, S. Denisov, De Branges canonical systems with finite logarithmic integral. Submitted. Available on arXiv 2. R. Bessonov, S. Denisov, A spectral Szeg˝o theorem on the real line, to appear in Adv. Math. Preprint available on arXiv 3. R. Bessonov, Scattering and Szeg˝o condition for one-dimensional Dirac operators, to appear in Constr. Approx. Preprint available on arXiv

Rate of Convergence of Critical Interfaces to SLE Curves Ilia Binder

Abstract In this short survey, we will discuss the use of the techniques of Complex Function Theory in establishing the results on the rate of convergence of critical interfaces of planar lattice models of Statistical Physics to SLE curves. In particular, we examine the exploration process for critical percolation. One of the main results is the fact that for any “reasonable” critical percolation model for which the convergence of the exploration process is established, the polynomial rate of convergence must automatically hold. So far, the result is unconditional for the critical site percolation on the hexagonal lattice and for some of its generalizations, which will also be discussed. In addition, we analyze a general framework for establishing these types of results for other models, such as Harmonic Explorer and the Ising model. The survey is based on joint projects with L. Chayes, H. Lei, and L. Richards.

1 Introduction: Deterministic Loewner Curves It was conjectured in the sixties that a number of two-dimensional lattice models of Statistical Physics have some sort of continuous limit in criticality when the mesh size tends to zero. This limit was also predicted to be conformally invariant. Motivated by these conjectures, Schramm introduced in [16] Schramm Loewner Evolution (SLEκ ), a one-parameter family of conformally invariant random curves in simply-connected planar domains. It rapidly became one of the active areas of research in Mathematical and Statistical Physics, Geometric Function Theory, and Probability. These curves are conjectured to be the scaling limits of the various interfaces in critical lattice models. Some of these conjectures have been verified in recent years, see [2, 8, 13, 14, 17, 18, 20]. Let us first discuss a version of deterministic Loewner Evolution, a classical way to describe a curve joining two boundary points of a simply-connected domain in conformally invariant terms (see [9] for a more classical definition). This version is called Chordal Loewner Evolution. I. Binder (B) Department of Mathematics, University of Toronto, Toronto, Canada e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. Abakumov et al. (eds.), Extended Abstracts Fall 2019, Trends in Mathematics 12, https://doi.org/10.1007/978-3-030-74417-5_7

43

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I. Binder

Since we want to describe the curve up to a conformal transformation, it is enough to consider in a canonical domain, say H, with two boundary points A = 0, B = ∞. Parameterize the curve and consider conformal mapping gt from H \ γ[0, t] back to H with hydrodynamic normalization at ∞: 2a(t) +O gt (z) = z + z



1 |z|2

 .

Let us re-parameterize the curve so that a(t) = t. Let λ(t) := gt (γ(t)) ∈ C(R+ ). Then the family (gt (z))t≥0 satisfies the Löwner equation: 2 ∂gt (z) = . ∂t gt (z) − λ(t) The opposite is almost true: given continuous λ(t), the solution to Löwner equation gives a conformal map from complement of a “curve” (which does not have to be a simple curve. In general, it does not even have to be a non-locally connected set). λ(t) is called its driving function.

2 Schramm–Loewner Evolution and Critical Interfaces According to Physics predictions, the scaling limit of critical interfaces γ should be random curves, i.e. probability measures (laws) on the space of curves, that satisfy: (1) Conformal invariance. (2) Domain Markov Property: The law of γ[t + T, ∞) is the same as the law of the exploration process in  \ γ[0, T ]. If we assume that these curves are almost surely Loewner curves, it can be rewritten in terms of its (random) driving function λ: the law of λ(t + T ) − λ(T ) is the same as the law of λ(t). This and scale invariance implies that λ(t) = B(κt) for some κ. B(t) is the standard Brownian motion started at 0. Definition (Schramm) A random curve driven by B(κt) is called Schramm-Löwner Evolution (SLEκ ). It is defined in an arbitrary simply-connected domain  as an image of the process in the halfplane by the corresponding conformal map. Rohde and Schramm in [15], SLE is a.s. a curve in H, which is • a simple curve, κ ∈ [0, 4]; • a self-touching curve, κ ∈ (4, 8); • a space filling curve, κ ≥ 8. The same is true for general simply connected domains [10]. The proofs of the convergence of critical lattice interfaces to SLE curves usually consist of three parts. First, the convergence is established for one observable of

Rate of Convergence of Critical Interfaces to SLE Curves

45

Fig. 1 Unforced crossing by γ[τ , 1]

a discrete model to a corresponding conformally-invariant observable for the SLE curve. Then it is established that the driving function of the discrete interface converges to the corresponding re-scaling of the Brownian motion. The last step is to establish the convergence of random interfaces themselves to SLE curves, in the uniform metric. While the first step is definitely model-dependent, the next two steps should be uniform for all (reasonable) models. The fundamental result in this direction was established by Kemppainen and Smirnov in [12]. In that paper they introduced the following geometric condition for a family of random curves. We call this condition KS-condition. Choose a parametrization for all curves γ : [0, 1] → C. The family of probability measures (Pn )n>0 satisfies: (KS) a geometric bound on unforced crossings if ∃C > 1 such that for any 0 < n < Cr ≤ R, for any stopping time 0 ≤ τ ≤ 1 and for any annulus A = A(z 0 , r, R) P (unforced crossing by γ[τ , 1]|γ[0, τ ])
0. KS condition is proved for a wide class of models, such as critical site percolation on hexagonal lattice, FK and Spin Ising models, and harmonic explorer. Because of this, KS condition lead to a significant simplification of the prove of the existence of the scaling limit for those models. The next natural question is to investigate the rate of convergence of a discrete model interface to the corresponding SLE curve. It is predicted in Physics literature that this rate should be polynomial in lattice step, i.e. that the interface on a lattice with mesh size n1 should be n −ψ -close (for some model dependent constant ψ > 0) to the corresponding SLE curve in the Prokhorov metric (with respect to uniform distance between curves). The first result of this type was established by Benes, Kozdron, and Viklund in [1, 11], where the convergence was established for Loop Erased Random Walk.

46

I. Binder

We aim to establish the general framework for proving polynomial rate of convergence of families of random curves interfaces to SLE. It starts with finding a discrete observable which converges polynomially fast to the corresponding observable for SLE curve. The second step is establishing the polynomial rate of the convergence of the driving function to the rescaled Brownian motion. Finally, one needs to establish the polynomially-fast convergence of the interfaces themselves. As before, the first step is model-dependent. In [6], Larissa Richards and I carry out Steps 2 and 3 for essentially any model satisfying KS condition. Please see below for the precise formulation.

3 Critical Site Percolation For the critical percolation on the hexagonal lattice the first step was done in [5]. Let me quickly describe the result that we proved. In his celebrated work on the critical site percolation on the hexagonal lattice [19], Smirnov established the existence and conformal invariance of the scaling limit of the exploration process: an interface between the open and closed sites for a domain with Dobrushin boundary conditions. As a corollary, one can identify this process with SLE6 . The crucial step in the proof was the convergence of the Cardy–Smirnov observable. Introduced in [19], they provide combinatorial description of the discrete approximation of the mapping ϕ to the standard equilateral triangle  with the vertices {1, e2πi/3 , e−2πi/3 }. More specifically H An (z) + e2πi/3 H Bn (z) + e−2πi/3 HCn (z) converge to ϕ , see Fig. 2. Cardy–Smirnov observables are a “complexification” of the Cardy’s formula for the probability of crossing a conformal rectangle, predicted by Cardy [7]. Cardy– Smirnov observables coincide with Cardy’s observable when z ∈ ∂. In [3, 4], we proved that Cardy–Smirnov observables also converge for the modified bond percolation when Mdim(∂) < 2. In [5], we established the polynomial rate of convergence for Cardy–Smirnov observable:

Fig. 2 Cardy–Smirnov observables

Rate of Convergence of Critical Interfaces to SLE Curves

47

Theorem ([5]) Let  be a bounded simply connected domain, A, B, C be prime ends at the boundary. Then for some universal constant ψ > 0 and for large enough n,   n  H (z) + e2πi/3 H n (z) + e−2πi/3 H n (z) − ϕ (z) < n −ψ . A B C For modified bond percolation, the estimate holds only when Mdim(∂) < 2. The proof relies on a careful analysis of the boundary behaviour of conformal maps and their discrete analytic approximations as well as a Percolation construction of the Harris systems.

4 Polynomial Rate of Convergence: A General Framework The second and the third step of the program are carried for a wide class of curves in [6]. We prove Theorem Suppose that the family  = (γ n (, A, B))n of probability measures on curves in simply-connected domains  joining two boundary prime ends A and B, which satisfies the KS Condition and the domain Markov property. Assume that for all simply-connected , there exists some T < ∞, s ∈ (0, 1) and n 0 () such that sup sup n 0 (t ) < ∞. Furthermore assume that for every n ≥ n 0 and all vertices n

t∈[0,T ]

v of n , the 1/n-discrete approximation of , the following holds: n (1) There is a discrete almost martingale observable Hn = H( n ,a n ,bn ) associated n n n with the curve γ . That is, Ht = H(nt ,γ n (t),bn ) is almost a martingale (for any fixed v) with respect to the (discrete) interface γ n growing from a n :

  n  H (v) − E[H n (v) |Dt ] ≤ n −s .  t (2) There is a continuous C 3 function h on the upper half plane H such that (a) ∂x h − ∂x2 h is not locally constant in any neighbourhood (b) H n is polynomially close to its continuous conformally invariant counterpart h:   n  H (v) − h ◦ −1 (v) ≤ n −s  where  is the map of H to  with hydrodynamic parametrization. Then there is a coupling of γ˜ n := −1 (γ n ) with Brownian motion B(t), t ≥ 0, with  P





˜ >n sup d∗ γ˜ (t), γ(t)

t∈[0,T ]

n

 −u

< n −u

48

I. Binder

for√ some u ∈ (0, 1) where γ˜ denotes the chordal SLEκ path for κ ∈ (0, 8) in H driven by κB(t) and both curves are parameterized by capacity. Moreover, if  is an α-Hölder domain, then under the same coupling SLE curve in the image is polynomially close to the original discrete curve:  P



−1



˜ >n sup d∗ γ (t), φ (γ(t)) n

t∈[0,T ]

 −v

< n −v

where v depends only on α and u. Note that the analogous statement holds for another version of Loewner chains, the radial Loewner chains. The main tool of the proof is the tip structure modulus. Defined by Viklund in [11], it is a geometric gauge of the regularity of a Loewner curve in the capacity parameterization. It is an analogue of a classical structure model defined by Warschawski in [21]. Informally, the tip structure modulus is the maximal distance the curve travels into a fjord with opening smaller than when viewed from the point toward which the curve is growing, see Fig. 3. The result of [11] that we use is the following sufficient condition for the polynomial closeness of Loewner curves: Theorem ([11]) Assume that γ1 , γ2 are two Lówner curves with driving functions W1 , W2 . Assume that for some r > 0, p > 0 1. (Driving functions are close) |W1 (t) − W2 (t)| < n −r , t ≤ T . 2. (Tip modulus estimate for both curves) ηti p (n − p ) ≤ Cn − pr . 3. A derivative estimate, just for γ2 .

Fig. 3 Tip structure modulus

Rate of Convergence of Critical Interfaces to SLE Curves

49

Then, for some q depending on parameters in 1–3, distsup (γ1 (t), γ2 (t)) ≤ Cn −q In [11], Viklund also establishes Condition 3 for any SLEκ with κ < 8. To establish the third step of our program, we prove in [6] that KS condition actually implies Condition 2. In our upcoming work with Dmitry Chelkak and Larissa Richards, we use our general framework to establish the polynomial rate of convergence of Ising interface and Harmonic Explorer to the corresponding SLE curves. Acknowledgements The research of Binder was partially supported by an NSERC Discovery grant.

References 1. C. Beneš, F.J. Viklund, M.J. Kozdron, On the rate of convergence of loop-erased random walk to SLE2 . Commun. Math. Phys. 318(2), 307–354 (2013) 2. I. Binder, L. Chayes, H.K. Lei, On the rate of convergence for critical crossing probabilities. Ann. Inst. Henri Poincaré Probab. Stat. 51(2), 672–715 (2015) 3. I. Binder, L. Chayes, H.K. Lei, On convergence to SLE6 I: conformal invariance for certain models of the bond-triangular type. J. Stat. Phys. 141(2), 359–390 (2010) 4. I. Binder, L. Chayes, H.K. Lei, On convergence to SLE6 II: discrete approximations and extraction of Cardy’s formula for general domains. J. Stat. Phys. 141(2), 391–408 (2010) 5. I. Binder, L. Chayes, H.K. Lei, On the rate of convergence for critical crossing probabilities. Ann. Inst. Henri Poincaré Probab. Stat. 51(2), 672–715 (2015) 6. I. Binder, L. Richards, Convergence rates of random discrete model curves approaching SLE curves in the scaling limit (2020). Preprint 7. J.L. Cardy, Critical percolation in finite geometries. J. Phys. A 25(4), L201–L206 (1992) 8. D. Chelkak, H. Duminil-Copin, C. Hongler, A. Kemppainen, S. Smirnov, Convergence of Ising interfaces to Schramm’s SLE curves. C. R. Math. Acad. Sci. Paris 352(2), 157–161 (2014) 9. P.L. Duren, Univalent Functions (Springer, Berlin, 1983) 10. C. Garban, S. Rohde, O. Schramm, Continuity of the SLE trace in simply connected domains. Israel J. Math. 187, 23–36 (2012) 11. F.J. Viklund, Convergence rates for loop-erased random walk and other Loewner curves. Ann. Probab. 43(1), 119–165 (2015) 12. A. Kemppainen, S. Smirnov, Random curves, scaling limits and Loewner evolutions. Ann. Probab. 45(2), 698–779 (2017) 13. G.F. Lawler, O. Schramm, W. Werner, Conformal invariance of planar loop-erased random walks and uniform spanning trees [mr2044671], in Selected Works of Oded Schramm. Selected Works in Probability and Statistics, Vol. 1, 2 (Springer, New York, 2011), pp. 931–987 14. G.F. Lawler, F. Viklund, Convergence of loop-erased random walk in the natural parametrization (2016) 15. S. Rohde, O. Schramm, Basic properties of SLE. Ann. Math. (2) 161(2), 883–924 (2005) 16. O. Schramm, Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118, 221–288 (2000) 17. O. Schramm, S. Sheffield, Harmonic explorer and its convergence to SLE4 . Ann. Probab. 33(6), 2127–2148 (2005)

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18. O. Schramm, S. Sheffield, Contour lines of the two-dimensional discrete Gaussian free field. Acta Math. 202(1), 21–137 (2009) 19. S. Smirnov, Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math. 333(3), 239–244 (2001) 20. S. Smirnov, Critical percolation and conformal invariance, in XIVth International Congress on Mathematical Physics (World Scientific Publishing, Hackensack, 2005), pp. 99–112 21. S.E. Warschawski, On the degree of variation in conformal mapping of variable regions. Trans. Am. Math. Soc. 69, 335–356 (1950)

Toeplitz and Hankel Operators on Bergman Spaces M. Bourass, O. El-Fallah, I. Marrhich, and H. Naqos

Abstract Let ϕ be a regular subharmonic weight on the unit disc and let A2ωϕ be the weighted Bergman space associated with ωϕ (z) = e−2ϕ(z) . We consider compact Hankel operators Hφ , with conjugate analytic symbols φ, acting on A2ωϕ . We give a lower and an upper estimates of the trace of h(|Hφ |), where h is a convex function . Next, we give asymptotic estimates of their singular values. We also consider the similar problem for Toeplitz operators.

1 Preliminaries We present in this note some recent results concerning the asymptotic behavior of the singular values of compact Toeplitz and compact Hankel operators with anti-analytic symbols acting on large Bergman spaces. All these results are contained in [6, 7]. Let ω =: e−2ϕ be a weight on the unit disc D. The weighted Bergman space A2ω associated with ω is the Hilbert space of holomorphic functions given by A2ω



= f ∈ Hol(D) :  f ω :=



1/2 | f (z)| d Aω (z) 2

D

 < ∞ , d Aω := ωd A,

M. Bourass · O. El-Fallah (B) · H. Naqos Faculty of sciences, Mohammed V University in Rabat, CeReMAR -LAMA-, B.P. 1014 Rabat, Morocco e-mail: [email protected] M. Bourass e-mail: [email protected] H. Naqos e-mail: [email protected] I. Marrhich Laboratoire Modélisation, Analyse, Contrôle et Statistiques, Faculty of sciences Ain-Chock, Hassan II University of Casablanca, Maarif, B.P. 5366 Casablanca, Morocco e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. Abakumov et al. (eds.), Extended Abstracts Fall 2019, Trends in Mathematics 12, https://doi.org/10.1007/978-3-030-74417-5_8

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where Hol(D) denotes the space of all holomorphic functions on D. In this note we consider the class of weights, W0 , introduced by Hu, Lv and Schuster in [11]. Namely, let L0 be the class given by L0 = {τ ∈ i p(D, R+ ) : lim − τ (z) = 0}. |z|→1

We say that ω =: e−ϕ ∈ W0 if ϕ ∈ C 2 is strictly subharmonic and there exists τ ∈ L0 such that τ 2  1/ϕ. The class W0 includes the weights considered in [5, 12, 14]. It is known and easy to check that A2ω is a reproducing kernel Hilbert space. It’s kernel will be denoted by K (z, w). The Bergman projection Pω is the orthogonal projection from L 2ω onto A2ω and it is given by  Pω ( f ) =

D

f (w)K (., w)d Aω ,

f ∈ L 2ω .

It should be noted that Pω can be extended to all functions f ∈ L 1ω such that f K z ∈ L 1ω . Let μ be a positive Borel measure μ on D. The Toeplitz operator acting on A2ω and induced by μ is given by  Tμ ( f )(z) =

D

f (ζ)K (z, ζ)ω(ζ)dμ(ζ).

If dμ = gd A, with g ∈ L 1ω , then Tμ ( f ) = Pω (g f ), for f such that g f ∈ L 2ω . Let φ ∈ A2ω be such that φK z ∈ A2ω , for all z ∈ D. The (big) Hankel operator Hφ with symbol φ is the densely defined operator on A2ω given by Hφ f = φ f − Pω (φ f ), where f =



ci K zi , ci ∈ C and z i ∈ D.

1≤i≤n

A direct computation gives the following useful formula (Hφ K a )(z) = (φ(z) − φ(a))K a (z), z, a ∈ D.

(1)

Note also that the integral representation of Hφ is given by  Hφ f (z) =

D

(φ(z) − φ(w)) f (w)K (z, w)d Aω (w), z ∈ D.

2 Toeplitz Operators Toeplitz operators on Bergman spaces were studied by several authors [3, 8, 12, 13, 15]. The boundedness, compactness and membership in Schatten classes S p ,

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p > 0, of Tμ can be expressed by the mean function μˆ of μ or in terms of the Berezin transform associated with Tμ . First, we recall kernel estimates obtained in [11]. For ω ∈ W0 there exists α1 > 0 such that |K (z, w)| 

eϕ(z) eϕ(w) , |z − w| < α1 τ (z), τ (z)τ (w)

and for each M > 0 there exists C > 0 such that |K (z, w)| ≤ C

eϕ(z) eϕ(w) τ (z)τ (w)



min{τ (z), τ (w)} |z − w|

M , z = w.

We say that a sequence (z j ) j∈N ⊂ D is a lattice for A2ω if there exists α ∈ (0, α1 ) and m > 1 such that • (D(ατ (z j ))) j≥1 and (D(mατ (z j ))) j≥1 are finite multiplicity coverings of D. / D(ατ (z k )) i f j = k and D(ατ (z)) ⊂ D(mατ (z j )), j ≥ 1. • zj ∈ z∈D(ατ (z j ))

It is standard that such lattices exist for ω ∈ W0 [8]. In the sequel, (z j ) j≥1 , α and m will be fixed. The mean function of μ with respect to ω is given by μ(z) ˆ := μ(D(z,ατ (z))) ˜μ (z) = Tμ k z , k z , where k z = K z /K z  . The Berezin transform of T is T μ 2 τ (z) is the normalized reproducing kernel of A2ω . Explicit relations between μˆ and T˜μ are given in [8]. We have the following result. Theorem 2.1 Let ω = e−2ϕ ∈ W0 . Let μ be a positive Borel measure on D. Then 1. Tμ is bounded ⇐⇒ μˆ is bounded on D ⇐⇒ T˜μ is bounded on D. 2. Tμ is compact ⇐⇒ lim − μ(z) ˆ = 0 ⇐⇒ lim − T˜μ (z) = 0. |z|→1

3. Let p > 0 and let dλω = ϕ d A. We have

|z|→1

Tμ ∈ S p ⇐⇒ μˆ ∈ L p (dλω ) ⇐⇒ T˜μ ∈ L p (dλω ). It should be noted that the description of membership in S p in terms of T˜μ is valid for the unweighted Bergman space on D only for p > 1/2. In what follows, we will consider measures μ for which the corresponding Toeplitz ˆ j )) j . operators is compact. Let (a j (μ)) j be the decreasing rearrangement of (μ(z For a positive compact operator on a Hilbert space, the decreasing sequence of the eigenvalues of T will be denoted by (λ j (T )) j . We have the following result Theorem 2.2 Let ω ∈ W0 . Let μ be a positive Borel measure on D such that Tμ is compact on A2ω . There exists B > 0, which depends only on ω, such that n n n  B p p2  p p a (μ) ≤ λ j (Tμ ) ≤ a (μ), B j=1 j p j=1 j j=1

p ∈ (0, 1), n ≥ 1.

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Proof For the proof we refer to [7]. As a consequence, it is easy to prove the following result.

Theorem 2.3 Let ω ∈ W0 . Let ρ : [1, +∞) → (0, +∞) be an increasing function such that ρ(x)/x A is decreasing for some A > 0. Let μ be a positive Borel measure on D such that Tμ defines a compact operator on A2ω . Then 1. λ j (Tμ ) = O (1/ρ( j)) ⇐⇒ a j (μ) = O (1/ρ( j)). 2. λ j (Tμ )  1/ρ( j) ⇐⇒ a j (μ)  1/ρ( j). Once more, one can prove, under the same hypothesis the following: λ j (Tμ )  1/ρ( j) ⇐⇒ μ˜ j  1/ρ( j), where (μ˜ j ) is the decreasing rearrangement of (T˜μ (z j )) j .

3 Hankel Operators Hankel operators with anti-analytic symbols on Bergman spaces were studied in several papers. We refer the reader to the following articles [1, 2, 4, 9, 10]. First, we introduce some spaces which play an important role in the theory of Hankel operators. Let us denote B ω, p = {φ ∈ Hol(D) : φ τ  L p (dλω ) < ∞},

p ∈ (0, ∞],

B0ω = {φ ∈ Hol(D) : lim − τ (z)|φ (z)| = 0}. |z|→1

We have the following classical result. Theorem 3.1 Let ω ∈ W0 . The following are true. 1. Hφ is bounded (resp. compact) if and only if φ ∈ B ω,∞ (resp. φ ∈ B0ω ). 2. Let p ≥ 1. Then Hφ ∈ S p ⇐⇒ φ ∈ B ω, p . Remark that B ω,1 is reduced to constant functions, for all ω ∈ W0 . Then, Theorem 3.1 implies that if Hφ ∈ S1 then Hφ = 0. More precisely, let pω =: inf{ p > 0 : B ω, p is not reduced to constant functions}. Clearly, from Theorem 3.1, we have that pω is the infimum of p > 0 for which there exists a non constant analytic function φ ∈ B ω,∞ such that Hφ ∈ S p . Note that the space B ω,2 coincides with the classical Dirichlet space for all ω ∈ W0 . In particular, we have pω ∈ [1, 2]. For example, if ω = e−2ϕ ∈ W0 is such that ϕ(z)  1/(1 − |z|2 )2+β , β > 0, then pω = 2(1+β) . 2+β

Toeplitz and Hankel Operators on Bergman Spaces

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3.1 Trace Estimates The following result plays an important role for estimating the singular values of compact Hankel operators. Theorem 3.2 Let ω = e−2ϕ ∈ W0 and let φ ∈ B0ω . Let h be an increasing convex function such that h(0) = 0. Then there exists B > 0, which depends only on ω, such that    

1  τω (z)|φ (z)| dλω (z) ≤ Tr (h(|Hφ |)) ≤ h h Bτω (z)|φ (z)| dλω (z). B D D Let T be a compact operator between Hilbert spaces. The decreasing sequence of the singular values of T (the eigenvalues of (T ∗ T )1/2 ) will be denoted by (s j (T )) j≥1 .  Let R+ φ,ω the decreasing rearrangement of the function τω |φ | with respect to dλω . Namely,  R+ φ,ω (x) := sup{t ∈ (0, τ φ ∞ ] : Rφ,ω (t) ≥ x}, where Rφ,ω is the distribution function given by Rφ,ω (t) := λω ({z ∈ D : τω (z)|φ (z)| > t}). As a first consequence of Theorem 3.2, we have Theorem 3.3 Let ω = e−2ϕ ∈ W0 and let φ ∈ B0ω;∞ . Let ρ be an increasing function such ρ(x)/x γ is decreasing for some γ < 1. Then 1. s j (Hφ ) = O(1/ρ( j)) ⇐⇒ R+ φ,ω ( j) = O(1/ρ( j)). 2. s j (Hφ )  1/ρ( j) ⇐⇒ R+ ( φ,ω j)  1/ρ( j).

3.2 Critical Decay In what follows, the operator Hz will play an important role in the study of Hankel operators with regular symbols. Note that Theorem 3.3 gives us an efficient way to estimate the singular values of Hz . Indeed, it is not difficult to estimate R+ z,ω ( j). 2+β

− 2(1+β) . Then, For example, if τω2 (z)  (1 − |z|2 )2+β , with β ≥ 0, then R+ z,ω ( j)  j 2+β

Theorem 3.3 implies that s j (Hz )  j − 2(1+β) . We assert, in the following theorem, that the critical decay of (sn (Hφ ))n is attained by the symbol φ = z. For the proof we refer to [6].

Theorem 3.4 Let ω = e−2ϕ ∈ W0 be such that ϕ is equivalent to a radial function and let φ ∈ B0ω . Then 1. s j (Hφ ) = o(s j (Hz )) =⇒ Hφ = 0.

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2. s j (Hφ ) = o(R+ z,ω ( j)) =⇒ Hφ = 0. The Hardy space H p , p ≥ 1, consists of the analytic functions f on D such that p

1 0≤r 0, then . We have the following result. pω = 2(1+β) 2+β Theorem 3.5 Let ω=e−2ϕ ∈W0 be such that ϕ(z)  (1 − |z|)−2−β where β > 0. Let φ ∈ B0ω . We have s j (Hφ ) = O( j −1/ pω ) ⇐⇒ φ ∈ H pω . We can obtain a more precise result if ω is radial. Indeed, in this case it is easy to see that z j is an eigenfunction of Hz∗ Hz . Then by a direct computation we have  s j (Hz ) =

z j+1 2 z j 2 − j−1 2 j 2 z  z 

1/2 ,

j ≥ 1.

Then using Pushnitski result on asymptotic orthogonality [16], combined with Theorem 3.5, we obtain the following result. Theorem 3.6 Let ω = e−2ϕ ∈ W0 be a radial weight such that ϕ(z)  (1 − 1 |z|)−2−β where β > 0. Let φ ∈ B0ω . Suppose that s j (Hz ) ∼ γ j − pω , for some γ ∈ (0, ∞). Then 1 s j (Hφ ) ∼ γφ  H pω j − pω , φ ∈ H pω . Proof For the proof we refer to [6].



Acknowledgements Research partially supported by “Hassan II Academy of Science and Technology” for Bourass and El-Fallah.

References 1. J. Arazy, S.D. Fisher, S. Janson, J. Peetre, Membership of Hankel operators on the ball in unitary ideals. J. Lond. Math. Soc. 2(3), 485–508 (1991) 2. J. Arazy, S.D. Fisher, J. Peetre, Hankel operators on weighted Bergman spaces. Am. J. Math. 110(6), 989–1053 (1988) 3. H. Arroussi, J. Pau, Reproducing kernel estimates, bounded projections and duality on large weighted Bergman spaces. J. Geom. Anal. 1–29 (2014) 4. S. Axler, The Bergman space, the Bloch space, and commutators of multiplication operators. Duke Math. J. 53(2), 315–332 (1986) 5. A. Borichev, R. Dhuez, K. Kellay, Sampling and interpolation in large Bergman and Fock spaces. J. Funct. Anal. 242(2), 563–606 (2007)

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6. M. Bourass, O. El-Fallah, H. Marrhich, I. Naqos, On the singular values of Hankel operators on Bergman spaces. Preprint 7. O. El-Fallah, M. El Ibbaoui, On the eigenvalues of Toeplitz operators on Fock and Bergman spaces. Preprint 8. O. El-Fallah, H. Mahzouli, I. Marrhich, H. Naqos, Asymptotic behavior of eigenvalues of Toeplitz operators on the weighted analytic spaces. J. Funct. Anal. 270(12), 4614–4630 (2016) 9. M. Engliš, R. Rochberg, The Dixmier trace of Hankel operators on the Bergman space. J. Funct. Anal. 257(5), 1445–1479 (2009) 10. P. Galanopoulos, J. Pau, Hankel operators on large weighted Bergman spaces. Ann. Acad. Sci. Fenn. Math. 37, 635–648 (2012) 11. Z. Hu, X. Lv, A.P. Schuster, Bergman spaces with exponential weights. J. Funct. Anal. 276(5), 1402–1429 (2019) 12. P. Lin, R. Rochberg, Trace ideal criteria for Toeplitz and Hankel operators on the weighted Bergman spaces with exponential type weights. Pacific J. Math. 173(1), 127–146 (1996) 13. D.H. Luecking, Trace ideal criteria for Toeplitz operators. J. Funct. Anal. 73(2), 345–368 (1987) 14. J. Marzo, J. Ortega-Cerdà, Pointwise estimates for the Bergman kernel of the weighted Fock space. J. Geom. Anal. 19(4), 890–910 (2009) 15. V.L. Oleinik, G.S. Perel’man, Carleson’s imbedding theorem for a weighted Bergman space. Math. Notes 47(6), 577–581 (1990) 16. A. Pushnitski, Spectral asymptotics for Toeplitz operators and an application to banded matrices, in The Diversity and Beauty of Applied Operator Theory (Springer International Publishing, Cham, 2018), pp. 397–412

Bounds for Zeta and Primes via Fourier Analysis Emanuel Carneiro

Abstract This is a brief summary of a seminar talk, with the same title, delivered at the CRM—Barcelona in October, 2019. We discuss here some of the recent bounds for objects related to the Riemann zeta-function and prime gaps via the use of Fourier analysis machinery. Certain interesting Fourier optimization problems come into play, naturally related to our number theoretical entities. This is based on a joint work with M. Milinovich and K. Soundararajan.

1 The Smallest Bandlimited Function Define the Fourier transform of an integrable function f : R → R by  f (t) =

 R

e−2πit x f (x) dx.

We say that a function is bandlimited if its Fourier transform is compactly supported. The following optimization problem is a classical one in Fourier analysis. Problem 1 Assume that f : R → R is a continuous, integrable and nonnegative function such that supp(  f ) ⊂ [−1, 1] and f (0) = 1. What is the minimal possible value of  f  L 1 (R) ? The exact solution of a constrained Fourier optimization problem like the one above usually requires two main tools. First, one needs to identify the correct extremality tool, with which one proves that the desired optimal value cannot surpass a certain threshold. Then, once the correct extremality tool is in place, it possibly gives hints on the potential extremal functions, and one is left with the task of actually constructing them, if possible. This would be the constructive tool. E. Carneiro (B) ICTP - The Abdus Salam International Centre for Theoretical Physics, Strada Costiera, 11, I, 34151 Trieste, Italy e-mail: [email protected]; [email protected] IMPA - Instituto de Matemática Pura e Aplicada, Rio de Janeiro 22460-320, Brazil © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. Abakumov et al. (eds.), Extended Abstracts Fall 2019, Trends in Mathematics 12, https://doi.org/10.1007/978-3-030-74417-5_9

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For instance, for Problem 1 above, the correct extremality tool is the Poisson summation formula. Under the hypotheses of the problem, an old result of Plancherel and Pólya [14] guarantees that f is of bounded variation and hence Poisson summation holds pointwise, yielding   f  L 1 (R) =

R

f (x) dx =  f (0) =



f (n) ≥ f (0) = 1.

(1)

n∈Z

If equality occurs in (1) then one must have f (n) = 0 for all n ∈ Z \ {0} and, given that f is nonnegative, this implies that f  (n) = 0 for all n ∈ Z \ {0}. At this point, well-known interpolation formulas for bandlimited functions with derivative data [16, Theorem 9] can be applied, and one arrives at the conclusion that the unique  2 extremal function is the Fejér kernel f (x) = sin (πx)/(πx) . Let us now propose an innocent variation of this problem, excluding the nonnegativity condition. Problem 2 Assume that f : R → R is a continuous and integrable function such that supp(  f ) ⊂ [−1, 1] and f (0) = 1. What is the minimal possible value C of  f  L 1 (R) ? One may wonder if the Fejér kernel is still optimal, but after a little bit of thinking one arrives at examples like f (x) = cos(2πx)/(1 − 16x 2 ) that verifies  f  L 1 (R) = 0.9259... As a matter of fact, this is still an open problem after several decades of research (this problem is also referred to as the C-L constant of Nikol’skii). Previous works on this problem include those of Andreev, Konyagin, and Popov [1] and Gorbachev [11]. The current best bounds are due to Hörmander and Bernhardsson [12] 0.92433603021... < C < 0.92433603046... In particular, it is known that an extremal function for this problem exists and is unique [4, 12]. Similar constrained Fourier optimization problems have been recently used in bounding the modulus and the argument of the Riemann zeta-function on the critical line under the Riemann hypothesis [2, 3, 5], and also in connection to the sphere packing problem [7, 8, 17].

2 Prime Gaps and RH Let pn denote the n-th prime. Assuming the Riemann hypothesis (RH), Cramér [6] proved that pn+1 − pn ≤ c, (2) lim sup √ pn log pn n→∞ where c is a universal constant. The order of magnitude of the gap between consecutive primes (under RH) has never been improved, although it is conjectured that

Bounds for Zeta and Primes via Fourier Analysis

61

pn+1 − pn = O(log2 pn ). The efforts over the last decades have then been concentrated in reducing the value of the constant c in (2). Building upon the works of Goldston [10] (c = 4) and of Ramaré and Saouter [15] (c = 8/5), Dudek [9, Theorem 1.3] arrived at (2) with constant c = 1. In [4] we overcame the barrier c = 1 in this problem by exploring novel connections to Fourier analysis. Theorem 1 (Carneiro, Milinovich and Soundararajan) Assume the Riemann hypothesis. Then pn+1 − pn ≤ c < 1. lim sup √ pn log pn n→∞ This result is established in [4, Theorem 4] with c = 0.84. The three main ingredients in the proof are: (i) the explicit formula; (ii) derived optimization problems in Fourier analysis and (iii) the Brun-Titchmarsh inequality. In this note let us present an overview of a slightly more modest bound using only the tools (i) and (ii) above.1 This already conveys some of the main ideas involved. We require the following explicit formula whose proof can be found in [13, Theorem 5.12]. Lemma 1 (Guinand-Weil explicit formula) Let h(s) be analytic in the strip |Im s| ≤ 1 + ε for some ε > 0, and assume that |h(s)|  (1 + |s|)−(1+δ) for some δ > 0 when 2 |Re s| → ∞. Then  ρ

 h

ρ− i

1 2



=h





 ∞ 1 1 1  1 iu +h − − h(0) log π + du h(u) Re + 2i 2π 2π −∞  4 2



1  (n)  log n − log n − h + h , √ 2π 2π 2π n

1 2i



n≥2

where ρ = β + iγ are the non-trivial zeros of ζ(s),   /  is the logarithmic derivative of the gamma function, and (n) is the Von-Mangoldt function defined to be log p if n = p m with p a prime number and m ≥ 1 an integer, and zero otherwise. Proof of Theorem 1 Let f (x) = cos(2πx)/(1 − 16x 2 ). Let 0 <  ≤ 1 and 1 < a be free parameters to be properly chosen later. We set g(z) :=  f (z) and apply the explicit formula to h(z) = g(z)a i z to get the following inequality  



    1  log a  g 1 a 1/2 + g − 1 a −1/2  ≤ |g(γ)| +  g − log π  2i  2i 2π  2π  γ 

  ∞   1  1 iu du  g(u) a iu Re +  + 2π −∞  4 2   



  1  (n)  log(n/a) log na .  g +  + g − √  2π 2π 2π n   n≥2

1

In fact, this was our initial approach for the result that later became [4, Theorem 4].

(3)

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Assume now that√ for a certain c > 0 there exists a sequence of x → ∞ such that the interval [x, x + c x log x] contains no primes (we want to take advantage of this information to make the sum over prime powers in (3) small). For each such x, we set a and  such that

√ [x, x + c x log x] = a e−2π , a e2π

(4)

and observe that (allowing the implicit constants in the big-O notation here to depend on c)

2

log x log x log x =c √ +O (5) 4π = log 1 + c √ x x x and



√ log x 1/2 a = x 1+c √ = x + O( x log x). x

(6)

The rest of the proof is an asymptotic analysis of the terms in (3) as x → ∞. Error terms. Since supp( g ) ⊂ [−, ], and the interval (4) has no primes, one can verify that 

  1  (n) log(n/a)

log na    g + g − √  = O(1).    2π 2π 2π n n≥2   a As a → ∞ we have  g − log = 0. Using Stirling’s formula 2π −1

 (s) 

= log s +

O(|s| ) we also find out that 

  ∞   1   1 iu iu  + du  = O(1). g(u) a Re  2π  4 2 −∞ Main terms. Inequality (3) is then reduced to 

 

  g 1 a 1/2 + g − 1 a −1/2  ≤ |g(γ)| + O(1).  2i  2i γ

Observe that

1 g 2i



= f  =

 2i

1 −1



 =

1 −1

 f (t) dt + 

=  f (0) + O(2 ).

eπt  f (t) dt 

1 −1

 πt  e −1  f (t) dt

(7)

Bounds for Zeta and Primes via Fourier Analysis

63

We may similarly estimate g(− 2i1 ) and, therefore, the left-hand side of (7) verifies

1 g 2i

a

1/2

  1 a −1/2 =  f (0) a 1/2 + a −1/2 + O(2 a 1/2 ). +g − 2i

(8)

Let N (x) denote the number of zeros ρ = β + iγ of ζ(s) with 0 < γ≤ x. Using the x x x fact that N (x) = 2π log 2π − 2π + O(log x), we evaluate the sum γ |g(γ)| using summation by parts to get  γ

1 |g(γ)| = 2π



∞

−∞

|g(x)| log+

  |x| dx + O g∞ + g  (x) log+ |x|1 , 2π

where log+ x = max{log x, 0} for x > 0. Recalling that g(x) =  f (x), this yields  γ

1 |g(γ)| = 2π



∞

−∞

| f (y)| log+ |y/2π| dy + O(1) (9)

log(1/2π)  f 1 + O(1). = 2π Conclusion. From (7), (8) and (9) we get  f (0) a 1/2 + O(2 a 1/2 ) ≤

log(1/2π)  f 1 + O(1). 2π

In virtue of (5) and (6) this becomes  f 1 c f (0) log x ≤ log x + O(1) 4π 4π along this sequence of x → ∞. This is only possible if c≤

 f 1 = 0.9259... f (0)

as we wanted to show.

References 1. N.N. Andreev, S.V. Konyagin, A.Y. Popov, Extremum problems for functions with small support. Math. Notes 60(3) (1996). (translated from Mat. Zametki) 2. E. Carneiro, V. Chandee, M.B. Milinovich, Bounding S(t) and S1 (t) on the Riemann hypothesis. Math. Ann. 356(3), 939–968 (2013) 3. E. Carneiro, A. Chirre, Bounding Sn (t) on the Riemann hypothesis. Math. Proc. Camb. Philos. Soc. 164(2), 259–283 (2018)

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4. E. Carneiro, M. Milinovich, K. Soundararajan, Fourier optimization and prime gaps. Comment. Math. Helv. 94(3), 533–568 (2019) 5. V. Chandee, K. Soundararajan, Bounding |ζ( 21 + it)| on the Riemann hypothesis. Bull. Lond. Math. Soc. 43(2), 243–250 (2011) 6. H. Cramér, Some theorems concerning prime numbers. Ark. Mat. Astron. Fysik. 15(5), 1–32 (1920) 7. H. Cohn, N. Elkies, New upper bounds on sphere packings. I. Ann. Math. (2) 157(2), 689–714 (2003) 8. H. Cohn, A. Kumar, S.D. Miller, D. Radchenko, M. Viazovska, The sphere packing problem in dimension 24. Ann. Math. 185, 1017–1033 (2017) 9. A. Dudek, On the Riemann hypothesis and the difference between primes. Int. J. Num. Theory 11(3), 771–778 (2015) 10. D.A. Goldston, On a result of Littlewood concerning prime numbers, Acta Arith. 40(3), 263– 271 (1981/82) 11. D.V. Gorbachev, An integral problem of Konyagin and the (C, L)-constants of Nikol’skii. Trudy Inst. Mat. i Mekh. UrO RAN 11(2), 72–91 (2005) 12. L. Hörmander, B. Bernhardsson, An extension of Bohr’s inequality. Boundary value problems for partial differential equations and applications, 179–194, RMA Res. Notes Appl. Math., 29, Masson, Paris (1993) 13. H. Iwaniec, E. Kowalski, Analytic Number Theory. Am. Math. Soc. Colloq. Publ. 53(2004) 14. M. Plancherel, G. Pólya, Fonctions entiéres et intégrales de Fourier multiples, (Seconde partie) Comment. Math. Helv. 10, 110–163 (1938) 15. O. Ramaré, Y. Saouter, Short effective intervals containing primes. J. Num. Theory 98(1), 10–33 (2003) 16. J.D. Vaaler, Some extremal functions in Fourier analysis. Bull. Am. Math. Soc. 12, 183–215 (1985) 17. M. Viazovska, The sphere packing problem in dimension 8. Ann. Math. 185, 991–1015 (2017)

On Zeros of Solutions of a Linear Differential Equation Igor Chyzhykov and Jianren Long

Abstract We are looking for a function a(z) analytic in the unit disc such that f  + a(z) f = 0 possesses a solution having zeros precisely at the points z k , and the resulting function a(z) has ‘minimal’ growth.

We deal with the following problem (cf. [1, Problem 2]). Problem. Let (z k ) be a sequence of distinct points in the unit disk D without limit points there. Find a function a(z), analytic in D such that the equation f  + a(z) f = 0,

(1)

possesses a solution having zeros precisely at the points z k . Estimate the growth of the resulting function a(z). In [2] and [3] the authors deal with cases where the coefficient a(z) belongs to a weighted Hardy space and the sequence (z n ) is separated. If a(z) is of positive finite order, sharp estimates of the growth of a(z) were obtained in [4]. We complement aforementioned results considering non-separated sequences (z k ) such that its counting function is of zero order while solutions lie outside Korenblum spaces. Obtained results are sharp in some sense. For an analytic function f in D we denote M(r, f ) = max{| f (z)| : |z| = r }, r ∈  (0, 1). Let n ζ (t) = |zk −ζ |≤t 1 be the number of the members of the sequence (z k ) satisfying |z k − ζ | ≤ t. We write 

r

Nζ (r ) = 0

(n ζ (t) − 1)+ dt. t

I. Chyzhykov (B) Faculty of Mathematical and Computer Sciences, University of Warmia and Mazury in Olsztyn, Słoneczna 54, 10-710 Olsztyn, Poland e-mail: [email protected] J. R. Long School of Mathematical Sciences, Guizhou Normal University, Guiyang, Guizhou 550025, China e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. Abakumov et al. (eds.), Extended Abstracts Fall 2019, Trends in Mathematics 12, https://doi.org/10.1007/978-3-030-74417-5_10

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Let ψ : [1, +∞) → R+ be a nondecreasing function. We define ˜ ψ(x) =



x 1

ψ(t) dt. t

Let, in addition, ψ have finite order in the sense of Pólya, i.e. ψ(2x) = O(ψ(x)), x → +∞.

(2)

Remark 1 Polya’s order ρ ∗ [ψ] of ψ ([5]) is characterized by the condition that for any ρ > ρ ∗ [ψ], we have ψ(C x) ≤ C ρ ψ(x) (x, C → ∞). ˜ Remark 2 In the case ψ(x) = ln p x, p ≥ 0 we get ψ(x) = 1 ρ ρ ˜ = (x − 1). case ψ(x) = x , ρ > 0 we have ψ(x)

1 p+1

ln p+1 x, and in the

ρ

For s = [ρ] + 1, where ρ = ρ ∗ [ψ], we consider a canonical product of the form ∞    P(z) = P(z, Z , s) = E An (z), s ,

(3)

n=1

whereE(w, 0) = 1 − w, E(w, s) = (1 − w) exp{w + w2 /2 + ... + w s /s}, s ∈ N, 1 − |z n |2 and An (z) = . This product is an analytic function in D with the zero 1 − z¯ n z  (1 − |z n |)s+1 < ∞. sequence Z = (z n ) provided that z n ∈Z

We are mostly interested in the case where ψ(r ) is a slowly growing function such ˜ ) = O(ψ(r )) as r → +∞. To deal with the problem, that ψ(r ) = O(log r ) and ψ(r we need a new interpolation result. The following theorem complements a theorem from [4]. Theorem 1 Let (z n ) be a sequence of distinct complex numbers in D. Assume that for some nondecreasing unbounded function ψ : [1, +∞) → R+ satisfying (2) we have that  1   1 − |z|  ≤ Cψ , ∃C > 0 ∀z ∈ D : n z 2 1 − |z| and either ∃C > 0 : ∀n ∈ N Nzn or

or

 1   1 − |z |  n ≤ C ψ˜ , 2 1 − |z n |

   ∀n ∈ N : − log (1 − |z n |)|P  (z n )| ≤ C ψ˜

1  , 1 − |z n |

On Zeros of Solutions of a Linear Differential Equation

67

∀n ∈ N : − log |Bn (z n )| ≤ C ψ˜



1  , 1 − |z n |

holds, where Bn (z) = P(z)/E(An (z), a), P is the canonical product defined by (3), s ≥ [ρ] + 1, where ρ is the Polýa order of ψ. Then for any sequence (bn ) satisfying ∃C > 0 : log |bn | ≤ C ψ˜



1  , n∈N 1 − |z n |

there exists an analytic function f in D with the properties f (z n ) = bn and ∃C > 0 : log M(r, f ) ≤ C ψ˜

 1  , r ∈ (0, 1). 1−r

(4)

Theorem 2 Let conditions of Theorem 1 be satisfied. Then there exists an analytic function a in D satisfying (4) such that (1) possesses a solution f having zeros precisely at the points z k , k ∈ N. Corollary 3 If for some ρ > 0 and β > 0 a sequence (z k ) satisfies the conditions ∃C > 0 : n zk ∃C > 0 : Nzk

 1 − |z |  1 k ≤ C logβ , 2 1 − |z k |

 1 − |z |  1 k ≤ C logβ+1 , 2 1 − |z k |

1 ), then there exists a function a analytic in D satisfying log M(r, a) = O(logβ+1 1−r r ∈ (0, 1) such that (1) possesses a solution f having zeros precisely at the points z k , k ∈ N.

This corollary is sharp in the following sense. Theorem 4 For arbitrary η1 , η2 > 0 there exists a sequence of distinct points {z n } in D with the following properties:   1 k| ≤ C logη1 1−|z , k ∈ N; (i) ∃C > 0: n zk 1−|z 2 k|   1−|z k | 1 1+η1 +η2 ≤ C log (ii) ∃C > 0: Nzk , k ∈ N; 2 1−|z k | (iii) (z k ) cannot be the zero sequence of a solution of (1) where log M(r, a) = O log1+η

1 1−r

, η < η2 .

We also obtain an estimate of the growth of a(z) under the assumption of finiteness of the density introduced by Borichev, Dhuez, and Kellay in [6] combined with a separation condition.

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References 1. J. Heittokangas, Solutions of f  + A(z) f = 0 in the unit disc having Blaschke sequence as zeros. Comput. Meth. Funct. Theory 5(1), 49–63 (2005) 2. J. Gröhn, A. Nikolau, J. Rättyä, Mean growth and geometric zero distribution of solutions of linear differential equations. J. d’Anal. Math. 134, 747–768 (2018) 3. J. Gröhn, Solutions of complex differential equation having pre-given zeros in the unit disc. Constr. Approx. 49, 295–306 (2019) 4. Igor Chyzhykov, Iryna Sheparovych, Interpolation of analytic functions of moderate growth in the unit disc and zeros of solutions of a linear differential equation. J. Math. Anal. Appl. 414, 319–333 (2014) 5. D. Drasin, D. Shea, Pólya peaks and the oscillation of positive functions. Proc. Am. Math. Soc. 34, 403–411 (1972) 6. A. Borichev, R. Dhuez, K. Kellay, Sampling and interpolation in large Bergman and Fock space. J. Funct. Anal. 242, 563–606 (2007)

On Riesz Bases of Exponentials for Convex Polytopes with Symmetric Faces Alberto Debernardi and Nir Lev

Abstract This is an extended abstract of our recent paper [3] where we prove that for any convex polytope  ⊂ Rd which is centrally symmetric and whose faces of all dimensions are also centrally symmetric, there exists a Riesz basis of exponential functions in the space L 2 (). The result is new in all dimensions d greater than one.

1 Orthogonal Bases of Exponentials Let  ⊂ Rd be a bounded, measurable set of positive measure. When is it possible to find a countable set  ⊂ Rd such that the system of exponential functions E() = {eλ }λ∈ , eλ (x) = e2πiλx ,

(1)

constitute a basis in the space L 2 ()? The answer depends on what we mean by a “basis”. The best one can hope for is to have an orthogonal basis of exponentials. The problem of which domains admit an orthogonal basis E() has been extensively studied and goes back to Fuglede [4], who conjectured that these domains could be characterized geometrically as the domains which can tile the space by translations. For example, it was proved in [8] that the Fuglede conjecture holds for convex domains in two-dimensional space. In particular, a disk or a triangle in the plane does not have an orthogonal basis of exponentials (this was shown already in [4]). In [5] the conjecture was proved also for three-dimensional convex polytopes. Finally, in a recent paper [12] the Fuglede conjecture for convex domains was settled affirmatively in all dimensions, i.e. it was proved that a convex domain  ⊂ Rd admits an orthogonal basis E() if and only if  can tile the space by translations. On the other hand, the original version of the conjecture (without imposing convexity or any other extra assumptions on the domain ) was disproved in [15]. A. Debernardi · N. Lev (B) Department of Mathematics, Bar-Ilan University, Ramat-Gan 5290002, Israel e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. Abakumov et al. (eds.), Extended Abstracts Fall 2019, Trends in Mathematics 12, https://doi.org/10.1007/978-3-030-74417-5_11

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The counterexample is composed of finitely many unit cubes in a special arithmetic arrangement.

2 Riesz Bases of Exponentials If a domain  does not have an orthogonal basis of exponentials, then a Riesz basis is the next best thing one can hope for. A system of vectors { f n } in a separable Hilbert space H is called a Riesz basis if any f ∈ H admits a unique expansion in a series f = cn f n , and the coefficients {cn } satisfy A f 2 ≤



|cn |2 ≤ B f 2

n

for some positive constants A, B which do not depend on f . There are also other, equivalent definitions of a Riesz basis in a separable Hilbert space H . For example, a system { f n } ⊂ H is a Riesz basis if and only if it is the image of an orthonormal basis under a bounded and invertible linear map, see [17]. If a domain  ⊂ Rd does not have an orthogonal basis of exponential functions, does it have at least a Riesz basis of exponentials? For most domains this question remains unanswered. To give an example for which we do know the answer, consider the case when  is the union of a finite number of intervals on R, say  = ∪nj=1 [a j , b j ]. It was proved in [2] that  has a Riesz basis of exponentials if all the endpoints a j , b j lie in Z. In [14] the existence of such a Riesz basis was proved for n = 2, i.e. the union of two arbitrary intervals (without any constraints on the endpoints). Finally, in the paper [10] a Riesz basis of exponentials was constructed for an arbitrary union of finitely many intervals in R (see also [11] for a multi-dimensional version of this result).

3 Convex Polytopes with Symmetric Faces The present work is concerned with the existence of Riesz bases of exponentials for convex polytopes in Rd . In [13], Lyubarskii and Rashkovskii established the existence of a Riesz basis of exponentials for convex, centrally symmetric polygons in R2 such that all the vertices of the polygon lie on the integer lattice Z2 (one may alternatively assume that the vertices lie on some other lattice, due to the invariance of the problem under affine transformations). The approach in [13] involves methods from the theory of entire functions of two complex variables. A similar result in higher dimensions was obtained in [6, Corollary 3], where the existence of a Riesz basis of exponentials was established for centrally symmetric polytopes in Rd with centrally symmetric facets, such that all the vertices of the

On Riesz Bases of Exponentials for Convex …

71

polytope lie on the lattice Zd . The proof is based on the fact that such a polytope multitiles the space by lattice translates (in connection with this result, see also [7, 9]). In our recent paper [3] we established the existence of a Riesz basis of exponential functions for convex, centrally symmetric polytopes with centrally symmetric faces, without imposing any extra constraints. Our main result can be stated as follows: Theorem 1 Let  ⊂ Rd be a convex polytope which is centrally symmetric and all of whose faces of all dimensions are also centrally symmetric. Then there is a set  ⊂ Rd such that the system of exponential functions E() is a Riesz basis in L 2 (). The result is new in all dimensions d greater than one. Our approach to the proof of Theorem 1 is inspired by the paper [16] due to Walnut. In that paper, the author applies a technique outlined in [1] in order to construct a system of exponentials E() that is shown to be complete in the space L 2 (), where  ⊂ R2 is a convex, centrally symmetric polygon. The set  constructed in [16] is the union of a finite number of shifted lattices in R2 . It is shown that if the convex polygon satisfies certain extra arithmetic constraints given in [16, Theorem 4.2], then E() is not only a complete system, but is in fact a Riesz basis, in L 2 (). In [16] the author does not provide any transparent description as to which convex, centrally symmetric polygons satisfy the extra constraints imposed in [16, Theorem 4.2]. One can verify though that these constraints are satisfied if and only if, possibly after applying an affine transformation, all the vertices of the polygon lie in Z2 . Hence the class of planar convex polygons for which a Riesz basis E() is constructed in [16] coincides with the class covered by the result in [13]. In [3] we extended the technique from [16] to all dimensions, and moreover we combined it with the Paley-Wiener theorem about the stability of Riesz bases under small perturbations. This allowed us to construct a Riesz basis E() for any convex, centrally symmetric polytope in Rd with centrally symmetric faces, without imposing any extra constraints. The set of frequencies  in our construction is no longer a union of finitely many shifted lattices, but it rather has a less regular structure.

4 Open Problems We now mention a few related open problems. One problem pertains to the symmetry assumption in Theorem 1. Does there exist a Riesz basis of exponentials for non-symmetric convex polytopes in Rd ? The answer is not known even in the simplest case of a triangle in the plane. Another question is concerned with the construction of exponential Riesz bases for convex domains other than polytopes. In particular, it is still an open problem of whether a ball in Rd , d ≥ 2, admits a Riesz basis of exponentials. (We note that in order for a convex domain  ⊂ Rd to admit an orthogonal basis of exponentials, it is necessary that  be a centrally symmetric polytope with centrally symmetric facets. These necessary conditions are however not sufficient for

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the existence of an orthogonal basis E(); a complete characterization is obtained in [12].) Finally we mention that, in sharp contrast to the situation concerning orthogonal bases, no single example is known of a set  ⊂ Rd (convex or non-convex) of finite measure that does not admit any Riesz basis of exponentials. Acknowledgements Research supported by ISF Grants No. 447/16 and 227/17 and ERC Starting Grant No. 713927.

References 1. H. Behmard, A. Faridani, D. Walnut, Construction of sampling theorems for unions of shifted lattices. Sampl. Theory Signal Image Process. 5(3), 297–319 (2006) 2. L. Bezuglaya, V. Katsnelson, The sampling theorem for functions with limited multi-band spectrum. Z. Anal. Anwendungen 12, 511–534 (1993) 3. A. Debernardi, N. Lev, Riesz bases of exponentials for convex polytopes with symmetric faces. J. Eur. Math. Soc. (JEMS) (to appear), arXiv:1907.04561 4. B. Fuglede, Commuting self-adjoint partial differential operators and a group theoretic problem. J. Funct. Anal. 16, 101–121 (1974) 5. R. Greenfeld, N. Lev, Fuglede’s spectral set conjecture for convex polytopes. Anal. PDE 10(6), 1497–1538 (2017) 6. S. Grepstad, N. Lev, Multi-tiling and Riesz bases. Adv. Math. 252, 1–6 (2014) 7. S. Grepstad, N. Lev, Riesz bases, Meyer’s quasicrystals, and bounded remainder sets. Trans. Am. Math. Soc. 370(6), 4273–4298 (2018) 8. A. Iosevich, N. Katz, T. Tao, The Fuglede spectral conjecture holds for convex planar domains. Math. Res. Lett. 10(5–6), 559–569 (2003) 9. M. Kolountzakis, Multiple lattice tiles and Riesz bases of exponentials. Proc. Am. Math. Soc. 143(2), 741–747 (2015) 10. G. Kozma, S. Nitzan, Combining Riesz bases. Invent. Math. 199(1), 267–285 (2015) 11. G. Kozma, S. Nitzan, Combining Riesz bases in d . Rev. Mat. Iberoam. 32(4), 1393–1406 (2016) 12. N. Lev, M. Matolcsi, The Fuglede conjecture for convex domains is true in all dimensions. Acta Math. (to appear), arXiv:1904.12262 13. Y. Lyubarskii, A. Rashkovskii, Complete interpolating sequences for Fourier transforms supported by convex symmetric polygons. Ark. Mat. 38(1), 139–170 (2000) 14. K. Seip, A simple construction of exponential bases in L 2 of the union of several intervals. Proc. Edinburgh Math. Soc. 38, 171–177 (1995) 15. T. Tao, Fuglede’s conjecture is false in 5 and higher dimensions. Math. Res. Lett. 11(2–3), 251–258 (2004) 16. D. Walnut, A sampling theorem for symmetric polygons, in 2017 International Conference on Sampling Theory and Applications (SampTA) (IEEE, 2017), pp. 18–21 17. R. Young, An Introduction to Nonharmonic Fourier Series, 1st edn. (Academic, Cambridge, 2001)

Remez-Type Inequalities and Their Applications Omer Friedland

Abstract Polynomial inequalities on measurable sets play an important rôle in many areas of analysis. In particular, Remez [16] established the following result: For a measurable set E ⊂ [a, b], |E| > 0, and a real polynomial P of degree n, one has sup |P| ≤ Cn sup |P|,

[a,b]

E

where Cn = Tn (2(b − a)/|E|), and Tn is the Chebyshev polynomial of degree n. In these notes we discuss some equivalent statements of this result and present some classical generalizations, like the Turán-Nazarov inequality. We shall extend these results and show that the Lebesgue measure in these inequalities can be replaced with a certain geometric invariant, which can be effectively estimated in terms of the metric entropy of a set, and may be non-zero for discrete sets and even finite sets. Finally, we provide some applications which are related to some uncertainty principles as the Logvinenko-Sereda theorem which give limitations on the simultaneous concentration of a function and its Fourier transform.

1 Introduction By Remez-type inequalities we mean the inequalities which share the following feature. For a given function f , the maximum of | f | on a certain domain U is bounded by the maximum of | f | on a subset  ⊂ U with || > 0, that is, sup | f | ≤ C sup | f |, U



(1)

where C > 0 is a constant. Such inequalities hold for many different classes of functions and different domains. For example, for polynomials it is the classical Remez O. Friedland (B) Institut de Mathématiques de Jussieu - Paris Rive Gauche, Sorbonne Université - Campus Pierre et Marie Curie, 4 place Jussieu, 75252 Paris, France e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. Abakumov et al. (eds.), Extended Abstracts Fall 2019, Trends in Mathematics 12, https://doi.org/10.1007/978-3-030-74417-5_12

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inequality, for exponential-polynomials it is the Turán-Nazarov inequality, and there are many other examples for analytic functions, p-valent functions, solutions of certain PDE’s, etc. One of the main questions concerning such inequalities is the behavior of the constant C in (1). It clearly depends on the domain U and its subset , but typically we ask further that the constant C does not depend on the function itself but rather on certain parameters which depend on “families”, that is, the class of functions to which f belongs (this, of course, can not always be achieved). For example, in the case of polynomials, these parameters are the degree d and the dimension n of the ambient space. As we mentioned, the constant C depends on , so one should expect the following behaviour: as  gets smaller (in the sense of measure) the constant C gets bigger. Thus, it may be seen as a way to measure the “information” we have on the function f restricted to . Indeed, we show that the Lebesgue measure in these inequalities can be replaced with a certain geometric invariant, which can be effectively estimated in terms of the metric entropy of a certain level-set, and may be non-zero for discrete sets and even finite sets. We also provide some applications which are related to some uncertainty principles as the Logvinenko-Sereda theorem which give limitations on the simultaneous concentration of a function and its Fourier transform. These notes are based on some recent joint works with Evgeny Abakumov and Yosef Yomdin, for more details see [1, 8].

2 Remez Inequality Our story begins with the Russian mathematician Tchebycheff (1821–1894) who was interested (among many other things) in extremal problems for polynomials. Let n be the set of all polynomials of degree n. Tchebycheff observed that Theorem 1 Let p ∈ n and sup[−1,1] | p| ≤ 1. Then, for any |x| > 1, we have | p(x)| ≤ |Tn (x)|, where Tn (x) = cos(n arccos x) is the Tchebycheff polynomial of degree n. In other words, the Tchebycheff polynomial Tn (x) is the fastest growing polynomial outside [−1, 1]. This observation provokes the following question: How large can a polynomial be, given that it is constrained to be “small” on a substantial portion of its domain? Tchebycheff showed that the maximum of | p| on the interval [−1, 1] is bounded by the maximum of its absolute value on a subinterval I ⊂ [−1, 1], up to a multiplicative constant depending only on n and the ratio of the lengths 2/|I |. Remez [16] generalized this inequality by replacing the interval I by an arbitrary measurable subset  ⊂ [−1, 1]. Theorem 2 Let p ∈ n and let  ⊂ [−1, 1] be a measurable set. Then, we have

Remez-Type Inequalities and Their Applications

 sup | p| ≤ Tn

[−1,1]

75

 4 − || sup | p|. || 

Multidimensional versions of this inequality were also proved, in particular by Brudnyi and Ganzburg [6] (with sharp constant which coincides with that in the one dimensional case), and Yomdin [20] showed that Remez inequality (and also its multidimensional version) may be true also for some sets  of Lebesgue measure zero and even for certain finite sets. The simplest proof of Remez inequality is due to Bojanov [4]. For σ > 0 and p ∈ n define the set M( p) = {x ∈ [−1, 1 + σ] : | p(x)| ≤ 1}. Clearly, M( p) consists of mutually disjoints closed subintervals. Denote n (σ) = { p ∈ n : |M( p)| ≥ 2}. Evidently, Tn (x) ∈ n (σ) for any σ > 0 (since |Tn (x)| ≤ 1 on [−1, 1]). So, the problem is to characterize the extremal polynomial in n (σ) which has a maximal uniform norm over [−1, 1 + σ]. With above notations, Bojanov [4] proved the following result. Theorem 3 We have sup p∈n (σ)  p∞ = Tn ∞ , where the supremum norm is taken over [−1, 1 + σ].

3 Turán-Nazarov Inequality  Let f (t) = nj=0 b j eα j t be an exponential polynomial with b j , α j ∈ C. The classical Turán inequality [17] bounds the maximum of | f | on an interval B by the maximum of its absolute value on any subinterval  of B. Nazarov [14] generalized it to any subset  of positive measure.  Theorem 4 ([14]) Let f (t) = nj=0 b j eα j t be an exponential polynomial with b j , α j ∈ C. Let B ⊂ R be an interval, and let  ⊂ B be a measurable set. Then sup | f | ≤ e

|B| max | Re α j |

B

  |B| n c sup | f |, || 

where c > 0 is an absolute constant. This inequality has many generalizations, for example, there is a version of TuránNazarov inequality for quasipolynomials in one or several variables due to Brudnyi [5, Theorem 1.7]. Another inequality which strengthens the above result will be discussed in the next section. Nazarov established a very deep result, based on some tools from complex analysis. On the other hand, the proof of Turán inequality is purely algebraic, and is based on the following result, which is called Turán’s first main theorem. It is an essential result for many different applications, for more details see Turán’s book [18].

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Theorem 5 (Turán’s first main theorem) Let n, m ∈ N and let z 1 , . . . , z n ∈ C with |z j | ≥ 1. Then, for any b1 , . . . , bn ∈ C we have |

  n  m+n n b j | ≤ 2e max | b j z kj |. m+1≤k≤m+n n j=1 j=1

n 

4 Discrete Turán-Nazarov Inequality The Turán-Nazarov inequality can be generalized and strengthened by replacing the Lebesgue measure of  ⊂ B with a simple geometric invariant ω f (), which we call the metric-span of . The metric-span depends on the function f . For example, in the case of polynomials, it depends on the degree d and the dimension n of the ambient space. The metric-span always bounds the Lebesgue measure from above, and it is strictly positive for sufficiently dense discrete (in particular, finite) sets . It can be effectively estimated in terms of the metric entropy of . With appropriate modifications the following approach works also in higher dimensions. Originally it was introduced by Yomdin [20] in order to obtain a Remez-type inequality for algebraic polynomials on discrete sets. Let ρ > 0 and denote by Vρ = {t ∈ B : | f (t)| ≤ ρ} the ρ-sublevel set of f in B. Denote by Mρ the number of connected components of Vρ . In other words, Vρ consists of a finite number of closed intervals i , and we have Mρ Vρ = ∪i=1 i . Note that Mρ depends on the function f . In this section we consider exponential polynomials as given in the Turán-Nazarov inequality. For ε > 0 and a bounded set U ⊂ R we denote by M(ε, U ) the minimal number of ε-intervals covering U . Thus, we have Mρ  M(ε, Vρ ) ≤ (|i |/ε + 1) = |Vρ |/ε + Mρ . i=1

This motivates the following definition of the metric-span ω f for exponential polynomials f . Definition 6 Let  ⊂ B, and set ρ = sup | f |. The metric-span ω f () of  is given by   ω f () = sup ε M(ε, ) − Mρ . ε>0

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This definition can be generalized to other classes of functions, e.g. in Sect. 5 we consider L p functions so that the support of their Fourier transform is bounded. In [8] the following generalization of the Turán-Nazarov inequality was obtained  Theorem 7 ([8]) Let f (t)= nj=0 b j eα j t be an exponential polynomial with b j , α j ∈ C. Let B ⊂ R be an interval, and let  ⊂ B. Then, sup | f | ≤ e

|B| max | Re λk |

B



c|B| ω f ()

n sup | f |, 

where c > 0 is an absolute constant. For any measurable  we always have ω f () ≥ ||, with equality if  is a sublevel set of f . Thus, Theorem 7 provides a true generalization and strengthening of the Turán-Nazarov inequality. There is another version of Turán-Nazarov inequality for quasipolynomials in one or several variables due to A. Brudnyi [5, Theorem 1.7]. While less accurate than the original one (in particular, the role of real and complex parts of the exponents is not separated) this result gives an important information for a wider class of quasipolynomials. We provide a strengthening of Brudnyi’s result along the same lines as above. The main observation behind the proof of the above result is the fact that Remeztype inequalities are essentially inequalities about sublevel sets. Moreover, the metricspan can be effectively defined for any function for which we have a uniform bound on the number of zeros on B. Therefore, this approach is rather general and can be extended to many different settings.

5 Logvinenko-Sereda Type Estimates One of the main manifestations of the uncertainly principle in mathematics is the fact that a function and its Fourier transform cannot both be localized to small sets (for details see e.g. [9]). This fact leads to the following classical definitions: Definition 8 Let S,  ⊂ Rn be measurable subsets. The pair (S, ) is called (a) Weakly annihilating if supp f ⊂ S and supp  f ⊂  implies f ≡ 0. (b) Strongly annihilating if ∃c > 0 so that ∀ f we have

f 2L 2 (Rn \) .  f 2L 2 (Rn ) ≤ c  f 2L 2 (Rn \S) +  

(2)

These notions have been extensively studied (see e.g. [2, 11–13, 15]). Typical questions we are interested in are: What is the relation between these two notions? What can be said about the subsets S and ? What about the constant c > 0? Clearly, any strongly annihilating pair is a weakly annihilating one. Also, a pair (S, ) is weakly annihilating if S and  are compact sets. Benedicks [3] showed that

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the pair (S, ) is weakly annihilating if S and  are sets of finite Lebesgue measure |S|, || < +∞, and Amrein-Berthier [2], under the same assumptions, showed that in fact the pair (S, ) is strongly annihilating. Nazarov [14] studied the constant c > 0 in (2), and obtained an optimal one-dimensional estimate c ≤ c1 ec2 |S||| (with absolute constants), which later was extended, by Jaming [10], to any dimension, under the assumption that one of the two subsets is convex (it is still open whether the convexity assumption can be dropped). These two last results rely heavily on the Turán-Navarov inequality (which we presented above). In fact, a pair (S, ) is strongly annihilating if and only if ∃c > 0 so that ∀ f whose Fourier transform is supported in ,  f  L 2 (Rn ) ≤ c f  L 2 (Rn \S) (for more details see e.g. Havin-Jöricke [9]). A complete description of all strongly annihilating pairs seems to be out of reach, however, a complete description of all support sets S forming a strongly annihilating pair with any bounded set  is given in terms of the following notion of relatively dense sets. Definition 9 A measurable subset E ⊂ Rn is relatively dense with constants L , γ>0 if for any x ∈ Rn we have |E ∩ (x + [0, L]n )| ≥ L n γ > 0.

(3)

Logvinenko-Sereda [13] provided the following classical description of strongly annihilating pairs. Theorem 10 ([13]) Let S,  ⊂ Rn be measurable subsets and  bounded. Then, the pair (S, ) is strongly annihilating if and only if the subset Rn \ S is relatively f ⊂ J , J ⊂ R is an interval, and dense. Moreover, if f ∈ L p (1 ≤ p ≤ ∞), supp  E ⊂ R is (L , γ)-relatively dense, then,  f  L p (R)

  L|J | + 1  f  L p (E) . ≤ exp c p γ

With the above notation, E is nothing but the complement of S in the corresponding space. Kovrijkine [12] improved this estimate, generalized it also to higher dimensions, and provided an optimal constant. Theorem 11 ([12]) Let 1 ≤ p ≤ ∞ and let f ∈ L p (Rn ) with supp  f ⊂ [0, R]n for n some R > 0. Let E ⊂ R be (L , γ)-relatively dense. Then,  f  L p (Rn ) ≤

 C(L R+1) c  f  L p (E) , γ

where c, C > 0 are absolute constants.

(4)

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In the context of control theory of partial differential equations, it is essential to understand how the constant in (4) depends on the set E. Thanks to this explicit dependence of this constant on the parameter R > 0, recently Egidi-Veselic [7], and Wang, Wang, Zhang and Zhang [19] have independently established that the heat equation is null-controllable in any positive time T > 0 for a measurable control subset E ⊂ Rn if and only if this control subset E is relatively dense.

References 1. E. Abakumov, O. Friedland, Y. Yomdin, Discrete Logvinenko-Sereda Type Estimates (2019) 2. W.O. Amrein, A.M. Berthier, On support properties of L p -functions and their Fourier transforms. J. Funct. Anal. 24(3), 258–267 (1977) 3. M. Benedicks, On Fourier transforms of functions supported on sets of finite Lebesgue measure. J. Math. Anal. Appl. 106(1), 180–183 (1985) 4. B. Bojanov, Elementary proof of the Remez inequality. Am. Math. Mon. 100(5), 483–485 (1993) 5. A. Brudnyi, Bernstein type inequalities for quasipolynomials. J. Approx. Theory 112(1), 28–43 (2001) 6. J.A. Brudny˘ı, M.I. Ganzburg, A certain extremal problem for polynomials in n variables, Russian. Izv. Akad. Nauk SSSR Ser. Mat. 37, 344–355 (1973) 7. M. Egidi, I. Veseli´c, Sharp geometric condition for null-controllability of the heat equation on Rd and consistent estimates on the control cost. Arch. Math. (Basel) 111(1), 85–99 (2018) 8. O. Friedland, Y. Yomdin, An observation on the Turán-Nazarov inequality. Stud. Math. 218(1), 27–39 (2013) 9. V. Havin, B. Jöricke, The uncertainty principle in harmonic analysis, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 28 (Springer, Berlin, 1994), xii+543 10. P. Jaming, Nazarov’s uncertainty principles in higher dimension. J. Approx. Theory 149(1), 30–41 (2007) 11. V. È. Kacnel’son, Equivalent norms in spaces of entire functions. Russian, Mat. Sb. (N.S.), 92(134), 34–54, 165 (1973) 12. O. Kovrijkine, Some results related to the Logvinenko-Sereda theorem. Proc. Am. Math. Soc. 129(10), 3037–3047 (2001) 13. Logvinenko, V. N., Sereda, Ju. F., Equivalent norms in spaces of entire functions of exponential type, Russian, Teor. Funkci˘ı Funkcional. Anal. i Priložen., Vyp. 20, 1974, 102– 111, 175 14. Nazarov, F. L., Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type, Russian, with Russian summary, Algebra i Analiz, 5, 1993, 4, 3-66, St. Petersburg Math. J., 5, 1994, 4, 663–717 15. Paneah, B., Support-dependent weighted norm estimates for Fourier transforms. II, Duke Math. J., 92, 1998, 2, 335–353 16. E.J. Remez, Sur une propriété des polynômes de Tchebyscheff. Comm. Inst. Sci. Kharkow. 13, 93–95 (1936) 17. P. Turán, Eine Neue Methode in Der Analysis Und Deren Anwendungen (Akadémiai Kiadó, Budapest, German, 1953), p. 196 18. Turán, P., On a new method of analysis and its applications, Pure and Applied Mathematics (New York), With the assistance of G. Halász and J. Pintz; With a foreword by Vera T. Sós; A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1984, xvi+584 19. Wang, G., Wang, M., Zhang, C., Zhang, Y., Observable set, observability, interpolation inequality and spectral inequality for the heat equation in Rn , arXiv:1711.04279, 2017 20. Y. Yomdin, Remez-type inequality for discrete sets. Israel J. Math. 186, 45–60 (2011)

Shift-Invariant Spaces of Entire Functions Karlheinz Gröchenig

Abstract We study certain shift-invariant spaces of entire functions and characterize their sets of sampling and of interpolation. The results are similar to those for PaleyWiener space. Furthermore we indicate that phase-retrieval works in these shiftinvariant spaces. We mention several open problems that might be of interest for complex analysts.

1 Shift-Invariant Spaces Shift-invariant spaces are a standard model for one-dimensional signals used in signal processing. They are defined as follows: fix a generating function φ and define the associated shift-invariant space  V (φ) =

f ∈ L (R) : f = 2



 ck φ(· − k) with c ∈  (Z) 2

k∈Z

We always make the basic assumption that  f 2  c2 .

(1)

This assumption is satisfied for all reasonable generators. In fact, it is easy to see that the stability condition (1) is satisfied, if and only if for two constants A, B > 0 A≤



ˆ − k)|2 ≤ B |φ(ξ

a.e. ξ ∈ R ,

k∈Z

where φˆ is the Fourier transform of φ.

K. Gröchenig (B) Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. Abakumov et al. (eds.), Extended Abstracts Fall 2019, Trends in Mathematics 12, https://doi.org/10.1007/978-3-030-74417-5_13

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In signal   processingthese spaces are motivated as follows: If f ∈ V (φ), then −2πikξ ˆ ˆ f (ξ) = φ(ξ). Assuming that φ is smooth so that φˆ decays quickly, k ck e we see that every f ∈ V (φ) has the same “essential” support, so that all functions in V (φ) are approximately bandlimited. By definition V (φ) is invariant under the integer shifts f → f (· − k), k ∈ Z. Sometimes V (φ) is also called a space with finite rate of innovation (Vetterli) or a spline-type space (Feichtinger). While at first encounter, shift-invariant spaces deal with functions of a real variable, the best understanding and optimal results have been achieved only for certain classes of generators and with complex variable methods. The two principal examples of generators are the cardinal sine function and the Gaussian: sin πx πx 2 φ(x) = e−πx

φ(x) =

(2) (3)

If φ(x) = sinπxπx , then the generated shift-invariant space is precisely the classical Paley-Wiener space P W = { f ∈ L 2 (R) : supp fˆ ⊆ [−1/2, 1/2]} . If φ(x) = e−πx , then 2

V (φ) = { f ∈ L 2 R : f =



ck e−π(x−k) , c ∈ 2 (Z)} , 2

k

but there is no intrinsic description of V (φ). Analyticity. For both generators we obtain shift-invariant spaces of entire functions. Precisely, if f ∈ P W , then f possesses an extension to an entire function of growth | f (x + i y)| = O(eπ|y| ) , thus functions in P W are entire of order 1 with type π. If f = then f possesses an extension to an entire function



ck e−π(x−k) ∈ V (φ), 2

| f (x + i y)| = O(eπ y ) , 2

thus functions in V (φ) are entire of order 2 with type π. This follows from the fact that   2 2 2 ck e−π(x+i y−k) = eπ y e−2πi x y ck e2πiky e−π(x−k) . f (x + i y) = k∈Z

k∈Z

With a small modification, the growth can be made more symmetric:

Shift-Invariant Spaces of Entire Functions

| f (z)eπz

2

/2

83

| = O(eπ y eπ(x 2

2

−y 2 )/2

) = O(eπ|z|

2

/2

).

We summarize this fact: Let φ(x) = e−πx . Then f ∈ V (φ) extends to an entire 2 2 function with growth | f (x + i y)| = O(eπ y ). The space V (φ)eπz /2 is a subspace ∞ of the Bargmann-Fock space F consisting of all entire functions with growth 2 |F(z)| = O(eπ|z| /2 ). 2

2 Sampling A set  ⊂ R is a set of stable sampling for a closed subspace V ⊆ L 2 (R), if there exist A, B > 0, such that  | f (λ)|2 ≤ B f 22 ∀f ∈ V . (4) A f 22 ≤ λ∈

Here A, B are the sampling constants and κ = B/A is the condition number of the sampling set. These constants are important because the sampling inequality (4) implies automatically a reconstruction algorithm, e.g., f ∈ V can be recovered from its samples on  by the iterative frame algorithm or by means of a dual frame [3]. To quantify how many samples are necessary for the complete and stable recovery of every function f ∈ V , one uses the lower Beurling density of  defined as D − () = lim inf inf

r →∞ x∈R

# ∩ [x, x + r ] . r

D − () represents the average number of samples per unit and, in its essence, is an information theoretic notion. For some important generators the sets of stable sampling can be characterized almost completely with the Beurling density. Theorem 1 Let φ be either sinπxπx or e−πx and  ⊆ R be relatively separated. (i) Landau-type: If  is a set of sampling, then D − () ≥ 1. (ii) Beurling-Kahane-type: If D − () > 1, then  is a set of sampling for V (φ). 2

The case of the Paley-Wiener space is well-known and goes back to the results of Landau, Beurling, and Kahane. The case of the Gaussian, the hyperbolic secant etc. is more recent and contained in [4]. There are a few other generators for which this result is also valid, namely generators whose Fourier transform possesses the following factorization: 2 ˆ φ(ξ) = e−γξ

N 

(1 + 2πiν j ξ)−1 ,

ν j ∈ R, γ > 0 ,

j=1

and the hyperbolic secant φ(x) = (eax + e−ax )−1 with a > 0.

(5)

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We observe that for all these generators the functions in V (φ) possess an extension from R to entire functions.

2.1 Sampling with Derivatives A useful generalization of Theorem 1 includes derivatives at every sampling point. In this case we use a weighted Beurling density with a weight function m :  → N . 1 y∈R r



D − (, m) := lim inf inf r →∞

m(λ).

λ∈∩[y,y+r ]

Here m(λ) − 1 is the number of derivatives at λ. We then have the following characterization [5]. Theorem 2 Let φ be either sinπxπx or e−πx and  ⊆ R be relatively separated, and let m :  → N be bounded. If D − (, m  ) > 1, then (, m  ) is a sampling set for V (φ), i.e., 2

A f 22 ≤

λ −1  m

| f (k) (λ)|2 ≤ B f 22

for all f ∈ V (φ) .

λ∈ k=0

Again the statement of Theorem 2 holds for the larger class of generators satisfying (5). Open problem. It would be interesting to understand sampling with derivatives when the multiplicity function m is unbounded. See [1] for a related problem in Fock space.

3 Interpolation A set  is a set of interpolation for V ⊆ L 2 (R), if for every a = (aλ ) ∈ 2 () there exists f ∈ V , such that ∀λ ∈  . f (λ) = aλ In this case f can be chosen so that  f 2 ≤ Ca2 for a constant depending only on . The appropriate information theoretic notion is the upper Beurling density of  ⊆ R defined as D + () = lim sup sup r →∞

x∈R

# ∩ [x, x + r ] . r

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85

Then we have the following result that is dual to Theorem 1. Theorem 3 ([4]) Let φ be either sinπxπx or e−πx and  ⊆ R. (i) Landau-type: If  is an interpolating set, then D + () ≤ 1. (ii) Beurling-Kahane-type: If D + () < 1, then  is an interpolating set for V (φ). 2

Open problems. 1. Analogous results for other generators are still missing. 2. At the critical density D + () = D − () = 1 anything may happen;  may or may not be sampling or interpolating. For the Paley-Wiener space the situation is understood and a complete characterization of all sets that are simultaneously sampling and interpolating is known. For the shift-invariant space V (φ) with Gaussian generator this question is not yet answered.

4 Phase-Retrieval in Shift-Invariant Spaces A question that is of some importance in signal processing is the question of phaseretrieval: Can you recover a function f from its phaseless or unsigned samples | f (λ)|, λ ∈ ? Clearly there is an obvious ambiguity, as the phaseless samples can determine f only up to a global factor α, |α| = 1, since | f (λ)| = |α f (λ)|. Phaseretrieval in general shift-invariant spaces has been studied for instance in [2]. The following theorem formulates the only sharp results that are known so far. Theorem 4 Let φ be either sinπxπx or e−πx and  ⊆ R be relatively separated. (i) If D − () > 2, then phase-retrieval is possible on  for all real-valued functions in V (φ). (ii) Phase-retrieval is impossible under the following conditions: f is complexvalued or D − () < 2. 2

This means that if f, g ∈ V (φ) are real-valued and | f (λ)| = |g(λ)| for all λ ∈ , then either g = f or g = − f . The result for P W is due to Thakur [7], the result for V (φ) with Gaussian generator is contained in [6]. The starting point for bandlimited functions is the observation that the product of two bandlimited functions is again bandlimited; precisely, if f ∈ P W f 2 = supp fˆ ∗ fˆ ⊆ [−1, 1]. real-valued, then f 2 ∈ P W2 , i.e., supp observation holds for f ∈ V (φ) with Gaussian generator: let f (x) =  A similar −π(x−k)2 c e . Then k k∈Z | f (x)|2 =



ck cl e−π(k

k,l∈Z

=

 n∈Z

2



+l 2 ) π(k+l)2 /2 −2π x− k+l 2

e

2

e

 2 2 2 2 ck cn−k e−πk e−π(n−k) eπn /2 e−2π(x−n/2) .

k∈Z

Thus | f |2 belongs to a different shift-invariant space with step size 1/2, consequently the sampling theorem for V (φ) is applicable.

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Open problem: Is it possible to prove sharp results also for shift-invariant spaces with other generators?

5 Further Problems (i) The results for P W and V (φ) are completely analogous. This raises the question how the spaces of entire functions P W and V (φ) are related? Discussions at the CRM 2 with Y. Belov, A. Baranov, and A. Borichev suggest that V (φ) with φ(x) = e−πx is isomorphic to the direct sum of two small Fock spaces. (ii) Quasi-shift-invariant spaces. A natural extension of the results about shiftinvariant spaces is to replace the integer shifts of a generator by nonuniform shifts and to define a quasi-shift-invariant space with generator φ and shifts  ⊆ R as follows: V ( f, ) = { f ∈ L 2 (R) : f (x) =



cγ φ(x − γ), c ∈ 2 ()} .

γ∈

Clearly, all questions about sampling and interpolation remain meaningful in these quasi- shift-invariant spaces. Assume first that {e2πiγx : γ ∈ } is a Riesz basis for L 2 ([−1/2, 1/2]), equivalently,  ⊆ R is sampling and interpolating for P W . Then clearly for φ(x) = sinπxπx V (φ, ) = { f (x) =

 γ∈

cγ

sin π(x − γ) , c ∈ 2 ()} = P W . π(x − γ)

So questions about sampling and interpolation for V (φ, ) are the same as for V (φ). 2 For Gaussian generator φ(x) = e−πx the quasi-shift-invariant space V (φ, ) differs from the shift-invariant space V (φ) = V (φ, Z), and currently there is no result about sampling and interpolation in V (φ, ). Our investigations suggest that the following result should hold true: If  is a set of sampling and interpolation for P W and D − () > 1, then  is sampling for V (φ, ). A resolution of this question would have many important consequences for sampling in shift-invariant spaces. Summary. Shift-invariant spaces with certain generators yield new spaces of entire functions. Although they behave similarly to Paley-Wiener space, the precise connection is not yet understood. While there is an elaborate theory of sampling and interpolation in shift-invariant spaces, optimal results are known only in few cases, and invariably complex-variable methods provide the decisive arguments. As these spaces are not well-known in complex analysis, they offer many open problems. Acknowledgements Most results are based on joint work with José Luis Romero, University of Vienna, and Joachim Stöckler, Technical University Dortmund. K. G. was supported in part by the project P31887-N32 of the Austrian Science Fund (FWF).

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References 1. A. Borichev, A. Hartmann, K. Kellay, X. Massaneda, Geometric conditions for multiple sampling and interpolation in the Fock space. Adv. Math. 304, 1262–1295 (2017) 2. C. Cheng, J. Jiang, Q. Sun, Phaseless Sampling and Reconstruction of Real-Valued Signals in Shift-Invariant Spaces. J. Fourier Anal. Appl. 25(4), 1361–1394 (2019) 3. R.J. Duffin, A.C. Schaeffer, A class of nonharmonic Fourier series. Trans. Am. Math. Soc. 72, 341–366 (1952) 4. K. Gröchenig, J.-L. Romero, J. Stöckler, Sampling theorems for shift-invariant spaces, Gabor frames, and totally positive functions. Invent. Math. 211(3), 1119–1148 (2018) 5. K. Gröchenig, J.-L. Romero, J. Stöckler, Sharp results on sampling with derivatives in shiftinvariant spaces and multi-window Gabor Frames. Constr. Approx. 51(1), 1–25 (2020) 6. K. Gröchenig, Phase-retrieval in shift-invariant spaces with Gaussian generator. J. Fourier Anal. Appl. 26(3), 52 (2020) 7. G. Thakur, Reconstruction of band limited functions from unsigned samples. J. Fourier Anal. Appl. 17(4), 720–732 (2011)

Describing Blaschke Products by Their Critical Points Oleg Ivrii

Abstract In this talk, I will discuss a question which originates in complex analysis but is really a problem in non-linear elliptic PDE. A finite Blaschke product is a proper holomorphic self-map of the unit disk, just like a polynomial is a proper holomorphic self-map of the complex plane. A celebrated theorem of Heins says that up to post-composition with a Möbius transformation, a finite Blaschke product is uniquely determined by its critical points. Dyakonov suggested that it may be interesting to extend this result to infinite degree, however, one must be careful since infinite Blaschke products may have identical critical sets. I will first explain how to parametrize inner functions F of finite entropy (i.e. which have derivative in the Nevanlinna class) in terms of Inn F  . The answer involves measures on the unit circle that do not charge Beurling–Carleson sets. Afterwards, I will discuss how one might parametrize arbitrary inner functions using 1-generated invariant subspaces of the weighted Bergman space A21 . The proofs rely on the Liouville correspondence which provides a bridge between complex analysis and non-linear elliptic PDE.

1 Introduction A finite Blaschke product can be described as a proper holomorphic self-map of the unit disk. It is easy to show that a finite Blaschke product can be factored as F(z) = eiθ

d  z − ai , 1 − ai z i=1

ai ∈ D.

(1)

O. Ivrii (B) Wladimir Schreiber Institute of Mathematical Sciences, Tel Aviv University, Tel Aviv-Yafo, Israel e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. Abakumov et al. (eds.), Extended Abstracts Fall 2019, Trends in Mathematics 12, https://doi.org/10.1007/978-3-030-74417-5_14

89

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O. Ivrii

The above representation privileges the zeros of the Blaschke product. In this talk, we take a less traveled route and study Blaschke products in terms of their critical points, the zeros of the derivative. Our starting point is the following celebrated theorem of Heins [6]: Theorem 1 ([6]) Given a set C of d − 1 points in the unit disk, there exists a Blaschke product F of degree d with critical set C. Moreover, F is determined uniquely up to post-composition with an element of Aut(D). Our aim is to extend Heins’ theorem to infinite degree in the context of inner functions. Loosely speaking, an inner function is a holomorphic self-map of the unit disk which extends to a measurable dynamical system on the unit circle. More precisely, we want the radial boundary values to exist a.e. on the unit circle and have absolute value 1. When one tries to parametrize inner functions by their critical points, one runs into a simple obstruction: different inner functions can have  same critical sets.  the (universal covering For example, F1 (z) = z (identity map) and F2 (z) = exp z+1 z−1 map of the punctured disk) have no critical points. Nevertheless, it is possible to say that F2 (z) has “boundary critical structure.” We will discuss two ways of making this precise.

2 Inner Functions of Finite Entropy For the first approach to defining the critical structure of inner functions, we restrict ourselves to the class J of inner functions of finite entropy. Analytically, one asks that the derivative lies in the Nevanlinna class:  2π 1 log |F  (eiθ )|dθ < ∞. (2) 2π 0 In order for the above formula to make sense, we require that F  has radial boundary values a.e. on the unit circle. Under the normalization F(0) = 0, the Lebesgue measure m on the unit circle is invariant under F, that is, m(F −1 (E)) = m(E) for any measurable set E ⊂ D. Craizer [2] showed that the measure-theoretic entropy of m is finite if and only if the integral (2) converges, in which case the entropy is given by (2). By a fundamental result of Ahern and Clark [1], for any inner function F ∈ J , F  admits a B S O decomposition into a Blaschke factor, a singular factor and an outer factor (in general, functions in the Nevanlinna class admit B(S1 /S2 )O factorizations). In this decomposition, the Blaschke factor describes the critical set of F while the singular factor describes the “boundary critical structure.” By recording S in addition to B, one can distinguish the two inner functions F1 (z) and F2 (z) from the introduction.

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Dyakonov [4] asked the following question: To what extent is an inner function F ∈ J determined by its critical structure Inn F  = B S? What are the possible critical structures of inner functions? Theorem 2 (I, 2017) An inner function in J is uniquely determined by its critical structure up to post-composition with a Möbius transformation. An inner function B Sμ can be represented as Inn F  for some F ∈ J if and only if μ is sufficiently concentrated (lives on a countable union of Beurling–Carleson sets). Dyakonov [3] studied the special case when the critical structure is trivial, while Kraus [10, Theorem 4.4] considered the case when the critical structure is a Blaschke product. A Beurling–Carleson set E is a closed subset of the unit circle which has measure  0 such that |I j | · log |I1j | < ∞, where {I j } are the complementary intervals. Remark The condition for a measure to not charge Beurling–Carleson sets appeared in the works of Korenblum [8, 9] and Roberts [14] on the study of cyclic functions in Bergman spaces.

3 Conformal Metrics and Liouville’s Theorem The Gaussian curvature of a conformal metric λ(z)|dz| is given by kλ = −

 log λ . λ2

2|dz| A simple computation shows that the hyperbolic metric λD = 1−|z| 2 has curvature ≡ −1, while the Euclidean metric |dz| is flat (has curvature 0). The importance of Gaussian curvature to complex analysis comes from Gauss Theorema Egregium which says that curvature is a conformal invariant. More precisely, if F : D → D is a holomorphic map then

λ F = F ∗ λD =

2|F  | 1 − |F|2

(3)

is a conformal metric of curvature ≡ −1 on D \ crit(F), but is only a pseudometric on D. Its logarithm u F = log λ F satisfies u F = e2u F + 2π



δc .

(4)

c∈crit(F)

Liouville observed that all solutions of the Gauss curvature equation with integral singularities arise in this way. Furthermore, the function F : D → D can be reconstructed from u F uniquely up to post-composition with an element of Aut(D). For

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more information on conformal metrics and Liouville’s theorem, we refer the reader to [11]. In principle, Liouville’s theorem allows one to translate problems from complex analysis to PDE and vice versa. In practice, it is difficult to find questions that are simultaneously interesting in both settings. It turns out that Dyakonov’s problem is one of these questions. The Gauss curvature equation u = e2u ,

u : D → R.

(5)

has a unique maximal solution u max = log λD which dominates all other solutions pointwise. The maximality of log λD is just a restatement of the Schwarz lemma. It turns out that Dyakonov’s question is equivalent to understanding solutions that are close to maximal in the sense that  lim sup (u max − u)dθ < ∞. (6) r →1

|z|=r

We now present a brief summary of the discussion from [7]. For each 0 < r < 1, we may view (u max − u)dθ as a positive measure on the circle of radius r . Since these measures have uniformly bounded mass, we can extract a weak limit as r → 1 and obtain a measure μ[u] on the unit circle. Since u max − u is subharmonic (its Laplacian is non-negative), we don’t need to pass to a subsequence, so that μ[u] is well-defined. One can show that the deficiency measure μ uniquely determines the solution u. Thus, the question becomes: which measures occur? The next step is to show that each measure μ on the unit circle can be uniquely decomposed into a constructible part and an invisible part: μ = μcon + μinv . A measure is constructible if it is the deficiency measure of some nearly-maximal solution, while a measure μ is invisible if no measure 0 < ν ≤ μ is constructible. In fact, u μcon is the minimal solution which exceeds the subsolution u max − Pμ , where Pμ is the Poisson extension of μ. It remains to relate constructible measures to Beurling–Carleson sets. Given a measure μ which lives on a Beurling–Carleson set, one can produce a nearly-maximal solution with deficiency μ by means of a simple construction (which we don’t discuss here). It is also not difficult to show that if the measure μ satisfies the modulus of continuity condition ωμ (t) ≤ Ct log(1/t), then it is invisible. Recall that the modulus of continuity ωμ (t) := max|I |=t μ(I ) measures the maximal mass of an arc of length t. The proof looks like this: since the measure μ is small, its Poisson extension is small, so that u max − Pμ is close to u max . Then, the minimal dominating solution of u max − Pμ must be u max itself. To get the full result, one uses a miraculous theorem due to Roberts [14] which says that if a measure does not charge Beurling–Carleson sets, it can be decomposed as a countable sum of measures which satisfy the t log(1/t)

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condition at a super-exponential set of scales. With help of the Roberts decomposition, one can show that the minimal dominating solution of u max − Pμ is u max in this case as well. The proof is a bit like fighting Diablo: if you have a healer who heals you faster than Diablo is dealing you damage, then you are invincible.

4 Invariant Subspaces of Bergman Space It may be desirable to parametrize all inner functions by their critical structure, not just the relatively small subset J . To set the stage, we recall Beurling’s theorem which allows one to parametrize zero structures of inner functions. The following tautological statement expresses that zero sets of functions in the Hardy space H 2 are Blaschke sequences: BP / S1 = {zero-based subspaces of H 2 }. On the left, we quotient Blaschke products by rotations, because a Blaschke product is determined by its zero set up to a unimodular factor. By definition, a (closed) subspace X ⊂ H 2 is zero-based if it is defined as the collection of functions in H 2 that vanish at a prescribed set of points. To write something non-trivial, we take the “closure” of the above statement and obtain a famous theorem of Beurling: Theorem 3 (Beurling 1949) Inn / S1 = {zero-based subspaces of H 2 } = {invariant subspaces of H 2 }. The collection of closed subspaces of a Banach space carries a natural topology where X n → X if for any convergent sequence xn → x with xn ∈ X n , the limit x ∈ X , and conversely, any x ∈ X can be approximated by a convergent sequence xn → x with xn ∈ X n . Beurling showed that the closure of the zero-based subspaces consists of all subspaces that are invariant under multiplication by z. The process of taking closure has been given the beautiful name asymptotic spectral synthesis by N. Nikol’skii. Let (H 2 ) denote the space of derivatives of H 2 functions. It is well known that the space (H 2 ) can be identified with the weighted Bergman space A21 which is the collection of all holomorphic function on the unit disk which satisfy 1/2



f A21 =

D

| f (z)|2 (1 − |z|)|dz|2

< ∞.

The starting point for our discussion is a beautiful result of Kraus [10] which says that critical sets of inner functions coincide with critical sets of H 2 functions. One can tautologically rewrite Kraus’ result as

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MBP / Aut(D) = {zero-based subspaces of A21 }. The exact definition of the class of maximal Blaschke products [11, 13] will not be important to us, only that if C is a critical set of some Blaschke product, then the maximal Blaschke product with critical set C is a particular Blaschke product with that critical set. For a function H ∈ A21 , the subspace generated by H is defined as the minimal closed subspace of A21 which contains H and is invariant under multiplication by z: [H ] = {H p : p polynomial}. In an important paper, Shimorin [15] showed that the closure of the zero-based subspaces in A21 consists of subspaces that can be generated by a single function. In light of Shimorin’s result, we can try to write down the “closure” of Kraus’ theorem: Conjecture 4 If an invariant subspace of A21 can be generated by a single function, it can be generated by the derivative of an essentially unique inner function. While I don’t have a complete proof of this conjecture, I hope to convince you that it is true. For more information on Bergman spaces, I refer you to the book by Hedenmalm, Korenblum and Zhu [5].

5 Canonical Solutions Given a function H ∈ A21 (D) \ {0}, we describe a simple procedure which produces an inner function I H with I H ∈ [H ]. Recipe. Let u H,n be the solution of the boundary value problem

u = |H |2 e2u , in D, u = n, on ∂D.

(7)

Form the canonical solution u H,∞ := limn→∞ u H,n . By Liouville’s theorem, 2|I  | u H,∞ = log |H1 | 1−|I for some holomorphic self-map I of the unit disk. |2 Remark The PDE (7) is understood weakly in the sense of distributions: we require u(z) and |H |2 e2u(z) to be in L 1loc (D), and ask that for any test function φ ∈ Cc∞ (D), compactly supported in the disk, 

 D

uφ |dz|2 =

D

|H |2 e2u φ |dz|2 .

(8)

The boundary data is interpreted in the sense of weak convergence of measures u(r eiθ )dθ → n dθ as r → 1.

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Remark In general, u H,∞ does not coincide with the pointwise-maximal solution of (7). For instance, if H has no zeros in the disk but generates a non-trivial invariant 2 subspace in A21 , then the maximal solution u H,max = log |H1 | 1−|z| 2 has the Liouville  =1∈ / [H ]. function I H,max (z) = z but I H,max

5.1 Why Is I  ∈ [H]? 2|I  |

By Liouville’s theorem, u H,n = log |H1 | 1−|Inn |2 for some holomorphic self-map In of the unit disk. We normalize In (0) = 0. Subharmonicity implies that u H,n ≤ n on D =⇒ |In /H | ≤ en /2 =⇒ In ∈ [H ], since invariant subspaces of A21 are closed under multiplication by bounded functions. We want to conclude that I  ∈ [H ] by virtue of the fact I  = lim In ; however, the In may not converge in norm. Nevertheless, we can argue as follows: since In A21 In H 2 ≤ 1 are uniformly bounded, we can pass to a subsequence that converges weakly. It remains to use the following fact from functional analysis: in a Banach space, a subspace is closed if and only if it is weakly closed.

5.2 The Case When H = I  In the case when H = I  is the derivative of an inner function, we can write down the canonical solution explicitly: u I  ,∞ = log

1 2|I  | 2 = log .  2 |I | 1 − |I | 1 − |I |2

2 blows up near the unit circle in a measure-theoretic In one direction, since log 1−|I |2 2 2 sense, log 1−|I |2 ≥ u I  ,n for any n ∈ R. Hence, log 1−|I ≥ u I  ,∞ . |2 

2|r I | 2 Conversely, one can approximate log 1−|I by the solutions log |I1 | 1−|r = |2 I |2 2r 2 log 1−|r I |2 that are bounded above. This gives the opposite inequality log 1−|I |2 ≤ u I  ,∞ .

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5.3 Why Is I An Inner Function (For General H)? Since I  ∈ [H ], there exists a sequence of polynomials pk such that H pk → I  in A21 . Dilating if necessary, we may assume that the pk have no zeros on the unit circle. |I  | for any n ∈ R. Taking By the comparison principle, u H pk ,n ≤ log |H1pk | · 1−|I |2 1 . Since this is true for any n ∈ R, k → ∞ gives u I  ,n ≤ log 1−|I |2 u I  ,∞ ≤ log

1 , 1 − |I |2

which forces I to be inner.

5.4 Inner Functions Embed into Invariant Subspaces The above argument shows that if [H1 ] ⊂ [H2 ], then |I H 1 | 1 − |I H1 |2

≤

|I H 2 | 1 − |I H2 |2

.

If [H1 ] = [H2 ], the equality |I H 1 | 1 − |I H1 |2

=

|I H 2 | 1 − |I H2 |2

forces I H1 = I H2 up to post-composition  with an element of Aut(D). At this stage, we have proved that Inn / Aut(D) ⊆ 1-generated invariant subspaces of A21 .

5.5

I0 Generates [H]

We now show that any 1-generated invariant subspace of A21 can be generated by the derivative of a bounded function. Previously, it was known that the generator could be chosen to be the derivative of a BMO function, see [5, Theorem 3.3]. 2|I  | Since u = log |H1 | 1−|I00 |2 has zero boundary data, its minimal harmonic majorant 2|r I  |

is 0. For 0 < r < 1, define u r = log |H1r | 1−|r I00 |2 , where Hr :=

2r  · I ∈ [I0 ], φr 0

φr (z) := Out 1−|r I0 |2 (z).

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Here, φr is the outer function with absolute value 1 − |r I0 |2 on the unit circle. By construction, the minimal harmonic majorant of u r is also 0. Since the |Hr | increase to |H |, we see that Hr → eiθ H in A21 for some θ ∈ [0, 2π). It follows that H ∈ [I0 ] and therefore, [H ] = [I0 ].

5.6 Does I  Generate [H]? We now complete the proof of the Conjecture 4 using a plausible-looking statement from PDE: Conjecture 5 Any solution of u = |I  |2 e2u that is ≤ u I  ,∞ can be approximated uniformly on compact subsets by solutions u k that are bounded from above. By the above “fact” we can approximate log

|Fk | 1 |I0 | 1 → log . |I  | 1 − |Fk |2 |I  | 1 − |I0 |2

Since each Fk ∈ [I  ], their limit I0 ∈ [I  ]. Therefore, [I  ] = [H ] as desired.

References 1. P.R. Ahern, D.N. Clark, On inner functions with H p -derivative. Michigan Math. J. 21(2), 115–127 (1974) 2. M. Craizer, Entropy of inner functions. Israel J. Math. 74(2), 129–168 (1991) 3. K.M. Dyakonov, A characterization of Möbius transformations. C. R. Math. Acad. Sci. Paris 352(2), 593–595 (2014) 4. K.M. Dyakonov, Inner functions and inner factors of their derivatives. Integr. Equ. Oper. Theory 82(2), 151–155 (2015) 5. H. Hedenmalm, B. Korenblum, K. Zhu, Theory of Bergman spaces, Graduate Texts in Mathematics 199 (Springer, Berlin, 2000) 6. M. Heins, On a class of conformal metrics. Nagoya Math. J. 21, 1–60 (1962) 7. O. Ivrii, Prescribing inner parts of derivatives of inner functions. J. d’Analyse Math. 139, 495–519 (2019) 8. B. Korenblum, Cyclic elements in some spaces of analytic functions. Bull. Amer. Math. Soc. 5, 317–318 (1981) 9. B. Korenblum, Outer Functions and Cyclic Elements in Bergman Spaces. J. Funct. Anal. 151(1), 104–118 (1993) 10. D. Kraus, Critical sets of bounded analytic functions, zero sets of Bergman spaces and nonpositive curvature. Proc. Lond. Math. Soc. 106(4), 931–956 (2013) 11. D. Kraus, O. Roth, Critical points, the gauss curvature equation and blaschke products, blaschke products and their applications. Commun. Fields Inst. 65, 133–157 (2012) 12. D. Kraus, O. Roth, Conformal metrics, Lecture Notes Mathematical Society, Lecture Notes Series, vol. 19, pp. 41–83 (2013) 13. D. Kraus, O. Roth, Maximal Blaschke Products. Adv. Math. 241, 58–78 (2013)

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14. J.W. Roberts, Cyclic inner functions in the Bergman spaces and weak outer functions in H p , 0< p 0 is small enough, with high probability, f is close to the identity map on K . Loosely speaking, their result says that the distortion on the A-squares and B-squares cancel out. We now consider a more general setup. Let

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{real-linear mappings} / {complex-linear mappings} ∼ = D be the space of Beltrami coefficients. For any probability measure λ on D, one can construct a random quasiconformal mapping f : C → C where on each cell of the square grid, one selects the real-linear map according to λ. The mapping will only be a genuine quasiconformal mapping if λ is supported on a compact subset of the unit disk. The argument of [2] based on the homogenization of iterated singular integrals shows that when the distortion is uniformly bounded above (i.e. when λ is supported on a compact subset of D), then the random quasiconformal mapping homogenizes to some real-linear mapping Aλ : C → C depending on λ. Our argument [5] based on extremal properties of quasiconformal mappings also applies to the case of unbounded distortion. Remark In general, we do not know how to compute the real-linear map Aλ explicitly. In the model case above where λ = (δ A + δ B )/2, one can use the 90◦ symmetry of the model. Of course, if λ = δ A is supported on a single real-linear mapping, then Aλ = A.

3 Circle Packing A circle packing P = {Ci } is a collection of circles in the plane with disjoint interiors. The tangency pattern of P is an embedded graph in the plane whose vertices are centers of circles in P and edges are line segments which connect centers of tangent circles. The Koebe–Andreev–Thurston Circle Packing Theorem [6, 11, 12] says that any finite  triangulation T of a topological disk admits a maximal circle packing P = Ci ⊂ D whose boundary circles are horocycles. Furthermore, this maximal packing is unique up to Möbius transformations. Let  be a simply-connected domain whose boundary is a C 1 curve. Consider the tiling of the plane by equilateral triangles with side length δ. Let Tδ be the set of vertices contained in . If δ is small, then Tδ will be a triangulation of a topological disk so it admits a maximal circle packing. Rodin and Sullivan [8] showed that for small δ, the maximal circle packing of Tδ approximates a conformal map ϕ :  → D. To be precise, fix two points z 1 , z 2 ∈ . For each i = 1, 2, let vi ∈ Tδ be the closest point to z i (in case of a tie, choose vi arbitrarily). Let P δ be the maximal circle packing of Tδ normalized so that Cv1 is centered at the origin while the center of Cv2 lies on (0, 1). Consider the piecewise linear map ϕP that takes vertices of Tδ to the centers of the corresponding circles and is linear on triangles. Rodin and Sullivan showed that as δ → 0+ , ϕP tends to the conformal map ϕ :  → D with ϕ(z 1 ) = 0 and 0 < ϕ(z 2 ) < 1. Instead of circle packing deterministic triangulations, one can also try to circle pack random triangulations. Randomly choose N ≥ 1 points in  with respect to Lebesgue measure and take the union of the Delauney triangles contained in .

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Based on numerical experiments, Kenneth Stephenson suggested that when N ≥ 1 is large, then with high probability, the maximal circle packing of a random Delauney triangulation approximates a conformal map ϕ :  → D. In the paper [5], we proved this conjecture: Theorem 1 Let  be a bounded simply-connected domain in the plane with C 1 boundary and ϕ :  → D be the conformal map with ϕ(z 1 ) = 0 and 0 < ϕ(z 2 ) < 1. Consider the random Delauney triangulation with N points. For any compact set K ⊂  and ε > 0, when N ≥ N0 (K , ε) is sufficiently large, ϕP − ϕ C(K ) < ε holds with probability at least 1 − ε.

4 A Lemma on Percolation One difficulty in the proof of Theorem 1 is that ϕP might have arbitrarily large local distortion as triangles close to equilateral can be mapped to thin ones and vice versa. Nevertheless, with high probability, the modulus of every rectangle R ⊂  whose sides have length ≥ ε is distorted by a bounded amount (independent of the number of points in the Delauney triangulation). To prove this, it is preferable to use discrete modulus since it only depends on the combinatorics of the Delauney triangulation in R. In general, the discrete and continuous moduli of ϕP (R) are unrelated, however, if the triangulation in question has bounded valence, the two notions of modulus agree up to a multiplicative constant. While a random Delauney triangulation may have vertices of arbitrarily large valence, they are quite rare and can be “avoided” using a percolation argument which we learned from Mathieu’s paper [7]. Fix the percolation parameter 0 < r < 1 and colour the cells of the square grid yellow with probability r and blue with probability 1 − r . For two points x, y ∈ R2 , their Euclidean distance |x − y| measures the minimal length of a curve that joins x to y. We are more interested in the chemical distance dchem (x, y) which minimizes the part of the length that is contained in the blue squares. By definition, dchem (x, y) ≤ |x − y|. The following lemma says that if the percolation parameter is sufficiently small, and the points x, y are at macroscopic distance from one another, then the chemical distance is equivalent to the Euclidean distance: Lemma 2 There exists a universal constant 0 < r0 < 1/2 so that if the percolation parameter 0 < r < r0 is sufficiently small, then with probability ≥ 1 − 1/N 2 , for two points x, y ∈ [−N , N ] × [−N , N ] with |x − y| ≥ log N , 1 |x − y| ≤ dchem (x, y) ≤ |x − y|. 2 This simple lemma can be interpreted as saying that random sets of small measure have little effect on the moduli of curve families:

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Lemma 3 Suppose  that f : [−N , N ] × [−N , N ] → C is a quasiconformal map whose dilatation ∂ f /∂ f  ≤ k < 1 is bounded by a definite constant on the blue squares but could be arbitrarily close to 1 on the yellow squares. If the condition of Lemma 2 holds, then the modulus of the conformal rectangle f [−N , N ] ×  [−N , N ] is bounded above by a constant depending only on k and r0 .  For the  proof, one needs to design a conformal metric ρ : f [−N , N ] × [−N , N ] → [0, ∞) with the following two properties:  (1) The ρ-length of any curve that connects the vertical sides of f [−N , N ] ×  [−N , N ] is at least  1. (2) The area A(ρ) = ρ(w)2 |dw|2 is uniformly bounded. We define ρ(w) = 0 on the images of the yellow squares and  ρ(w) =

K · Jac f −1 (w) N

on the images of the blue squares. By construction, A(ρ) ≤ K while the ρ-length of any curve connecting the vertical sides is at least 1/2. Replacing ρ with 2ρ produces the desired metric.

References 1. K. Astala, T. Iwaniec, G.J. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane (Princeton University Press, Princeton, 2009) 2. K. Astala, S. Rohde, E. Saksman, T. Tao, Random Quasiconformal Maps and Homogenization of Iterated Singular Integrals, preprint 3. M. Biskup, Recent progress on the random conductance model. Probab. Surv. 8, 294–373 (2011) 4. M. Biskup, T. Prescott, Functional CLT for random walk among bounded random conductances. Electron. J. Probab. 12, paper no. 49, 1323–1348 (2007) 5. O. Ivrii, V. Markovi´c, Homogenization of random quasiconformal mappings and random Delauney triangulations, submitted 6. P. Koebe, Kontaktprobleme der konformen Abbildung (Hirzel, 1936) 7. P. Mathieu, Quenched invariance principles for random walks with random conductances. J. Stat. Phys. 130(5), 1025–1046 (2008) 8. B. Rodin, D. Sullivan, The convergence of circle packings to the Riemann mapping. J. Diff. Geom. 26(2), 349–360 (1987) 9. A. Rousselle, Quenched invariance principle for random walks on Delaunay triangulations, Electron. J. Probab. 20, paper no. 33, 32 (2015) 10. V. Sidoravicius, A-S. Sznitman, Quenched invariance principles for walks on clusters of percolation or among random conductances. Probab. Theory Relat. Fields 129, 219–244 (2004), Nov. 2 11. K. Stephenson, Introduction to Circle Packing: The Theory of Discrete Analytic Functions (Cambridge University Press, New York, 2005) 12. W.P. Thurston, The Geometry and Topology of 3-Manifolds, Princeton Lecture Notes, 1978– 1981

Toeplitz Operators Between Distinct Abstract Hardy Spaces Alexei Karlovich

Abstract This note contains the extended abstract of the talk presented by the author on the conference “Spaces of Analytic Functions: Approximation, Interpolation, Sampling” held in Barcelona on 25th–29th of November of 2019. This talk is based on the joint work of the author and Eugene Shargorodsky [9].

1 Classical Hardy Spaces For 1 ≤ p ≤ ∞, let L p := L p (T) represent the standard Lebesgue space on the unit circle T in the complex plane C with respect to the normalized Lebesgue measure dm(t) = |dt|/(2π). For f ∈ L 1 , let 1  f (n) := 2π



π −π

f (eiϕ )e−inϕ dϕ, n ∈ Z,

be the sequence of the Fourier coefficients of f . For 1 ≤ p ≤ ∞, the classical Hardy spaces H p are defined by   f (n) = 0 for all n < 0 . H p := f ∈ L p :  We refer to [5, 13] for the theory of classical Hardy spaces.

A. Karlovich (B) Departamento de Matemática, Faculdade de Ciências e Tecnologia, Centro de Matemática e Aplicações, Universidade Nova de Lisboa, 2829–516 Quinta da Torre, Caparica, Portugal e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. Abakumov et al. (eds.), Extended Abstracts Fall 2019, Trends in Mathematics 12, https://doi.org/10.1007/978-3-030-74417-5_16

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2 The Riesz Projection Consider the operators S and P, defined for a function f ∈ L 1 and an a.e. point t ∈ T by  f (t) + (S f )(t) f (τ ) 1 p.v. dτ , (P f )(t) := , (1) (S f )(t) := πi τ − t 2 T respectively, where the integral is understood in the Cauchy principal value sense. The operator S is called the Cauchy singular integral operator. It is well known that the operators P and S are bounded on L p if p ∈ (1, ∞) and are not bounded on L p if p ∈ {1, ∞} (see, e.g., [2, Sect. 1.42]). Note that using the elementary equality   eiθ ϑ−θ 1 1 + i cot , θ, ϑ ∈ [−π, π], = eiθ − eiϑ 2 2 one can write for f ∈ L 1 and ϑ ∈ [−π, π], (S f )(eiϑ ) =

1 p.v. π



π

−π

f (eiθ )eiθ dθ =  f (0) + i(C f )(eiϑ ), eiθ − eiϑ

where the operator C, called the Hilbert transform, is defined for f ∈ L 1 by  1 p.v. (C f ) eiϑ := 2π



π

−π

 ϑ−θ dθ, ϑ ∈ [−π, π]. f eiθ cot 2

Hence the definition of P f for f ∈ L 1 in terms of the Cauchy singular integral operator given by the second equality in (1) is equivalent to the following definition in terms of the Hilbert transform and the zeroth Fourier coefficient of f (cf. [2, Sect. 1.43]): 1 1 P f := ( f + iC f ) +  f (0). 2 2 If f ∈ L 1 is such that P f ∈ L 1 , then

f (n) =  P f (n) for n ≥ 0,

f (n) = 0 for n < 0 P

(2)

(see, e.g., [9, Sect. 2.6]). If 1 < p < ∞, then the operator P projects the Lebesgue space L p onto the Hardy space H p . The operator P is called the Riesz projection.

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3 The Brown–Halmos Theorem For a ∈ L ∞ , the Toeplitz operator Ta with symbol a on H p , 1 < p < ∞, is defined by Ta f = P(a f ), f ∈ H p . The theory of Toeplitz operators has its origins in the classical paper by Otto Toeplitz [15]. We refer to [2, 12] for the theory of Toeplitz operators on classical Hardy spaces H p , 1 < p < ∞.  For f ∈ L p and g ∈ L p , where 1/ p + 1/ p  = 1, put   f, g :=

T

f (t)g(t) dm(t).

(3)

For n ∈ Z and τ ∈ T, put χn (τ ) := τ n . Then the Fourier coefficients of a function f (n) =  f, χn  for n ∈ Z. The space of all bounded f ∈ L 1 can be expressed by  linear operators from a Banach space E to a Banach space F is denoted by B(E, F). We adopt the standard abbreviation B(E) for B(E, E). Brown and Halmos [3, Theorem 4] (see also [12, Part B, Theorem 4.1.4]) proved the following. Theorem 1 If A ∈ B(H 2 ) and there exists a sequence {an }n∈Z of complex numbers such that Aχ j , χk  = ak− j for all j, k ≥ 0, a (n) = an for all n ∈ Z. then there is a function a ∈ L ∞ such that A = Ta and  Moreover, Ta B(H 2 ) = a L ∞ . An analogue of this result is true for Toeplitz operators acting on H p , 1 < p < ∞ (see [2, Theorem 2.7]). The author [6, 7] extended the Brown–Halmos theorem to the setting of abstract Hardy spaces H [X ] built upon reflexive Banach function spaces X , on which the Riesz projection P is bounded. Tolokonnikov [16] was the first to study Toeplitz operators acting between different Hardy spaces H p and H q . In particular, [16, Theorem 4] contains a description of all symbols generating bounded Toeplitz operators from H p to H q for 0 < p, q ≤ ∞. Recently Le´snik [10, Theorem 4.2] generalized Theorem 1 to the setting of Toeplitz operator acting between abstract Hardy spaces H [X ] and H [Y ], where X ⊂ Y are distinct rearrangement-invariant Banach function spaces such that X is separable and Y has nontrivial Boyd indices. The aim of [9] is to remove from [10, Theorem 4.2] the assumption that spaces X and Y are rearrangement-invariant.

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4 Banach Function Spaces Let L 0+ be the subset of functions in L 0 whose values lie in [0, ∞]. The characteristic (indicator) function of a measurable set E ⊂ T is denoted by I E . Following [1, Chap. 1, Definition 1.1], a mapping ρ : L 0+ → [0, ∞] is called a Banach function norm if, for all functions f, g, f n ∈ L 0+ with n ∈ N, for all constants a ≥ 0, and for all measurable subsets E of T, the following properties hold: (A1) ρ( f ) = 0 ⇔ f = 0 a.e., ρ(a f ) = aρ( f ), ρ( f + g) ≤ ρ( f ) + ρ(g), (A2) 0 ≤ g ≤ f a.e. ⇒ ρ(g) ≤ ρ( f ) (the lattice property), (A3) 0 ≤ f n ↑ f a.e. ⇒ ρ( f n ) ↑ ρ( f ) (the Fatou property), (A4) m(E) < ∞ ⇒ ρ(I E ) < ∞,  (A5) f (t) dm(t) ≤ C E ρ( f ) E

with a constant C E ∈ (0, ∞) that may depend on E and ρ, but is independent of f . When functions differing only on a set of measure zero are identified, the set X of all functions f ∈ L 0 for which ρ(| f |) < ∞ is called a Banach function space. For each f ∈ X , the norm of f is defined by f X := ρ(| f |). The set X under the natural linear space operations and under this norm becomes a Banach space (see [1, Chap. 1, Theorems 1.4 and 1.6]). If ρ is a Banach function norm, its associate norm ρ is defined on L 0+ by 



ρ (g) := sup

T

f (t)g(t) dm(t) : f ∈

L 0+ ,

ρ( f ) ≤ 1 , g ∈ L 0+ .

It is a Banach function norm itself [1, Chap. 1, Theorem 2.2]. The Banach function space X  determined by the Banach function norm ρ is called the associate space (Köthe dual) of X . The associate space X  can be viewed as a subspace of the (Banach) dual space X ∗ .

5 Abstract Hardy Spaces Let X be a Banach function space. Following [17, p. 877], we consider the abstract Hardy space H [X ] built upon the space X , which is defined by   H [X ] := f ∈ X :  f (n) = 0 for all n < 0 . It is clear that if 1 ≤ p ≤ ∞, then H [L p ] is the classical Hardy space H p .

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Lemma 2 ([9, Lemma 1.1]) If the operator P defined by (1) is bounded on a Banach function space X over the unit circle T, then its image P(X ) coincides with the abstract Hardy space H [X ] built upon X . Since X ⊂ L 1 , this lemma follows immediately from formula (2) and the uniquiness theorem for Fourier series.

6 Pointwise Multipliers Let X and Y be Banach function spaces over the unit circle T. Following [11], let M(X, Y ) denote the space of pointwise multipliers from X to Y defined by M(X, Y ) := { f ∈ L 0 : f g ∈ Y for all g ∈ X } and equipped with the natural operator norm f M(X,Y ) = M f B(X,Y ) = sup f g Y . g X ≤1

Here M f stands for the operator of multiplication by f defined by (M f g)(t) = f (t)g(t) for t ∈ T. In particular, M(X, X ) ≡ L ∞ . Note that it may happen that the space M(X, Y ) contains only the zero function. For instance, if 1 ≤ p < q ≤ ∞, then M(L p , L q ) = {0}. The continuous embedding L ∞ ⊂ M(X, Y ) holds if and only if X ⊂ Y continuously. For example, if 1 ≤ q ≤ p ≤ ∞, then L p ⊂ L q and M(L p , L q ) ≡ L r , where 1/r = 1/q − 1/ p.

7 Main Result Let X be a Banach function space over the unit circle T and let X  be its associate space. For f ∈ X and g ∈ X  , let  f, g be defined by (3). The main result of [9] and the talk is the following extension of the Brown–Halmos Theorem 1. Theorem 3 (à la Brown–Halmos) Let X, Y be two Banach function spaces over the unit circle T. Suppose that X is separable and the Riesz projection P is bounded on the space Y . If A ∈ B(H [X ], H [Y ]) and there exists a sequence {an }n∈Z of complex numbers such that (4) Aχ j , χk  = ak− j for all j, k ≥ 0, a (n) = an for all n ∈ Z. then there is a function a ∈ M(X, Y ) such that A = Ta and  Moreover,

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a M(X,Y ) ≤ Ta B(H [X ],H [Y ]) ≤ P B(Y ) a M(X,Y ) . The above theorem and the fact that M(X, X ) ≡ L ∞ (see [11, Theorem 1]) immediately imply the following. Corollary 4 Let X be a separable Banach function spaces over the unit circle T and let the Riesz projection P be bounded on X . If A ∈ B(H [X ]) and there is a sequence {an }n∈Z of complex numbers satisfying (4), then there exists a function a ∈ L ∞ such a (n) = an for all n ∈ Z. Moreover, that A = Ta and  a L ∞ ≤ Ta B(H [X ]) ≤ P B(X ) a L ∞ . The proof of Theorem 3 given in [9, Sect. 4.2] follows the scheme of the proof of [2, Theorem 2.7]. Let us mention two its ingredients (see Lemmas 5 and 6 below), whose proof is not trivial in the case of abstract Hardy spaces H [X ] built upon Banach function spaces X , which are not necessarily rearrangement-invariant.

8 Density of Analytic Polynomials in Abstract Hardy Spaces

For n ∈ Z+ := {0, 1, 2, . . . }, a function of the form nk=−n αk χk , where αk ∈ C for all k ∈ {−n, . . . , n}, is called a trigonometric polynomial of order n. The set of

nall trigonometric polynomials is denoted by P. Further, a function of the form k=0 αk χk with αk ∈ C for k ∈ {0, . . . , n} is called an analytic polynomial of order n. The set of all analytic polynomials is denoted by P A . Lemma 5 ([8, Theorem 1.5]) If X is a separable Banach function space over the unit circle T, then the set of analytic polynomials P A is dense in the abstract Hardy space H [X ] built upon the space X .

9 Formulae for the Norm in a Banach Function Space Let X be a Banach function space over the unit circle T and X  be its associate space. It follows from [1, Chap. 1, Theorem 2.7 and Lemma 2.8] that for every f ∈ X , f X = sup{| f, g| : g ∈ X  , g X  ≤ 1}. If we additionally suppose that the Riesz projection P is bounded on the space X , then the space X  can be substituted by the set of all trigonometric polynomials in the above formula.

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Lemma 6 ([9, Corollary 3.8]) Let X be a Banach function space over the unit circle T. If the Riesz projection P is bounded on X , then for every f ∈ X , f X = sup{| f, p| : p ∈ P, p X  ≤ 1}. The proof of Lemma 6 is essentially based on the following pointwise estimate for the Hilbert transform. For a set G ⊂ [−π, π], we use the following notation I∗G

 iθ e :=



1, θ ∈ G, 0, θ ∈ [−π, π] \ G.

Let |G| denote the Lebesgue measure of G. Lemma 7 ([9, Lemma 3.2]) For every measurable set E ⊂ [−π, π] with 0 < |E| ≤ π/2, there exists a measurable set F ⊂ [−π, π] with |F| = π such that    √   ∗  iϑ  |E|  (CI ) e  > 1 log for a.e. ϑ ∈ E. 2 sin F π  2  Recall that an inner function is a function u ∈ H ∞ (D) such that |u(eiθ )| = 1 for a.e. θ ∈ [−π, π]. In turn, the proof of the above lemma essentially uses the following. Theorem 8 ([14, Lemma 5.1], [4, Theorem 7.2]) If E ⊂ T is a measurable set and γ ⊂ T is an arc such that m(E) = m(γ), then there exists an inner function u satisfying u(0) = 0 and such that the sets u −1 (γ) and E are equal almost everywhere. Acknowledgements This work was partially supported by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the project UIDB/MAT/00297/2020 (Centro de Matemática e Aplicações).

References 1. C. Bennett, R. Sharpley, Interpolation of Operators (Academic, Boston, 1988) 2. A. Böttcher, B. Silbermann, Analysis of Toeplitz Operators, 2nd edn. (Springer, Berlin, 2006) 3. A. Brown, P.R. Halmos, Algebraic properties of Toeplitz operators. J. Reine Angew. Math. 213, 89–102 (1963/64) 4. I. Chalendar, S.R. Garcia, W.T. Ross, D. Timotin, An extremal problem for characteristic functions. Trans. Amer. Math. Soc. 368, 4115–4135 (2016) 5. P.L. Duren, Theory of H p Spaces (Academic, New York, 1970) 6. A. Yu. Karlovich, Norms of Toeplitz and Hankel operators on Hardy type subspaces of rearrangement-invariant spaces. Integr. Equ. Oper. Theory 49, 43–64 (2004) 7. A. Yu. Karlovich, Toeplitz operators on abstract Hardy spaces built upon Banach function spaces. J. Funct. Spaces 2017 (2017), Article ID 9768210, 8 pages 8. A. Karlovich, E. Shargorodsky, More on the density of analytic polynomials in abstract Hardy spaces, in The Diversity and Beauty of Applied Operator Theory. Operator Theory: Advances and Applications, vol. 268 (2018), pp. 319–329

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9. A. Karlovich, E. Shargorodsky, The Brown-Halmos theorem for a pair of abstract Hardy spaces. J. Math. Anal. Appl. 472, 246–265 (2019) 10. K. Le´snik, Toeplitz and Hankel operators between distinct Hardy spaces. Studia Math. 249, 163–192 (2019) 11. L. Maligranda, L.E. Persson, Generalized duality of some Banach function spaces. Indag. Math. 51, 323–338 (1989) 12. N.K. Nikolski, Operators, Functions, and Systems: An Easy Reading, vol. 1 (Hardy, Hankel, and Toeplitz. American Mathematical Society, Providence, 2002) 13. N.K. Nikolski, Hardy Spaces (Cambridge University Press, Cambridge, 2019) 14. Y. Qiu, On the effect of rearrangement on complex interpolation for families of Banach spaces. Rev. Mat. Iberoam. 31, 439–460 (2015) 15. O. Toeplitz, Zur Theorie der quadratischen und bilinearen Formen von unendlichvielen Veränderlichen. I. Teil: Theorie der L-Formen. Math. Ann. 70, 351–376 (1911) 16. V.A. Tolokonnikov, Hankel and Toeplitz operators in Hardy spaces. J. Soviet Math. 37, 1359– 1364 (1987) 17. Q. Xu, Notes on interpolation of Hardy spaces. Ann. Inst. Fourier 42, 875–889 (1992)

Polynomial Hermite Padé m-System and Reconstruction of the Values of Algebraic Functions Aleksandr Komlov

Abstract For an arbitrary tuple of m + 1 analytic germs at some fixed point, we introduce the polynomial Hermite–Padé m-system, which consists of m tuples of polynomials. We find weak asymptotics of such polynomials in the case where Hermite–Padé m-system is constructed by a tuple of germs of meromorphic functions on some (m + 1)-sheeted compact Riemann surface R under an additional condition on R. As a corollary, we get a new method of reconstruction of values of an algebraic function f of order m + 1, determined by its initial germ f 0 , on all Nuttall’s sheets of its Riemann surface except the last one via polynomials of Hermite–Padé m-system constructed by the tuple [1, f 0 , f 02 , . . . , f 0m ].

1 Polynomial Hermite–Padé m-System Let f 0,∞ (z) ≡ 1, f 1,∞ (z), . . . , f m,∞ (z) be m + 1 arbitrary analytic germs at infinity. Fix a natural number k ∈ {1, . . . , m}. For each n ∈ N define a tuple of kth polynomials of Hermite–Padé m-system, constructed by the  [1, f 1,∞ , . . . , f m,∞ ] of germs  tuple polynomials Pn;i1 ,...,ik , 0  i 1 < at the point ∞, as follows. It is a family of m+1 k i 2 < · · · < i k  m, such that deg Pn;i1 ,...,ik  (m + 1 − k)n, at least one of Pn;i1 ,...,ik is not identically 0 and for each tuple of indices 0 < j1 < · · · < jk  m the following relation holds true:     k  1 l (−1) Pn;0, j1 ,..., jl−1 , jl+1 ,..., jk f jl ,∞ (z) = O kn+1 as z → ∞. Pn; j1 ..., jk + z l=1 (1)

A. Komlov (B) Steklov Mathematical Institute of RAS, Moscow, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. Abakumov et al. (eds.), Extended Abstracts Fall 2019, Trends in Mathematics 12, https://doi.org/10.1007/978-3-030-74417-5_17

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  Actually, (1) may be regarded as a system of (n(m + 1) + 1) mk homogeneous linear     coefficients of polynomials Pn;i1 ,...,ik . So polynoequations on n(m + 1) mk + m+1 k mials Pn;i1 ,...,ik always exist but they are not unique, because system (1) is underdetermined. Notice that the coefficients of this system are some linear combinations of Taylor coefficients of the germs f 1,∞ (z), . . . , f m,∞ (z). Recall the definitions of the classical Hermite–Padé polynomials of the first and of the second types. Hermite–Padé polynomials of the first type, constructed by the tuple [1, f 1,∞ , . . . , f m,∞ ] of germs at ∞, of order n are polynomials Q n,i , 0  i  m, such that deg Q n,i  n, at least one of Q n,i is not identically 0 and the following relation is satisfied:   m  1 as z → ∞. (2) Q n, j (z) f j,∞ (z) = O m(n+1) z j=0 Hermite–Padé polynomials of the second type, constructed by the tuple [1, f 1,∞ , . . . , f m,∞ ] of germs at ∞, of order n are polynomials qn,i , 0  i  m, such that deg qn,i  mn, at least one of qn,i is not identically 0 and for each 1  j  m the following relation is satisfied:  qn,0 (z) f j,∞ (z) − qn, j (z) = O

1 z n+1

 as z → ∞.

(3)

Notice that in the case k = 1 conditions (1) and (3) coincide. So the first polynomials of Hermite–Padé m-system are precisely Hermite–Padé polynomials of the second type. Furthermore, for k = m condition (1) “almost coincides” with condition (3). More precisely, if we put Pn;0,1,..., j−1, j+1,...,m := (−1) j Q n; j , then the lefthand sides of (1) (when k = m) and (2) coincide exactly, but the orders of contact of 0 in the right-hand sides are m(n + 1) and mn + 1, respectively. So Hermite–Padé polynomials of the first type are (up to a sign) a particular case of mth polynomials of Hermite–Padé m-system. Actually we are interested in the weak asymptotics of polynomials of Hermite–Padé m-system (in the spirit of Stahl’s theorem for standard Padé polynomials, see [3, 4]). Therefore this difference of contact orders in the righthand sides (which is equal to m − 1 and hence independent on n) does not matter to us. Remark 1 It is possible to define a Hermite–Padé m-system for a tuple of m + 1 C, not only at the point ∞ as in germs at an arbitrary point z 0 of Riemann sphere definition above. In this case, kth polynomials of Hermite–Padé m-system at z 0 ∈ C are defined by the relations analogous to (1): the left-hand side remains the same while the right-hand side is replaced by O((z − z 0 )n(m+1)+1 ) as z → z 0 (that does not depend on k). But we will consider Hermite–Padé m-systems only for tuples of germs at ∞. Nevertheless, all results that we discuss here hold true in the general case (with appropriate change of formulations).

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2 Hermite–Padé m-System for Germs of Functions That Are Meromorphic on an m-Sheeted Compact Riemann Surface Let R be a compact (connected) Riemann surface, π : R → C be an (m + 1)-sheeted holomorphic branched covering of the Riemann sphere C, m ∈ N, and let  be the set of critical values of π. For points of R we will use “bold” symbols, and for their projections we will use the corresponding “usual” symbols. For example, z ∈ R, and π(z) = z. Denote by M (R) the space of meromorphic functions on R. Let f 1 , f 2 . . . , f m ∈ M (R) and assume that 1, f 1 , f 2 , . . . , f m are independent over the field of rational functions C(z). Let ◦ be a point on R that is not critical for the projection π. Without loss of generality we suppose that ◦ ∈ π −1 (∞) and denote ∞ (0) := ◦. Let f 1,∞ (z), . . . , f m,∞ (z) be the germs at the point ∞ (0) of functions f 1 (z), . . . , f m (z), respectively. More precisely, f j,∞ (z) := f j (π0−1 (z)), where π0−1 is the mapping inverse to π in a neighborhood of ∞ (0) . For simplicity we suppose that all germs f j,∞ (z) are holomorphic at ∞. In the rest of the paper we study Hermite– Padé m-system (1) constructed by the tuple [1, f 1,∞ , . . . , f m,∞ ] of these germs. We use the notation Pn, j1 ,..., jk for the corresponding kth polynomial of such Hermite– Padé m-system, without specifying the tuple of germs by which it is constructed. One ought to emphasize that the numbers m in the definition of Hermite–Padé msystem (1) and in the definition of the covering π are the same. We will find the limit distribution of the zeros and the asymptotics of the ratios of the kth polynomials of the Hermite–Padé m-system under the following additional condition on R. It can be assumed that the Riemann surface R is the standard compactification of the Riemann surface of an (m + 1)-valued global analytic function (GAF) π −1 (·) defined on the

as the standard compactification domain C \ . For fixed k, we define the surface R of the Riemann surface of all possible unordered collections of k distinct germs of C \  (for more details, see the function π −1 (·) considered at the same points z ∈

is connected. §4). We will require that the surface R Following Nuttall [1, 2], we introduce the partition of R into sheets determined by the point ∞ (0) (see [5] for details). Let u(z) be a harmonic function on R \ π −1 (∞) with the following logarithmic singularities at π −1 (∞): u(z) = −m log |z| + O(1), z → ∞ (0) ,

Let z ∈ C and let

u(z) = log |z| + O(1), z → π −1 (∞) \ ∞ (0) .

(4)

u 0 (z) ≤ u 1 (z) ≤ · · · ≤ u m (z)

(5)

be the ordered values of u at the points of the set π −1 (z). (In the case where z ∈  we write down the value of u(z) in (5) as many times as the order of z as a critical point for the projection π.) Let us define the jth sheet R( j) of the Nuttall partition of R, j = 0, . . . , m. If u j−1 (z) < u j (z) < u j+1 (z), then we include in R( j) such a

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point z( j) ∈ π −1 (z) that u(z( j) ) = u j (z); for j = 0 we consider only one inequality u 0 (z) < u 1 (z), and for j = m we also consider one inequality u m−1 (z) < u m (z). Otherwise the points of the set π −1 (z) are not included in R( j) . (For z = ∞, we replace u(z) by u(z) − log |z|.) It is clear that always ∞ (0) ∈ R(0) . For each j = C : u j−1 (z) = u j (z)}. It is shown in [5] that F j is a (real) 1, . . . , m, put F j := {z ∈ one-dimensional piecewise analytic set. It yields immediately that π(∂R( j) ) = F j ∪ F j+1 , where ∂R( j) is the boundary of the sheet R( j) ; and we put F0 = Fm+1 = ∅. Put F := ∪mj=1 F j . Following [5], we introduce the matrix A := f j (z(k) ) m k, j=0 (k is the row number, j is the column number) for z ∈ C \ F. For each 0  j1 < · · · < jk  m, let M j1 ,..., jk (z) be the minor of A that corresponds to the columns with numbers j1 , . . . , jk and the rows with numbers 0, 1, . . . , k − 1. It is clear that M j1 ,..., jk ∈ M ( C \ F). It

is connected, then M j1 ,..., jk does can be shown that if for a given k the surface R not vanish identically in any neighbourhood in C \ F. Then it is not difficult to see that for any 0  j1 < · · · < jk  m and 0 ≤ i 1 < · · · < i k ≤ m, the fraction C \ Fk . M j1 ,..., jk (z)/Mi1 ,...,ik (z) is a meromorphic function on cap ∗ → for the weak-∗ convergence and the symbol −→ for We will use the symbol − the convergence in (logarithmic) capacity. Let ddc be the standard analogue of the Laplace operator on C (which, in the general case, maps currents of degree 0 to currents of degree 2).

is connected. Theorem 2 Suppose that for a given k = 1, . . . , m the surface R Then (1) There exists a number L ∈ N such that for an arbitrary neighborhood V of the compact set Fk and for all sufficiently large n, n > N = N (V ), the number of zeroes of the polynomials Pn;i1 ,...,ik outside of V is not greater than L. (2) We have   k−1  1 c ∗ c dd log |Pn;i1 ,...,ik (z)| − → −dd u s (z) (6) n s=0 in the space of signed measures on C as n → ∞.

is connected. Theorem 3 Suppose that for a given k = 1, . . . , m the surface R Then for any compact set K ⊂ C \ Fk , Pn; j1 ,..., jk (z) cap M j1 ,..., jk (z) , z ∈ K , as n → ∞. −→ Pn;i1 ,...,ik (z) Mi1 ,...,ik (z)

(7)

Moreover, for any ε > 0 we have 

1/n Pn; j1 ,..., jk (z) (z) M j ,..., j 1 k · eu k (z)−u k−1 (z)  1 + ε → 0. (8) cap z ∈ K : − Pn;i1 ,...,ik (z) Mi1 ,...,ik (z)

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Unfortunately, for some Remark 4 (About the condition of connectedness of R) natural surface R this condition fails. For example, let R be a compactification of √ the Riemann surface of the GAF w(z) = 4 (z − 1)/z and let π : (z, w) → z. Then it

consists of two connected is not difficult to see that for k = 2 the Riemann surface R

components. Nevertheless, the following sufficient condition of connectedness of R

for shows that after a “small perturbation” of R we can achieve connectedness of R all k = 1, . . . , m. Assume that a projection π : R → C is such that all its critical points are of the first order and that for each point z ∈ C there is at most one critical

for all k = 1, . . . , m point of π over it (i.e. in the set π −1 (z)). Then the surfaces R are connected. Remark 5 Note that the analogues of Theorems 2 and 3 for Hermite–Padé polynomials of the first type, i.e., practically, for mth polynomials of Hermite–Padé msystem, are proven in [5]. At the same time the respective analogues for Hermite–Padé polynomials of the second type, i.e. for 1st polynomials of Hermite–Padé m-system, were proven in [2] in some general case. In this work, as in [5], we use ideas of the Nuttall approach, see [1, 2].

3 Reconstruction of the Values of an Algebraic Function Let f be an (m + 1)-sheeted algebraic function on Cand let some its germ f ∞ at −s ∞∈ C be given by Taylor expansion at ∞: f ∞ (z) = ∞ s=0 cs z . (For convenience here we suppose that f ∞ is holomorphic at ∞.) We are going to reconstruct the values of f on all sheets of the Nuttall partition of its Riemann surface except the mth one via polynomials of Hermite–Padé m-system. Consider f as a meromorphic function on its Riemann surface R, which is (m + 1)-sheeted branched covering of C with the natural projection π(z) = z. Let the given germ f ∞ correspond to the germ of f at the point ∞ (0) ∈ π −1 (∞). Since f is an (m + 1)-sheeted algebraic function, the functions 1, f, f 2 , . . . , f m are independent over C(z). Let us consider Hermite–Padé m-system, constructed by the 2 m , . . . , f∞ ] at ∞. Note that in this case M0,...,k−2,k (z)/M0,...,k−1 (z) = tuple [1, f ∞ , f ∞ k−1 (s) f (z ) (the Nuttall partition is determined by ∞ (0) ). Therefore we have the s=0 following corollary from Theorem 3.

is connected. Let Corollary 6 Let k ∈ {1, . . . m} and for this k the surface R Pn;0,...,k−2,k (z), Pn;0,...,k−1 (z) be corresponding kth polynomials of Hermite–Padé m2 m , . . . , f∞ ] at ∞. Then for any compact system, constructed by the tuple [1, f ∞ , f ∞ set K ⊂ C \ Fk we have Pn;0,...,k−2,k (z) cap  −→ f (z(s) ), z ∈ K , as n → ∞. Pn;0,...,k−1 (z) s=0 k−1

(9)

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Thus, considering kth polynomials of this Hermite–Padé m-system, we reconstruct the sum of the values of f on first k sheets of the corresponding Nuttall

for all k = 1, . . . , m are partition of the Riemann surface of f . So, if the surfaces R connected, we sequentially reconstruct the values of f on all sheets of the Nuttall partition, except the mth (the last) sheet, outside the set π −1 (F). A sufficient condi for all k = 1, . . . , m is given in Remark 4. tion for connectedness of the surfaces R Note that to compute polynomials of Hermite–Padé m-system we need to solve the systems of homogeneous linear equations (1), whose coefficients are linear combinations of Taylor coefficients ck of the initial germ f ∞ . One ought to remark that the idea of reconstruction of sums of values of a function on its Nuttall sheets instead of values themselves via appropriate polynomials was firstly suggested in [6]. Remark 7 Clearly, we can reconstruct the values of an (m + 1)-sheeted algebraic C (not only z 0 = ∞). For this function from its germ f z0 at an arbitrary point z 0 ∈ purpose we need to consider a Hermite–Padé m-system constructed by the tuple [1, f z0 , f z20 , . . . , f zm0 ] at the point z 0 , see Remark 1.

4 Ideas of the Proof of Theorems 2 and 3 First of all, we fix k ∈ {1, . . . , m} in Theorems 2 and 3. Now, we introduce an auxiliary

associated with R. Since compact (in general, disconnected) Riemann surface R, π: R → C is an (m + 1)-sheeted branched covering of C, we may consider R as the Riemann surface of some (m + 1)-sheeted algebraic function w(z). Since  is the set of critical values of π, we understand R as the standard compactification of the graph

is the standard compactification of the (in of w(z) over C \ . The Riemann surface R general, disconnected) Riemann surface of all sets of k different germs of   unordered

is a m+1 -sheeted (in general, disconnected) w(·) at the same points z ∈ C \ . So R k

→ branched covering of C with the set of branch points . By

π: R C we denote

we will use corresponding the standard projection, generated by π. For points of R

and

“bold” symbols with “tilde” (for example,

z ∈ R, π (

z) = z). Since for z ∈ C\ F all inequalities in (5) are strong, for such z we have π −1 (z) := {z(0) , z(1) , . . . , z(m) },

decomposes into over C \ F the Riemann surface R where z( j) ∈ R( j) .Therefore, 

( j1 j2 ... jk ) , 0 ≤ j1 < j2 < · · · < jk ≤ m. The sheet sheets R disjoint union of m+1 k

( j1 j2 ... jk ) consists of unordered sets {wz( j1 ) (·), wz( j2 ) (·), . . . , wz( jk ) (·)}, where wz( jl ) (·) R C \ F. We denote by

z( j1 j2 ... jk ) the is the germ of w(·) at the point z( jl ) , for all z ∈ ( j1 j2 ... jk )

that lies above z ∈ C \ F. Note that for z ∈ C \ Fk , we point of a sheet R

(01...k−1) over C \ Fk (not have u k−1 (z) < u k (z) in (5). So we can define the sheet R only over C \ F). Going further, we give another definition of kth polynomials of Hermite–Padé m-system in terms of new meromorphic functions on R constructed by f 1 , . . . , f m . Recall that the matrix A := f j (z(l) ) l,m j=0 (l is the row number, j is the column number). For each 0 ≤ j1 < j2 < · · · < jk ≤ m and 0 ≤ l1 < l2 < · · · < lk ≤ m, we k denote by M lj11,...,l ,..., jk (z) the minor of the matrix A that is composed by the columns with

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k numbers j1 , . . . , jk and the rows with numbers l1 , l2 , . . . , lk . We denote by Alj11,...,l ,..., jk (z) l1 ,...,lk k the algebraic complement of the minor M lj11,...,l ,..., jk (z) (i.e., A j1 ,..., jk (z) is the complel1 ,...,lk mentary minor to M j1 ,..., jk (z) with the sign (−1) j1 ···+ jk +l1 ···+lk ). As the matrix A, all l1 ,...,lk k minors M lj11,...,l ,..., jk (z) and all algebraic complements A j1 ,..., jk (z) are defined for each z∈ C \ F. For z ∈ C \ F we also introduce the matrix Mw (z) := w j (z(l) ) l,m j=0 (l is the row number, j is the column number). For each 0 ≤ l1 < l2 < · · · < lk ≤ m we denote by Mwl1 ,...,lk (z) the minor of the matrix Mw (z) that corresponds to the columns with numbers 0, 1, . . . , k − 1 and the rows with numbers l1 , l2 , . . . , lk . z) and For each 0 ≤ j1 < j2 < · · · < jk ≤ m, let us define the functions M j1 ,..., jk (

\

z) for

z∈R π −1 (F) as follows A j1 ,..., jk (

M j1 ,..., jk (

z(l1 ...lk ) ) :=

z(l1 ...lk ) ) := A j1 ,..., jk (

k M lj11,...,l ,..., jk (z)

Mwl1 ,...,lk (z)

,

(10)

k l1 ,...,lk (z) Alj11,...,l ,..., jk (z)Mw

det A

.

(11)

Note that (det A)2 is a meromorphic function on C and det A ≡ 0, since the functions 1, f 1 , f 2 , . . . , f m are independent over C(z). Therefore, the definition (11) is correct. The following proposition is readily verified: Proposition 8 For any 0 ≤ j1 < . . . < jk ≤m, the functions M j1 ,..., jk (

z) and A j1 ,..., jk (

z)

extend as meromorphic functions to the whole R. Now we can give a new definition of kth polynomials of Hermite–Padé m-system. Theorem 9 There exists p ∈ N (that doesn’t depend on n) such that all kth polynomials of Hermite–Padé m-system Pn; j1 ,..., jk satisfy the following relation: 

Pn; j1 ,..., jk (z)A j1 ,..., jk (

z(l1 ...lm ) ) = O

0≤ j1 0 for γn,m almost all V ∈ G(n, m). (3) If dim A > 2m, then PV (A) has non-empty interior for γn,m almost all V ∈ G(n, m). The conditions dim A ≤ m and dim A > m in (1) and (2) are of course necessary. The condition dim A > 2m in (3) is necessary if m = 1. I don’t know if it is necessary when m > 1. In the case m = 1 the example can be obtained with Besicovitch sets, first in the plane, showing that there is no theorem in the plane, and then taking cartesian products. More precisely, let B ⊂ R2 be a Borel set of measure zero which contains a countable dense set of lines in every direction. Then its complement has full Lebesgue measure and none of its projections has interior points. How much more one can say about the size of the sets of exceptional planes? Kaufman [8] proved in 1968 the first item of the following theorem in the plane (generalized in [11]), Falconer [3] in 1982 the second and Peres and Schlag [23] in 2000 the third. Recall that the dimension of G(n, m) is m(n − m). To get a better feeling of this notice that in the case m = 1 the three upper bounds are n − 2 + dim A, n − dim A and n + 1 − dim A. Theorem 2 Let A ⊂ Rn be a Borel set. (1) If dim A ≤ m, then dim{V ∈ G(n, m) : dim PV (A) < dim A} ≤ m(n − m) − m + dim A. (2) If dim A > m, then

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dim{V ∈ G(n, m) : Lm (PV (A)) = 0} ≤ m(n − m) + m − dim A. (3) If dim A > 2m, then dim{V ∈ G(n, m) : Int(PV (A)) = ∅} ≤ m(n − m) + 2m − dim A. The bounds in (1) and (2) are sharp by the examples which Kaufman and I constructed in 1975 in [9]. I don’t know if the bound in (3) is sharp. Another, seemingly very difficult, problem is estimating the dimension of the set in (1) when dim A is replaced by some u < dim A. We still have by the same proof dim{V ∈ G(n, m) : dim PV (A) < u} ≤ m(n − m) − m + u, but this probably is not sharp when u < dim A. In any case it is far from sharp in the plane when u = dim A/2: Theorem 3 Let A ⊂ R2 be a Borel set. Then dim{e ∈ S 1 : dim Pe (A) ≤ dim A/2} = 0.

(1)

To get some idea where dim A/2 comes from, notice that the case of the inequality dim M Pe (A) < dim M A/2 is very easy for the upper Minkowski dimension dim M , and even more is true: there can be at most one direction e for which dim M Pe (A) < dim M A/2. This is an easy exercise. However for the Hausdorff dimension the exceptional set can always be uncountable. Theorem 3 is due to Bourgain, [1, 2]. Bourgain’s result is more general and it includes a deep discretized version. The proof uses methods of additive combinatorics. D. M. Oberlin gave a simpler Fourier-analytic proof in [20], but with dim Pe (A) ≤ dim A/2 replaced by dim Pe (A) < dim A/2. Using combinatorial methods He [6] proved analogous higher dimensional results. More generally, it might be true, and has been conjectured by Oberlin [20], that Kaufman’s estimate dim{e ∈ S 1 : dim Pe (A) < u} ≤ u

(2)

could be extended for dim A/2 ≤ u ≤ dim A to dim{e ∈ S 1 : dim Pe (A) < u} ≤ 2u − dim A.

(3)

This would be sharp, as the constructions in [9] show. Theorem 3 is the only case where this is known.

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3 Plane Sections and Radial Projections What can we say about the dimensions if we intersect a subset A of Rn , dim A > m, with (n − m)-dimensional planes? The inequalities, dim(A ∩ (V + x)) ≤ dim A − m for Hm almost all x ∈ V ⊥ , dim(A ∩ (V + x)) ≤ dim A − m for γn,n−m almost all V ∈ G(n, n − m) are fairly easy. The lower bounds are not as obvious, but we have the following result, originally proved by Marstrand in the plane in [10] and then in general dimensions in [11]: Theorem 4 Let m < s ≤ n and let A ⊂ Rn be Hs measurable with 0 < Hs (A) < ∞. Then (1) For Hs almost all x ∈ A, dim(A ∩ (V + x)) = s − m for γn,n−m almost all V ∈ G(n, n − m), (2) for γn,n−m almost all V ∈ G(n, n − m), Hm ({x ∈ V ⊥ : dim(A ∩ (V + x)) = s − m}) > 0. These statements are essentially equivalent. Clearly, this generalizes part (2) of Theorem 1. Here are exceptional set estimates related to both statements. The first of these is due to Orponen [21]: Theorem 5 Let m < s ≤ n and let A ⊂ Rn be Hs measurable with 0 < Hs (A) < ∞. Then there is a Borel set E ⊂ G(n, n − m) such that dim E ≤ m(n − m) + m − s and for V ∈ G(n, n − m) \ E, Hm ({x ∈ V ⊥ : dim(A ∩ (V + x)) = s − m}) > 0. The bound m(n − m) + m − s = dim G(n, n − m) + m − s is the same as in Theorem 2(2). Since it is sharp there, it also is sharp here. The second estimate is due to Orponen and the author [19]: Theorem 6 Let m < s ≤ n and let A ⊂ Rn be Hs measurable with 0 < Hs (A) < ∞. Then there is a Borel set B ⊂ Rn such that dim B ≤ m and for x ∈ Rn \ B, γn,n−m ({V ∈ G(n, n − m) : dim A ∩ (V + x) = s − m}) > 0. This probably is not sharp. I expect that the sharp bound for dim B in the case m = n − 1 would again be 2(n − 1) − s, as for the orthogonal projections and as in Orponen’s radial projection theorem 8 below. Moreover, one could hope for an

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exceptional set estimate including both cases, that is, estimate on the dimension of the exceptional pairs (x, V ). The proof of Theorem 6 is based on the following theorem of Orponen and the author in [19]: Theorem 7 Let A and B be Borel subsets of Rn . (1) If dim A > m and dim B > m, then   γn,m {V ∈ G(n, m) : Lm (PV (A) ∩ PV (B)) > 0} > 0. (2) If dim A > 2m and dim B > 2m, then γn,m ({V ∈ G(n, m) : Int(PV (A) ∩ PV (B)) = ∅}) > 0. (3) If dim A > m, dim B ≤ m and dim A + dim B > 2m, then for every  > 0, γn,m ({V ∈ G(n, m) : dim(PV (A) ∩ PV (B)) > dim B − }) > 0. I give a sketch of the proof of Theorem 6 in the plane. Suppose that it is not true and that there is a set B with dim B > 1 such that through the points of B almost all lines meet A in a set of dimension less than s − 1. On the other hand, by Theorem 4 typical lines through the points of A meet A in a set of dimension s − 1. By Fubinitype arguments and using Theorem 7 we can find such typical lines meeting both A and B leading to a contradiction. For radial projections πx : Rn \ {x} → S n−1 , πx (y) =

y−x , |y − x|

Orponen proved in [22] the following sharp estimate for the exceptional set of x ∈ Rn : Theorem 8 Let A ⊂ Rn be a Borel set with dim A > n − 1. Then there is a Borel set B ⊂ Rn with dim B ≤ 2(n − 1) − dim A such that for every x ∈ Rn \ B, Hn−1 (πx (A)) > 0.

4 General Intersections The following theorem was proved in [13]: Theorem 9 Let s and t be positive numbers with s + t > n and t > (n + 1)/2. Let A and B be Borel subsets of Rn with Hs (A) > 0 and Ht (B) > 0. Then for almost all g ∈ O(n),

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Ln ({z ∈ Rn : dim A ∩ (g(B) + z) ≥ s + t − n}) > 0.

(4)

The condition t > (n + 1)/2 comes from some Fourier transform estimates. Probably it is not needed. This was preceded by the papers of Kahane [7] and the author [12] in which it was shown that the above theorem is valid for any s + t > n provided larger transformation groups are used. For example, it suffices to add also typical dilations x → r x, r > 0. Here we really need the inequality dim A ∩ (g(B) + z) ≥ s + t − n, the opposite inequality can fail very badly: for any 0 ≤ s ≤ n there exists a Borel set A ⊂ Rn such that dim A ∩ f (A) = s for all similarity maps f of Rn . This follows from [4]. The reverse inequality holds if dim A × B = dim A + dim B, see [14], Theorem 13.12. This latter condition is valid if, for example, one of the sets is Ahlfors-David regular, see [14], 8.12. For such reverse inequalities no rotations g are needed (or, equivalently, they hold for every g). The following two exceptional set estimates were proven in [16]: Theorem 10 Let s and t be positive numbers with s + t > n + 1. Let A and B be Borel subsets of Rn with Hs (A) > 0 and Ht (B) > 0. Then there is a Borel set E ⊂ O(n) such that dim E ≤ 2n − s − t + (n − 1)(n − 2)/2 = n(n − 1)/2 − (s + t − (n + 1)) and for g ∈ O(n) \ E, Ln ({z ∈ Rn : dim A ∩ (g(B) + z) ≥ s + t − n}) > 0.

(5)

Notice that n(n − 1)/2 is the dimension of O(n). The condition s + t > n + 1 is not needed in the case where one of the sets has small dimension and in this case we have a better upper bound for dim E, although we then need a slight technical reformulation: Theorem 11 Let A and B be Borel subsets of Rn with dim A = s, dim B = t and suppose that s ≤ (n − 1)/2. If 0 < u < s + t − n, then there is a Borel set E ⊂ O(n) with dim E ≤ n(n − 1)/2 − (s + t − n) such that for g ∈ O(n) \ E, Ln ({z ∈ Rn : dim A ∩ (g(B) + z) ≥ u}) > 0.

(6)

The formulation in [16] is slightly weaker, but it easily implies the above. What helps here is the following sharp decay estimate for quadratic spherical averages for Fourier transforms of measures with finite energy:

Hausdorff Dimension Exceptional Set Estimates for Projections …

 |v|=1

135

| μ(r v)|2 dv ≤ C(n, s)Is (μ)r −s , r > 0, 0 < s ≤ (n − 1)/2.

Such an estimate is false for s > (n − 1)/2. Let us speculate about the possible sharp estimates in the plane. In Theorem 10 we have the upper bound 4 − (s + t) and in Theorem 11 we have 3 − (s + t). Could the second estimate be valid whenever s + t > 2? This would mean that the dimension is 0 when s + t > 3. Could the exceptional set even be countable then? I don’t think so, but I don’t have a counter-example. Anyway, it need not be empty whatever the dimensions are. That is, using only translations we cannot say much for general sets. The following example follows from [12]: there are compact subsets A and B of Rn such that dim A = dim B = n and A ∩ (B + z) contains at most one point for every z ∈ Rn . Some new results related to this topic were proven in [17] and [18]. Acknowledgements The author was supported by the Academy of Finland through the Finnish Center of Excellence in Analysis and Dynamics Research.

References 1. J. Bourgain, On the Erdös-Volkmann and Katz-Tao ring conjectures. Geom. Funct. Anal. 13, 334–365 (2003) 2. J. Bourgain, The discretized sum-product and projection theorems. J. Anal. Math. 112, 193–236 (2010) 3. K.J. Falconer, Hausdorff dimension and the exceptional set of projections. Mathematika 29, 109–115 (1982) 4. K.J. Falconer, Classes of sets with large intersection. Mathematika 32, 191–205 (1985) 5. K.J. Falconer, T. O’Neil, Convolutions and the geometry of multifractal measures. Math. Nachr. 204, 61–82 (1999) 6. W. He, Orthogonal projections of discretized sets, arXiv:1710.00759, to appear in J. Fractal Geom. 7. J.-P. Kahane, Sur la dimension des intersections, in Aspects of Mathematics and Applications. North-Holland Mathematical Library, vol. 34, pp. 419–430 (1986) 8. R. Kaufman, On Hausdorff dimension of projections. Mathematika 15, 153–155 (1968) 9. R. Kaufman, P. Mattila, Hausdorff dimension and exceptional sets of linear transformations. Ann. Acad. Sci. Fenn. A Math. 1, 387–392 (1975) 10. J.M. Marstrand. Some fundamental geometrical properties of plane sets of fractional dimensions, Proc. London Math. Soc. 4(3), 257–302 (1954) 11. P. Mattila, Hausdorff dimension, orthogonal projections and intersections with planes. Ann. Acad. Sci. Fenn. A Math. 1, 227–244 (1975) 12. P. Mattila, Hausdorff dimension and capacities of intersections of sets in n-space. Acta Math. 152, 77–105 (1984) 13. P. Mattila, On the Hausdorff dimension and capacities of intersections. Mathematika 32, 213– 217 (1985) 14. P. Mattila, Geometry of Sets and Measures in Euclidean Spaces (Cambridge University Press, Cambridge, 1995) 15. P. Mattila, Fourier Analysis and Hausdorff Dimension (Cambridge University Press, Cambridge, 2015)

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16. P. Mattila, Exceptional set estimates for the Hausdorff dimension of intersections. Ann. Acad. Sci. Fenn. A Math. 42, 611–620 (2017) 17. P. Mattila, Hausdorff dimension and projections related to intersections, arXiv:2005.04947 18. P. Mattila. Hausdorff dimension of intersections with planes and general sets, arXiv:2005.11790 19. P. Mattila, T. Orponen, Hausdorff dimension, intersections of projections and exceptional plane sections. Proc. Am. Math. Soc. 144, 3419–3430 (2016) 20. D.M. Oberlin, Restricted Radon transforms and projections of planar sets. Canad. Math. Bull. 55, 815–820 (2012) 21. T. Orponen, Slicing sets and measures, and the dimension of exceptional parameters. J. Geom. Anal. 24, 47–80 (2014) 22. T. Orponen, A sharp exceptional set estimate for visibility. Bull. London. Math Soc. 50, 1–6 (2018) 23. Y. Peres, W. Schlag, Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions. Duke Math. J. 102, 193–251 (2000)

Generic Boundary Behaviour of Taylor Series in Banach Spaces of Holomorphic Functions Jürgen Müller

Abstract For several classical Banach spaces X of functions holomorphic in the unit disc we consider the question to which extent, for some f ∈ X , the sequence (Sn f ) of the partial sums of the Taylor series about 0 behaves erratically on compact sets outside of the unit disc. The problem is closely related to a question about simultaneous polynomial approximation.

1 Universality of Taylor Sections Let C∞ be the extended complex plane. For an open set  ⊂ C∞ we denote by H () the Fréchet space of functions holomorphic in  (and vanishing at ∞ if ∞ ∈ ) endowed with the topology of locally uniform convergence. If 0 ∈  and f ∈ H (), we write with Pk (z) := z k Sn f :=

n 

aν ( f )Pk

ν=0

for the n-th partial sum of the Taylor expansion of f about 0. A classical question in complex analysis is how the partial sums Sn f behave outside the disc of convergence and in particular on the boundary of the disc. Based on Baire’s category theorem, it can be shown that for functions f holomorphic in the unit disc D generically the sequence (Sn f ) turns out to be “maximally divergent” outside of D. In order to clarify in which sense maximal divergence is understood, we introduce some notations. For E compact in C we write A(E) := {h ∈ C(E) : h holomorphic in E ◦ }.

J. Müller (B) University of Trier, FB IV, Mathematics 54286 Trier, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. Abakumov et al. (eds.), Extended Abstracts Fall 2019, Trends in Mathematics 12, https://doi.org/10.1007/978-3-030-74417-5_20

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If E has connected complement then, according to Mergelyan’s theorem, A(E) is the closure of the set of polynomials in C(E), where C(E) is endowed with the uniform norm || · || E . For  an infinite subset of N and f ∈ H (D), we say that the sequence (Sn f ) is -universal on E if for all h ∈ A(E) a subsequence of (Sn f )n∈ tends to h uniformly on E, in other words, if {Sn f : n ∈ } is dense in A(E). In 1996, Nestoridis proved that for each  a residual set of functions f ∈ H (D) have the property that (Sn f ) is -universal on each compact set E ⊂ C \ D with connected complement (see [22]). For corresponding results we refer in particular to the expository articles [10] and [16]. For results on universal series in a more general framework see also [2]. If X is a Fréchet space of functions in H (D) which is continuously embedded in H (D) we say that a compact set E ⊂ C \ D is a set of universality for X if for all infinite sets  ⊂ N there is a residual set of functions f ∈ X with the property that the sequence (Sn f ) is -universal on E. Moreover, we call E a set of σ-universality  for X , if an increasing sequence of sets of universality E k exists with E = k∈N E k . In this case, Baire’s theorem implies that for each  there is a residual set of f ∈ X having the property that each continuous function on E is the pointwise limit of some subsequence of (Sn f )n∈ . Nestoridis’ result implies that the unit circle T is a set of σ-universality for H (D). The situation changes if we consider classical Banach spaces of holomorphic functions in D. Our aim is to study the generic limit behaviour of the Taylor sections Sn f outside of D and in particular on T for functions in Hardy spaces H p , in Bergman spaces A p and in the Dirichlet space D. We recall that, with m denoting the normalised arc length measure on T and m 2 the normalised area measure on D and for 1 ≤ p < ∞,  H p = { f ∈ H (D) :  f  p := sup | f (r ·)| p dm < ∞}, r 1 and each f ∈ H p the partial sums Sn f converge to f ∗ almost everywhere on T. Moreover, due to results of Kolmogorov, for functions in H 1 convergence in measure still holds and hence each subsequence of (Sn f ) has a sub-subsequence that converges almost everywhere to the boundary function f ∗ . In the case of the classical Dirichlet space D := D 2 , more can be said. We recall that a set E ⊂ C is called polar if it has vanishing logarithmic capacity, and that a property is said the be satisfied quasi everywhere if it is satisfied up to a polar set. If f belongs to D

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then existence of f ∗ is guaranteed quasi everywhere (Beurling’s theorem; see e.g. [8, Theorem 3.2.1]) and convergence of (Sn f ) to f ∗ holds in all points ζ ∈ T where f ∗ (ζ) exists (see e.g. [20, p. 12]). In contrast, generically in A p functions do not have non-tangential limits almost everywhere. However, from a result of Shkarin ([23]; cf. also [9, Corollary 2]) it follows that for all f ∈ A1 and for all arcs E ⊂ T the sequence (Sn f ) has at most one limit function h ∈ C(E) on E. Concerning possible sets of universality, we conclude from the above results: • No closed E ⊂ T with m(E) > 0 is a set of universality for any H p . • No closed E ⊂ T of positive capacity is a set of universality for D. • No arc E ⊂ T is a set of σ-universality for any A p . In the converse direction we have (see [3, 21]) Theorem 1 Closed sets E ⊂ T with m(E) = 0 are sets of universality for all H p and closed polar sets E ⊂ T are sets of universality for D. Moreover, there are sets of full measure E ⊂ T which are sets of σ-universality for all A p . The proof of Theorem 1 is based on results on simultaneous approximation by polynomials. We write X ⊕ Y for the (external) direct sum of two Fréchet spaces X and Y and we say that a compact set E ⊂ C \ D is a set of simultaneous approximation for the Fréchet space X ⊂ H (D) if the pairs (P|D , P| E ), where P ranges over the set of polynomials, form a dense set in the sum X ⊕ A(E). Lemma 2 Let X be a Fréchet space continuously embedded in H (D) and let E ⊂ C \ D. If E is a set of simultaneous approximation for X then E is a set of universality for X . Proof Since X is continuously embedded in H (D), the mappings X f → ak ( f ) Pk | E ∈ A(E) are continuous. For fixed infinite set  ⊂ N0 we consider the family (Sn )n∈ (more precisely f → Sn f | E ) of continuous linear mappings from X to A(E). The Universality Criterion (see e.g. [10, Theorem 1] or [11, Theorem 1.57]) implies that it is sufficient—and necessary—to show that for each pair ( f, g) ∈ X ⊕ A(E) and each ε > 0 there exist a polynomial P and an integer n ∈  so that || f − P|| X < ε and ||g − Sn P|| E < ε. Since Sn P = P for all polynomials P and all large n (depending on the degree of P), this is satisfied if E is a set of simultaneous approximation for X . While the simultaneous approximation condition is sufficient for the universality property to hold, is turns out to be not necessary in general: Obviously, no non-empty set E ⊂ T is a set of universality for the disc algebra A(D). On the other hand, finite sets E ⊂ T are (exactly the) sets of universality for A(D) (see [13], [6, Theorem 3.3] for the sufficiency and [4, Corollary 3.5] for the necessity).

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2 Simultaneous Approximation by Polynomials Let now X = (X,  ·  X ) be a Banach space of functions continuous on some set M ⊂ C. Moreover, we suppose that the polynomials are dense in X and that the 1/k sequence (||Pk || X )k is bounded. In this case we briefly speak of a regular space  X . By X we denote the (norm) dual of X and by H (0) the linear space of germs of functions holomorphic at 0. Then the Cauchy transform K X : X  → H (0) with respect to X is defined by (K X φ)(w) =

∞ 

φ(Pν )w ν

ν=0

for |w| sufficiently small and φ ∈ X  . Since the polynomials form a dense set in X , the Cauchy transform K X is injective. Then the range X ∗ = R(K X ) of K X is called Cauchy dual of X . The following consequence of the Hahn-Banach theorem (see [14, Theorem 1.2], [6, Lemma 2.7])) is the basis for our subsequent considerations. Lemma 3 Let X and Y be regular. Then X ∗ ∩ Y ∗ = {0} if and only if the pairs (P, P), where P ranges over the set of polynomials, form a dense set in the sum X ⊕ Y. Let E ⊂ C be compact with connected complement. Since A(E) is a subspace of C(E), according to the Hahn-Banach theorem and the Riesz representation theorem, each φ ∈ A(E) can be represented (not uniquely) by some complex Borel measure μ supported on E. If μ is an arbitrary representing measure of φ then K A(E) (φ) is the germ at 0 of the Cauchy transform  μ ∈ H (C∞ \ (1/E)) given by   μ(w) = E

dμ(z) (w ∈ C∞ \ (1/E)). 1 − zw

Similarly, for 1 < p < ∞ and φ ∈ (H p ) the Cauchy transform is of the form  (K H p φ)(w) = ∗

T

h(z) dm(z) (w ∈ D), 1 − zw ∗

for some h ∈ H p , where p ∗ denotes the conjugate exponent. Then K H p (φ) ∈ H p ∗ and, more precisely, the Cauchy dual (H p )∗ equals H p in this sense (cf. [5] or [7]). ∗ In the same way, (D p )∗ = A p (see [5, p. 88]). By proving uniqueness of the Cauchy transforms involved and applying Lemma 3, one can show Theorem 4 Closed sets E ⊂ T with m(E) = 0 are sets of simultaneous approximation for all H p and closed polar sets E ⊂ T are sets of simultaneous approximation for D.

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The first statement, originally proved by Havin (see [12], cf. [14]) is a consequence of the F. and M. Riesz theorem. As I learnd during the SAFAIS conference, the second one is due to Khrushchev and Peller ([15]). It was reinvented in [21] with a proof using Tumarkin’s theorem and the fact that polar sets are removable for A2 (2D \ (1/E)). As I realized recently, this proof contains a flaw, which fortunately can be corrected by a very nice argument given by Koosis ([19]) and published already shortly after [15]. The first two statements of Theorem 1 follow immediately from Theorem 4 and Lemma 2. The third statement of Theorem 1 is a consequence of Lemma 2 and the following result on simultaneous approximation in A p , which is based on a deep theorem of Khrushchev on uniqueness of Cauchy transforms (see [14, Theorem 4.1]), and the fact that an increasing sequence (E k ) of Cantor sets E k ⊂ T satisfying (see [3]). Recall that a closed Carleson’s condition with m(E k ) → 1 as k → ∞ exists set E ⊂ T satisfies Carleson’s (entropy) condition if I m(I ) log(1/m(I )) < ∞, where the sum ranges over all components I ⊂ T of T \ E. From [3, Theorem 2.2] one obtains Theorem 5 Each Cantor set E ⊂ T of positive measure satisfying Carleson’s condition is a set of simultaneous approximation for all A p with 1 < p < ∞. In the very recent paper [18], Khrushchev proved two new results on universality of Taylor series.1 Both are based on results on simultaneous approximation that can be seen as extensions of Theorem 4: Firstly, Corollary 1 in [18] says that closed sets E ⊂ T with m(E) = 0 are sets of simultaneous approximation for the quotient space C(T)/B, where B := { f ∈ C(T) :  f (n) = 0 for n ∈ N0 }. According to the duality results of Fefferman, Sarason and Stein (see e.g. [5, Sect. 3.5]), this can also be seen as a result on simultaneous approximation for the space V M O A. Secondly, replacing logarithmic capacity by corresponding p-capacities, sets of vanishing p-capacity 1/ p are sets of simultaneous approximation for the analytic Besov spaces P+ B p , where 1 < p < ∞ (for p = 2, the Dirichlet space case is recovered). Hence, sets E ⊂ T of vanishing m-measure are sets of universality for V M O A and sets of vanishing 1/ p p-capacity are sets of universality for P+ B p . The latter statement is essentially [18, Theorem 3.2]). In all above results, simultaneous approximation is restricted to the case of compact subsets of the unit circle T. We briefly discuss the situation of more general sets E ⊂ C \ D, which was considered only recently and is not yet fully clarified. It is known that compact sets E ⊂ C \ D with connected complement are sets of universality even for the space A∞ (D) of functions in H (D) having the property that each derivative has a continuous extension to D (see e.g. [16, Theorem 4.2]). In view of Theorem 1, a reasonable guess is that compact sets E ⊂ C \ D with connected complement and touching T in a set of vanishing m-measure would be sets of universality for H p . It turns out that this in not true. More precisely, using an idea of Bayart (see [1]), in [6] the following result was proved: 1

The paper appeared online after the SAFAIS conference and I became aware of it while writing the extended abstract.

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Theorem 6 Let E ⊂ C \ D be a compact set containing a rectifiable arc γ : [0, 1] → C \ D with γ(0) = γ(1) and γ(0), γ(1) ∈ T. Then for all f ∈ H 1 the sequence (Sn f ) is not universal on E. Theorem 6 shows that even nice compact sets E touching T in only two points, as e.g. the subarc of |z − 1| = 1 lying outside the unit disc, cannot be sets of universality for H 1 . A result in the converse direction is also given in [6]. Recall that, according to the Cantor-Bendixson theorem, each compact set A ⊂ C can be decomposed in unique way as union of a perfect set, called the perfect kernel of A, and a countable set. For compact E ⊂ C \ D and ζ ∈ E ∩ T let C E (ζ) denote the component of E that contains ζ and let PE be the union of all C E (ζ) with ζ ranging over the perfect kernel of E ∩ T. Then E is said to satisfy the kernel condition if PE has vanishing area measure and if, in addition, the set of all ζ ∈ E ∩ T with the property that C E (ζ) has positive area measure has positive distance to the perfect kernel of E ∩ T. Theorem 7 Let E ⊂ C \ D be a compact set with connected complement which satisfies the kernel condition. Moreover, suppose that no component of E touches T in more than one point. If m(E ∩ T) = 0 then E is a set of simultaneous approximation for all H p and if E ∩ T is polar then E is a set of simultaneous approximation for D. The first part of Theorem 7 is proved in [6], the second can be proved in essentially the same way, where in the final step the uniqueness argument leading to part 2 of Theorem 4 has to be repeated. According to the results of Sect. 1, the conditions on E ∩ T turn out to be necessary and, due to Theorem 6, the condition that no component of E touches T in more than one point is a natural one. It is not known if the kernel condition is necessary. The condition is satisfied in particular if the perfect kernel of E ∩ T is empty, which means that E ∩ T is countable, and also if PE has vanishing area measure and no component C E (ζ) has positive area measure.

References 1. F. Bayart, Boundary behavior and Cesàro means of universal Taylor series. Rev. Mat. Complut. 19, 235–247 (2006) 2. F. Bayart, K.-G. Grosse-Erdmann, V. Nestoridis, C. Papadimitropoulos, Abstract theory of universal series and applications. Proc. Lond. Math. Soc. 96, 417–463 (2008) 3. H.-P. Beise, J. Müller, Generic boundary behaviour of Taylor series in Hardy and Bergman spaces. Math. Z 284, 1185–1197 (2016) 4. L. Bernal-González, A. Jung, J. Müller, Banach spaces of universal Taylor series in the disc algebra. Integr. Equ. Oper. Theory 86, 1–11 (2016) 5. J.A. Cima, W.T. Ross, The Backward Shift on the Hardy Space (American Mathematical Society, Providence, RI, 2000) 6. G. Costakis, A. Jung, J. Müller, Generic behavior of classes of Taylor series outside the unit disk. Constr. Approx. 49, 509–524 (2019) 7. P. Duren, Theory of H p Spaces (Dover Publications, Mineola, 2000)

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8. O. El-Fallah, K. Kellay, J. Mashreghi, T. Ransford, A Primer on the Dirichlet Space (Cambridge University Press, Cambridge, 2014) 9. S. Gardiner, Taylor series, universality and potential theory, in New Trends in Approximation Theory, Fields Institute Communications, ed. by J. Mashreghi, M. Manolaki, vol. 81, pp. 247– 264 (Paul M, Gauthier, 2018) 10. K.G. Grosse-Erdmann, Universal families and hypercyclic operators. Bull. Am. Math. Soc. 36, 345–381 (1999) 11. K.G. Grosse-Erdmann, A. Peris Manguillot, Linear Chaos (Springer, London, 2011) 12. V.P. Havin, Analytic representation of linear functionals in spaces of harmonic and analytic functions continuous in a closed region. Dokl. Akad. Nauk SSSR, 151, 505-508 (1963). English transl. in Soviet Math. Dokl. 4 (1963) 13. G. Herzog, P. Kunstmann, Universally divergent Fourier series via Landau’s extremal functions. Comment. Math. Univ. Carolin. 56, 159–168 (2015) 14. S.V. Hrušˇcev, The problem of simultaneous approximation and removal of singularities of Cauchy type integrals. (Russian) Spectral theory of functions and operators. Trudy Mat. Inst. Steklov. 130, 124-195 (1978), English transl. in Proceedings of the Steklov Institute of Mathematics 4 (1979) 15. S.V. Hrušˇcev, V. Peller, Hankel operators, best approximation, and stationary Gaussian processes. Russian Math. Surv. 67, 61–144 (1982) 16. J.P. Kahane, Baire’s category theorem and trigonometric series. J. Anal. Math. 80, 143–182 (2000) 17. A.S. Kechris, Classical Descriptive Set Theory (Springer, New York, 1995) 18. S. Khrushchev, A continuous function with universal Fourier series on a given closed set of Lebesgue measure zero. J. Approx. Theory (2020). https://doi.org/10.1016/j.jat.2019.105361 19. P. Koosis, A theorem of Khrushchev and Peller on restrictions of analytic functions having finite Dirichlet integral to closed subsets of the unit circumference, in Conference on Harmonic Analysis in honor of Antoni Zygmund, vol II, Wadsworth, Belmont (1983) 20. J. Korevaar, Tauberian Theory—A Century of Developments (Springer, Berlin, 2004) 21. J. Müller, Spurious limit functions of Taylor series. Anal. Math. Phys. 9, 875–885 (2019) 22. V. Nestoridis, Universal Taylor series. Ann. Inst. Fourier (Grenoble) 46, 1293–1306 (1996) 23. S. Shkarin, Pointwise universal trigonometric series. J. Math. Anal. Appl. 360, 754–758 (2009)

Szegö-Type ASD for “Multiplicative Toeplitz” Operators Nikolai Nikolski

Abstract This is a brief overview of Asymptotic Spectral Distributions (ASD) for Toeplitz and Toeplitz-like operators, including the multi-tori Tn , T∞ and arbitrary compact abelian groups, with an emphasis on the “multiplicative Toeplitz” operators. Several new elements in the last theme mostly follow (and develop) the preprint Nikolski and Pushnitski (Szegö-type limit theorems for “multiplicative Toeplitz” operators and non-Følner approximations, 2020)[14]. The text below represents an abridged version of my three lectures in the Winter School “Spaces of Analytic Functions: Approximation, Interpolation, Sampling” at the CRM, Barcelona, December 2–5, 2019. It is to celebrate 100 Years of Szegö’s Asymptotic Spectral Distribution Theorem ([18]). The full version of the lectures may be published elsewhere.

1 Toeplitz Operators Classical Toeplitz operators are defined on the sequence space l 2 (Z+ ) by infinite matrices whose entries depend on the difference of the indices,   T = c( j − k)

j,k∈Z+

,

where Z+ = {0, 1, 2, ...} and c is a function on the group of integers Z. It is known that T is bounded on l 2 (Z+ ) if and only if c is the Fourier transform of a bounded ˆ k∈Z function ϕ ∈ L ∞ (T) (the symbol of T ) on the unit circle T: c(k) = ϕ(k), (Toeplitz, 1911; all elementary properties of Toeplitz operators can be found, for example, in [13]); we will write T = T (ϕ).

N. Nikolski (B) Institut de Mathématiques, Université de Bordeaux, Talence, France e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. Abakumov et al. (eds.), Extended Abstracts Fall 2019, Trends in Mathematics 12, https://doi.org/10.1007/978-3-030-74417-5_21

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Changing the standard 0 − 1 basis of l 2 for another orthonormal basis (in another Hilbert space), we obtain unitarily equivalent Toeplitz-type operators posessing therefore the same spectral properties (for example, the Toeplitz operators on the Hardy space H 2 (T) with the basis (z k )k∈Z+ , or the Wiener–Hopf integral operators on L 2 (R+ ) having Toeplitz matrices with respect to the basis of the Laguerre polynomials). The First Szegö Limit Theorem for real valued ϕ ∈ L ∞ (T) tells us that, for all f ∈ C[inf ϕ, sup ϕ],  1 Trace ( f (TN (ϕ))) = f ◦ ϕ dm, N →∞ N T  ˆ − j)}0≤k, j j (N ) and lim N →∞ A j (N ) = ∞ for any j. Its Q+ counterpart is, of course, {pα : α ∈ σ N }. D. Determinants of dilated systems. Bohr’s lift for dilated functions f (kx) is yet another realization of the same idea as the mapping B of Sect. 5 but adapted to the dilations instead of the “multiplicative Toeplitz” matrices. Namely, for an orthogonal series f =



√ an 2 sin(πnx),

f 2L 2 (0,1) =



|an |2 < ∞

n≥1

(which gives a 2-periodic odd extension of f to R), we define ˜ f (ζ) = B



an ζ α(n) ,

ζ ∈ T∞ ,

n≥1

˜ : L 2 (0, 1) → in the sense of the L 2 (T∞ ) convergence. This is a unitary mapping B 2 ∞ 2 H (T ), unitarily equivalent to the restriction of B of Sect. 5 to l (N). This restricted ˜ transforms the dilation operation Dk : L 2 (0, 1) → L 2 (0, 1), B Dk f (x) = f (kx),

k ∈ N,

into the multiplication by a multi-monomial, ˜ ˜ k = M(ζ α(k) )B. BD The following corollary of Theorem 6 can be useful for the Hilbert space geometry of dilated systems (for an information on these systems, see, for example, [10–12]).

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Corollary 8 Let h ∈ L 2 (0, 1) be extended to R as an odd 2-periodic function such ˜ ∈ L ∞ (T∞ ), and let h k = Dk h, k = 1, 2, ..., be dilated functions. Denote by that Bh G the Gram matrix ((h n , h m )) and   G N = (h n , h m )

n,m∈σ N

,

where {σ N }, σ N ⊂ N, is a Følner sequence for Q+ (for example, as in Example C ˜ > 0 or ˜ 2 dm = −∞. Then above), and assume that either inf T∞ |Bh| log |Bh| T∞

1/|σ N |  = exp lim det (G N )

N →∞

 T∞

 ˜ 2 dm . log |Bh|

Indeed, G is a “multiplicative Toeplitz” matrix, (h n , h m ) = (h nk , h mk ) for all k, m, n ∈ N, and ˜ 2 , z α(m)−α(n) ) L 2 (T∞ ) . (h n , h m ) L 2 (0,1) = (|Bh| ˜ 2 and f (x) = log x) gives the result. Now, Theorem 6 (with ϕ = |Bh|



E. A Dirichlet series interpretation. The following H. Bohr’s Lemma (see [9]) allows us to interpret all results established on T∞ in terms of Dirichlet almostperiodic series. Lemma 9 Let p(it) = ( p1−it , p2−it , ..., ps−it , ...) ∈ T∞ , t ∈ R be (one of the) Kronecker “solenoid map” R → T∞ . Then, if F ∈ C(T∞ ), the function t −→ F(p(it)) is a Bohr almost periodic function on R, and clos F(p(iR)) = F(T∞ ). Moreover, for every F ∈ L 1 (T∞ , m), there exists a limit 1 T →∞ 2T





T

lim

−T

F(p(it)) dt =

F dm. T∞

In particular, the mapping U : F(ζ) −→ F(p(it)) is a unitary map ⎛ U⎝

⎞



aα ζ α ⎠ =

α∈Z(∞)



aa r −it ,

U : L 2 (T∞ , m) → A P2 (iR)

r ∈Q+

(with r = pα ) onto the space of Besikovitch almost periodic functions having the Fourier spectrum in the group  = log Q+ . Moreover, U maps the Hardy space H 2 (T∞ ) onto the space of Dirichlet series A P2 (C+ ) holomorphic in the half-plane C+ = {z ∈ C : Re z > 0}, ⎛ U⎝



α∈Z+ (∞)

⎞ aα ζ α ⎠ =

  an , |an |2 < ∞ (n = pα ). it n n∈N n∈N

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F. On the ζ-function mean integrability on the vertical lines. Applying Bohr’s Lemma of Example E and the last statement of Sect. 6, we get the following corollary (slightly improving an example from [14]). Corollary 10 Let {σ N } N ∈N be a Følner sequence in Z+ (∞), γ ∈ (1, ∞) and ϕγ =  ζ α(n) n∈N n γ . Then  1 ϕγ (p(it)) = = ζ(γ + it), n γ+it n∈N

and  1 Trace ( f (|Tσ N (ϕγ )|)) = f (|ϕγ |) dm N →∞ |σ N | T∞  T 1 f (|ζ(γ + it)|) dt, = lim T →∞ 2T −T lim

f ∈ C[inf |ϕγ |, sup |ϕγ |].

In particular, these limits exist for f (x) = x s , s ∈ R, and f (x) = log x. 1 ≤ |ζ(γ + ζ(γ) it)| ≤ ζ(γ) < ∞ for every t ∈ R and γ > 1 (the second inequality is obvious, and the 1 = first one follows, for example, from the Euler formula for the reciprocal ζ(γ + it)  μ(n) , where μ(n) stands for the Möbius function taking values +1, −1, 0).  n γ+it n∈N Indeed, using previous comments, it remains only to add that 0
4 is also unknown [19].

8 Asymptotic Spectral Densities as Spectral Approximations (Cum Grano Salis) For simplicity, we mostly consider here the classical setting of the torus T, as in Sect. 1 (using the same notation). A few remarks below have a goal to recall to the interested reader some obstacles in order to consider the above ASD theorems as a tool for an approximation to the spectrum of the limiting infinite dimensional operator (as they are sometimes regarded, see [7] for a discussion). In general, the spectral approximation meaning of the ASD theorems is particularly interesting for the “multiplicative Toeplitz” operators defined by their matrices (c(m/n))m,n∈N . In this case, the range of the symbol ϕ(T∞ ) = σ(M(ϕ)) can be hardly made explicit, not speaking on the spectra σ(T0 (ϕ)) of the corresponding Toeplitz operators on the Hardy spaces H20 (G) (see Sect. 6). The dependences of

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σ(T0 (ϕ)) on the generating subsemigroup 0 is unknown (even for  = Zn , n > 1; think on various 0 = Zn ∩ V where V is a convex cone in Rn with a non-empty interior). All operators below are supposed selfadjoint. A. Finite dimensional sections are the same, the operator  spectra are different. Indeed, as it is remarked, PN M(ϕ)Ran PN = PN T (ϕ)Ran PN , and so   σ(PN M(ϕ)Ran PN ) = σ(PN T (ϕ)Ran PN ), but in general, σ(M(ϕ)) = σ(T (ϕ)) even for the selfadjoint case (ϕ = ϕ). For example, for ϕ = χ E − χT\E with E ⊂ T, 0 < m E < 1, σ(M(ϕ)) = {−1, 1}, but σ(T (ϕ)) = [−1, 1]. (For the general problem of the approximation of spectra and, in particular, the phenomenon of the “spectral pollution”, we can quote [5, 7, 8]). Moreover, even in a situation where σ(M(ϕ)) = σ(T (ϕ)), the spectral resolutions E M(ϕ) (·) and E T (ϕ) (·) can be very different: if ϕ = const, E T (ϕ) (·) is always absolutely continuous with respect to the Lebesgue measure on R ([15]), whereas E M(ϕ) (·) is an arbitrary projection-valued measure on ϕ(T) (without point masses of finite rank, 0 < rank E M(ϕ) ({x}) < ∞). On more general groups , a unique operator M(ϕ) is related to many T0 (ϕ), 0 − 0 = , and the relations between σ(M(ϕ)) and σ(T0 (ϕ)) are unknown. B. Szegö’s ASD theorem proves the weak convergence of measures μ N (ϕ) =

1  δλ j,N → m ϕ = ϕ−1 (m) (weak convergence), N

m ϕ being the push-forward of the Lebesgue measure m with respect to ϕ, m ϕ () = m(ϕ−1 ()). It can be shown that m ϕ () = (E M(ϕ) ()1, 1) L 2 (T) for every  ⊂ [inf ϕ, sup ϕ], and that m ϕ is one of the scalar spectral measures of M(ϕ) : L 2 (T) → L 2 (T) (i.e., it is mutually absolutely continuous with respect to the orthoprojection valued spectral resolution E M(ϕ) (·) of M(ϕ), even if the constant function 1 is not necessarily a cyclic vector of M(ϕ)). Therefore, the limits lim N μ N (ϕ) = m ϕ and lim N μ N (ψ) = m ψ (ϕ, ψ ∈ L ∞ (T)) can coincide m ϕ = m ψ , without defining unitarily equivalent operators M(ϕ) and M(ψ). Indeed, if m ϕ = m ψ , the unitary equivalence M(ϕ) ∼ M(ψ) holds if and only if the (von Neumann) spectral dimension functions coincide, dim M(ϕ) (x) = dim M(ψ) (x) m ϕ -a.e. on σ(M(ϕ)) = supp (m ϕ ). But for a multiplication operator M(ϕ), dim M(ϕ) (x) is the Banach indicatrix function Nϕ (x) = car d{t ∈ T : ϕ(t) = x} (for m ϕ -a.e. x). It entails that for every ϕ “without Lebesgue jumps” (i.e., m ϕ (I ) > 0 for every non-empty open interval I ⊂ [inf ϕ, sup ϕ]) there exists a strictly monotone equimesurable function ψ such that m ϕ = m ψ and dim M(ψ) (x) = 1 m ϕ -a.e.,

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whereas dim M(ϕ) (x) are arbitrary (at least for m ϕ without point masses).  Finally, one can remember (... for more turba et confusio ...) that in the approximating measures μ N (ϕ) and μ N (ψ) (N = 1, 2, ...), all eigenvalues are counted with their multiplicities; where they are lost?.. and how to make the limits m ϕ remember them? C. Geometrically, σ(TN (ϕ)) converge (for N →∞) to σ(T (ϕ)), not to σ(M(ϕ)). Here, we mean the convergence with respect to the Hausdorff uniform distance d(·, ·) between two compacts σ, τ ⊂ C, d(σ, τ ) = max{x ∈ σ, y ∈ τ : dist (x, τ ) + dist (y, σ)}. Recall that for real valued ϕ = ϕ, we have  σ(T (ϕ)) = I and σ(PN T (ϕ)Ran PN ) ⊂ I, whereas σ(M(ϕ)) = ϕ(T) (the essential range of ϕ), where I := [inf ϕ, sup ϕ]. Both spectra coincide with the corresponding essential spectra. The difference σ(T (ϕ))\σ(M(ϕ)) consists of a union of open intervals where some eigenvalues of TN (ϕ) must appear (in order to ensure σ(TN (ϕ)) → σ(T (ϕ))), the phenomenon is known as a “spectral pollution”. Due to the Szegö ASD (as in Example B above), the density of these “erroneous” eigenvalues tends to 0 as N → ∞. For more discussions of the phenomenon, we can quote (anew) [5, 7, 8] and the references therein; see also the surveys [4, 6]. Acknowledgements This work was partially supported in frameworks of the agreement No. 07515-2019-1620 between Russian Ministry of Science and Higher Education and St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Science.

References 1. P. Ara, F. Lledó, D. Yakubovich, Følner sequences in operator theory and operator algebras. Operator Theory: Advances and Applications, vol. 242 (Birkhäuser–Springer, Basel, 2014), pp. 1–24 2. W. Arveson, C ∗ -algebras and numerical linear algebra. J. Funct. Anal. 122(2), 333–360 (1994) 3. E. Bédos, On Følner nets, Szegö’s theorem and other eigenvalue distribution theorems. Expo. Math. 15, 193–228 (1997) 4. A. Böttcher, I. Bogoya, S. Grudskii, E. Maksimenko, Asymptotics of the eigenvalues and eigenvectors of Toeplitz matrices. Mat. Sbornik 208(11), 4–28 (2017); [Engl. transl.: Sb. Math. 208(11), 1578–1601 (2017)] 5. E. Davies, M. Plum, Spectral pollution. IMA J. Numer. Anal. 24(3), 417–438 (2004) 6. P. Deift, A. Its, I. Krasovsky, Toeplitz matrices and Toeplitz determinants under the impetus of the Ising model: some history and some recent results. Comm. Pure Appl. Math. 66(9), 1360–1438 (2013) 7. R. Hagen, S. Roch, B. Silbermann, C ∗ -Algebras and Numerical Analysis (Marcel Dekker Inc., Basel, NY, 2001) 8. A. Hansen, On the approximation of spectra of linear operators on Hilbert spaces. J. Funct. Anal. 254(8), 2092–2126 (2008)

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9. D. Li, H. Queffélec, Introduction to Banach Spaces: Analysis and Probability, vol. 2 (Cambridge University Press, Cambridge, 2018) 10. N. Nikolski, Binomials whose dilations generate H 2 (D). Algebra i Analiz. 29(6), 159–177 (2017) 11. N. Nikolski, The current state of the dilation completeness problem, King’s College London Analysis Seminar, November 15, (2018); preprint: a PDF is available upon request, [email protected] 12. N. Nikolski, Hardy Spaces (Cambridge University Press, Cambridge, 2019) 13. N. Nikolski, Toeplitz Matrices and Operators (Cambridge University Press, Cambridge, 2020) 14. N. Nikolski, A. Pushnitski, Szegö-type limit theorems for “multiplicative Toeplitz” operators and non-Følner approximations, Algebra i Analiz, 2020, 32(6), 101–123; arXiv:2001.01474 [math.FA] 15. M. Rosenblum, The absolute continuity of Toeplitz’s matrices. Pacif. J. Math. 10, 987–996 (1960) 16. D. SeLegue, A C ∗ -algebraic extension of the Szegö trace formula. Talk given at the GPOTS. Arizona State University, Tempe, May 22 (1996) 17. B. Simon, Orthogonal Polynomials on the Unit Circle, Part 1: Classical Theory, Colloquium Publications, vol. 54, issue 1. American Mathematical Society, Providence (2005) 18. G. Szegö, Beitrage zur Theorie der Toeplitzsche Formen, Part I: Math. Zeit. 6, 167–202 (1920); Part II: 9, 167–190 (1921) 19. E.C. Titchmarsh, The Theory of the Riemann Zeta-Function, 2nd edn. (Oxford University Press, Oxford, 1986) 20. O. Toeplitz, Zur Theorie der Dirichletschen Reihen. Am. J. Math. 60(4), 880–888 (1938) 21. S. Verblunsky, On positive harmonic functions (second paper). Proc. London Math. Soc. 40(2), 290–320 (1936)

Around Uncertainty Principle A. M. Olevskii

Abstract Below I outline the content of the mini-course given in October 2019 in CRM in the framework of Research Program in Analysis.

1 Introduction The uncertainty principle plays a fundamental role in quantum mechanics. In the language of mathematics, it says that a function and its Fourier transform cannot be both sharply concentrated near a point, or, in a stronger form, on “small” sets.  be its Fourier transform: Let F denote an L 2 -function on R, and let F   F(w) := F(t) e−2πiwt dt. One says that F is supported by a set S if F = 0 almost everywhere (a.e.) on  If S is a set of finite the complement S c . The spectrum Q of F is the support of F. measure then the Fourier transform is a continuous function, so it is natural to define  takes non-zero values. The Lebesgue measure Q as the closure of the set where F of a set S will be denoted by |S|. Clearly, only the case when S is unbounded is interesting. Otherwise the function  is analytic and the spectrum is the whole R. F Definition 1 One says that a pair of sets (A, B) annihilates if F = 0 whenever F  is supported on B. is supported on A and F Definition 2 A pair of sets (A, B) strongly annihilates if there is a number C such that every F ∈ L 2 (R) satisfies the condition  F2 ≤ C

 Ac

|F|2 +

Bc

 2 . | F|

A. M. Olevskii (B) Tel Aviv University, P.O. Box 39040, 6997801 Tel Aviv, Israel e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. Abakumov et al. (eds.), Extended Abstracts Fall 2019, Trends in Mathematics 12, https://doi.org/10.1007/978-3-030-74417-5_22

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This means that a non-trivial function F cannot have the support and the spectrum both essentially concentrated on A and B correspondingly. Some basic results: 1. If S and Q are sets of finite measure then they annihilate, and even strongly annihilate (Benedicks, 1974, Amrein-Bertier, 1985). 2. T. Wolff and co-authors [6] proved that if sets S, Q are locally small in a certain sense (but may have infinite measure, and even positive density) then they strongly annihilate. We send the reader to the monograph [3] which surveys results in the subject published till the 90s. Let us mention also a recent paper [2] where a version of uncertainty principle for a fractal set is established.

2 Spectral Gaps There are examples of functions showing that the uncertainty principle cannot go too far. In other words, there are non-trivial functions with the support and the spectrum both “small” in a certain sense. Kargaev (1982) constructed a set S of finite measure such that the Fourier transform of the indicator function 1 S vanishes on an interval I . One calls such I a spectral gap of F. The proof is based on delicate arguments involving the implicit functions theorem in Banach spaces. Kargaev and Volberg (1992) modified this approach and produced a function F ∈ L 1 (R), supported on a set S of finite measure and having a sequence of spectral gaps of infinite total length. There is another approach to construction of non-annihilating “small” sets, inspired by the classical Menshov “correction” theorem. It was developed by Arutjunyan (1984) and by Kislyakov (1984). The technique there is quite hard. Recently, jointly with Fedor Nazarov, we have found a new simple construction of functions with the support and the spectrum both “small”: Theorem 1 ([4]) There is a set S ⊂ R of finite measure such that the spectrum Q of its indicator function F = 1 S satisfies the condition |Q ∩ (−r, r )| = o(r ), r → ∞. That is, the density dens(Q) is zero. Remark 1 Moreover, the left part can be made o(w(r )), where w is a given function growing to infinity arbitrarily slowly. This means that the spectrum is as small as possible (remember that |Q| must be infinite). This result was presented in our mini-course. In the construction, F is obtained as a sum of certain sequence of highly oscillating band-limited functions.

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The following simple model illustrates the idea of the construction. Take a random walk on the segment [0, 1] such that at the moment n the trajectory makes a jump from the position xn on h n = xn (1 − xn ) to the left or to the right with equal probabilities. It is not difficult to prove that when n tends to infinity, the trajectory goes almost surely to one of the ends. So the limit is a (random) indicator function. Some interesting open problems could be mentioned: – in Theorem 1, is it possible to get the uniform density of Q (in the sense of Beurling) equal to zero? – is it possible to get a closed set S?

3 Uniqueness Sets for Paley–Wiener Spaces Here we discuss a concept of independent interest, related to the previous subject. Given a set S of finite measure, consider the space of all functions f which are Fourier transforms of L 2 -functions F supported on S. It is the so-called Paley-Wiener space P W S with spectrum S. In the classical case it consists of entire functions of the exponential type. We are mainly interested in the case when S is unbounded. Then only continuity of f is granted. Definition 3 A set  is called a uniqueness set for the space P W S if f | = 0 implies that f ≡ 0. The following theorem was proved in a joint paper with Alexander Ulanovskii. Theorem 2 ([5]) For every set S, |S| < ∞, there is a uniformly discrete set  which is a uniqueness set for P W S . For the proof, we “project” a set S, |S| < 1, on the circle T = [0, 1], that is, reduce S by modulo 1, and consider “multiplicity” w(t) of the projection as a weight. The key idea is to choose infinitely many pairwise disjoint sets  j of integer frequencies such that the corresponding systems of exponentials are complete in the weighted space L 2 on T . Then we prove that the union of the appropriately perturbed sets  j is a uniqueness set for P W S . The connection between uniqueness and annihilation comes from the following remark: if  is a uniqueness set for P W S , which is disjoint with a set Q, then the pair (S, Q) annihilates. This gives us a way to prove annihilation. For example, if (S, Q) both are of finite measure then one can construct the corresponding . On the other hand, Theorem 1 shows that the conditions |S| < ∞, dens(Q) = 0 do not imply the annihilation. It turns out that under an extra geometric condition they do. This gives a new version of the uncertainty principle: Theorem 3 ([1]) Assume that:

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(i) The set S is of finite measure and has a periodic gap (for example, let S avoid intervals (n, n + ), n ∈ Z, for some positive ); (ii) dens(Q) = 0. Then the pair (S, Q) annihilates. The main point of the proof is: due to the periodicity condition and using the famous Beurling-Malliavin theorem, one can find the above  j in a specific form, which allows us to get the set  disjoint with a given set Q of density zero. It should be mentioned also that the strong annihilation in Theorem 3 does not hold in general, see [1].

References 1. 2. 3. 4.

T. Amit, A. Olevskii, On the Annihilation of Thin Sets, arXiv:1711.04131v2 J. Bourgain, S. Dyatlov, Fourier dimension and spectral gaps, GAFA, 27 (2017), 744–771 V.P. Havin, B. Jöricke, The Uncertainty Principle in Harmonic Analysis (Springer, Berlin, 1994) F. Nazarov, A. Olevskii, A function with support of finite measure and “small” spectrum, 50 Years with Hardy Spaces. A Tribute to Victor Havin, Operator Theory: Advances and Applications, vol. 261 (2018), pp. 387–391 5. A. Olevskii, A. Ulanovskii, Functions with Disconnected Spectrum. Sampling, Interpolation, Translates (Amer. Math. Soc, 2016) 6. C. Shubin, R. Vakulian, T. Wolff, Some harmonic analysis questions suggested by AndersonBernulli models. GAFA 8, 932–964 (1998)

Inner Functions, Completeness and Spectra Alexei Poltoratski

Abstract This course focused on applications of inner functions in completeness problems and spectral analysis. Such applications traditionally stem from the key role played by inner functions in Nagy-Foias functional model theory. More recent problems in this area concern finite rank perturbations of linear operators and spectral problems for differential equations.

1 Inner Functions and Clark Theory Recall that a bounded analytic function in the upper half-plane C+ = {z > 0} or the unit disk D = {|z| < 1} is called inner if its non-tangential boundary values belong to the unit circle T = {|z| = 1} almost everywhere on the boundary of the domain. We will need the following definitions and notations. A function on the real line is called Poisson-summable if it is summable with respect to the measure (x), d(x) = d x/(1 + x 2 ). A measure μ on R is Poisson-finite if μ(x)/(1 + x 2 ) ˆ if it is a is finite. We say that μ is a Poisson-finite measure on the extended line R sum of a Poisson-finite measure on R and a finite point mass at infinity. ˆ then its Schwarz integral is defined as If μ is a Poisson-finite measure on R Sμ(z) =

1 πi

  R

 1 c t dμ(t) + z, − 2 t−z 1+t πi

where c = μ({∞}). An analytic function f in C+ is outer if it can be represented as f (z) = eSh(z)

A. Poltoratski (B) Department of Mathematics, University of Wisconsin, Van Vleck Hall, 480 Lincoln Drive, 53706 Madison, WI, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. Abakumov et al. (eds.), Extended Abstracts Fall 2019, Trends in Mathematics 12, https://doi.org/10.1007/978-3-030-74417-5_23

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for some real Poisson-summable function h on R. One of the most important facts of complex function theory is that all “reasonable” analytic functions (functions from the Smirnov class to be exact) admit inner/outer factorizations, f = I O where I is inner and O is outer. Similar formulas and results can be presented for the unit disk, see [2, 4]. A theorem of Beurling from 1949 characterizes all invariant subspaces of the shift operator S : f (z) → z f (z) in the Hardy space H 2 (D) as subspaces of the form θ H 2 where θ is inner. It follows that for the adjoint operator S ∗ , S∗ f =

f − f (0) , z

the invariant subspaces are those of the form K θ = H 2  θ H 2 , where θ is inner. This result led to the key role played by inner functions in the Nagy-Foias model, see [9]. Our first goal is to present the basics of the so-called Clark theory of families of measures on T (or R) which stems from finite dimensional perturbation theory. Let θ be inner in D. To simplify the formulas we will assume that θ(0) = 0. Then for every α ∈ T there exists unique probability measure σα on T such that its Schwarz integral Sσα in the unit disk satisfies α+θ = Sσα = α−θ



¯ 1 + ξz dσα (z). ¯ 1 − ξz

Here existence follows from the Herglotz representation theorem. Via this statement, for each such θ one can consider a family of Clark measures {σα }α∈T on the unit circle, see [16] for the basics and further references. Notice that any probability measure σ is a Clark measure σ1 for some inner θ, θ(0) = 0. This way we obtain a parametrization of the set of all such θ by probability measures on the unit circle. Many objects of complex function theory stem from functional analysis and spectral theory. Clark measures also belong to this category due to their connection to perturbation problems. Let Sθ denote a compression of S onto K θ , i.e., Sθ : K θ → K θ , Sθ f = Pθ S f, where Pθ is the orthogonal projection from H 2 to K θ . As was shown in [1], σα is the spectral measure of the unitary operator Uα f = Sθ f + α f, θ/z = z( f − f, θ/z θ/z) + α f, θ/z

corresponding to the cyclic vector θ/z of Sθ . The operators Uα present all possible unitary rank one perturbations of the model operator Sθ . Also, the operators Uα , α = 1

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are all possible rank one perturbations of U1 corresponding to the subspaces spanned by θ/z and U1 (θ/z) = 1. Analogously, one can define Clark families of measures on the real line corresponding to inner functions in C+ . Such measures will represent spectral measures of rank one perturbations of self-adjoint operators or, equivalently, spectral measures of a self-adjoint differential operator on an interval with varying boundary condition at one of the endpoints. Due to these connections Clark measures can be used in various constructions and proofs related to finite-rank perturbation theory and spectral theory of differential equations.

2 Normalized Cauchy Transform By the spectral theorem, for each operator Uα defined above there must exist a unitary operator Vα : L 2 (σα ) → K θ such that

Vα Mz Vα∗ = Uα ,

where Mz is the operator of multiplication by z in L 2 (σα ). As was shown in [1], Vα has the following form. For μ ∈ M(T) and f ∈ L 1 (|μ|) we denote by K f μ the Cauchy integral in the unit disk.  K f μ(z) =

f (ξ)dμ(ξ) . ¯ 1 − ξz

Let Cμ f denote the normalized Cauchy transform Cμ f =

Kfμ . Kμ

The Clark operator Vα can be defined via the formula Vα f = Cσα f. The property that the operator is unitary is a deep and useful generalization of the Parseval identity. The problem of finding a description for the inverse Clark operator Vα∗ was posed by D. Sarason in the late 1980s. It was solved in [10] by showing that each function f ∈ K θ has non-tangential boundary values, f ∗ , σα -a.e. for every α and Vα∗ f = f ∗ . This connection allows one to use Clark model in problems of convergence of Cauchy-Stiltjes and similar singular integrals. As an immediate corollary we obtain that for any finite complex measure μ on T and f ∈ L 1 (|μ|), the (meromorphic)

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function Cμ f has non-tangential boundary values |μ|-a.e on T which coincide with f at |μs |-a.e. point (here μs denotes the singular part of μ). Various problems on the maximal non-tangential operator associated with the normalized Cauchy transform Cμ and similar singular operators remain open, see [11].

3 Toeplitz Operators Some of the most important applications conserning inner functions come from the theory of Toeplitz operators. The most basic definition of Toeplitz operators is as follows. Let φ be a bounded funciton on R and let P+ be the Riesz projector, the orthogonal projector from L 2 (R) onto H 2 (C+ ). The Toeplitz operator in H 2 (C+ ) with symbol φ is defined as Tφ f = P+ (φ f ). In many applications the symbol of a Toeplitz operator has the form φ = I J¯ where I and J are inner functions, see [5]. In some applications the operator has to be restricted (compressed) to one of the model subspaces K θ . Such compressions are called truncated Toeplitz operators. Operators with unbounded symbols produce unbounded operators. If μ is a measure on R one can define Tμ via the relation 

 R

Tμ f (x) g(x)d ¯ x=

f (x)g(x)dμ(x), f, g ∈ H 2 .

Similar formulas and definitions can be given in the case of the unit disk and unit circle. Truncated Toeplitz operators whose symbols are spectral measures of differential equations play an important role in inverse spectral problems, see [7]. Another important application of Toeplitz operators lies in the area of problems on completeness and basis properties for systems of special functions in various spaces. Let us illustrate this connection with the following simple example. The Hilbert space K θ is a reproducing kernel Hilbert space, i.e., for any λ ∈ C+ point evaluation f → f (λ) is a bounded linear functional on K θ which can be realized via the inner product f (λ) = f, kλ , where kλ ∈ K θ is called the reproducing kernel corresponding to the point λ. The general problem of completeness of reproducing kernels is to determine for which sequences  ⊂ C+ the corresponding sequence of {kλ }λ∈ is complete in K θ . For specific choices of θ this problem becomes equivalent to classical problems on completeness of exponential functions (see below), special functions, solutions of various differential equations, etc. Let  = {λn } be a sequence of points in C+ (or D) and let E = {kλn } be the sequence of reproducing kernels in K θ . The sequence E is incomplete in K θ if and only if there exists f ∈ K θ orthogonal to all kλn , i.e., if and only if there exists f of the form f = B g, where g ∈ H 2 and B is the Blaschke product with zeros at . Using the fact that K θ is the kernel of the Toeplitz operator Tθ¯ , we obtain that g

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belongs to the kernel of TB θ¯ . All in all, the system E is complete in the space if and only if the opetator TB θ¯ is injective. Numerous problems of completeness of systems of special functions in various spaces can be reduced to the above problem of completeness of reproducing kernels in a model space via the Fourier transform or its generalizations, such as the WeylTitchmarsh transform associated with differential operators, see [5]. The classical Beurling–Malliavin problem of completeness of exponentials in L 2 on an interval can be shown to be equivalent to such a problem with θ(z) = ei z . A generalization of the Beurling–Malliavin theory using the language of Toeplitz operators was obtained in [5, 6]. Many problems in this area remain open.

4 Toeplitz Version of BM Theory To state the generalized version of the Beurling–Malliavin (BM) theorem from [6] we need to introduce the notion of BM intervals. Let γ be a continuous function R → R such that γ(∓∞) = ±∞, i.e., lim γ(x) = +∞,

x→−∞

lim γ(x) = −∞.

x→+∞

The family B M(γ) is defined as the collection of the connected components of the open set   x | γ(x) = max γ . [x,+∞)

Note that max[x,+∞) γ is the smallest decreasing majorant of γ. In what follows, we will measure how far a function is from decreasing by the ‘size’ of its BM intervals. If κ ≥ 0, then we say that γ is (κ)-almost decreasing if 

γ(∓∞) = ±∞,

(dist(l, 0) + 1)κ−2 |l|2 < ∞.

(1)

l∈B M(γ)

In relation to the standard terminology in the classical BM theory (the case when κ = 0), the family B M(γ) is short if γ is almost decreasing and long otherwise. Let κ ≥ 0, and let U = eiγ and V = eiσ be smooth unimodular functions on R such that (x → ∞). (2) γ  (x)  −|x|κ , σ  (x)  |x|κ , Consider the family of symbols U V¯ a = eiγa , and define the transition parameter

γa = γ − aσ,

(a ∈ R),

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c ≡ c(U, V ; κ) = inf{a | γa is (κ)-almost decreasing} ∈ (−∞, +∞]. We denote by N p [φ] the kernel of the Toeplitz operator with symbol φ in the H p space. Theorem 1 Let J be a meromorphic inner function, and suppose that a unimodular function V satisfies (arg V ) (x)  |x|κ ,

x → ∞, κ ≥ 0.

Denote c = c(J, V ; κ). Then for all p ≤ ∞, N p [J V¯ a ] = 0 for all a < c and

N p [J V¯ a ] = 0 for all a > c.

This result is equivalent to the BM theorem in the case κ = 0. For other values of κ it can be applied to treat completeness problems of special functions and spectral problems for differential operators beyond the classical BM theory. Many cases of the theorem, including for instance the case κ < 0, remain open, see [6].

5 Toeplitz Order An attempt to systemathize problems on Toeplitz operators with symbols equal to ratios of inner functions, appearing in various fields of analysis and mathematical physics, was recently undertaken in [12]. A natural partial order induced by Toeplitz operators on the set of inner functions translates many of such problems into one universal language and reveals possible directions for further research. The following definition of Toeplitz order was given in [12]. If θ is an inner function we define its Toeplitz dominance set D(θ) as ¯ ] = 0}. D(θ) = {I inner | N [θI The order by inclusion for Toeplitz dominance sets defines a pre-order on inner functions. We say that I ∼ J if D(I ) = D(J ), defining an equivalence relation on the set of inner functions. We say that I > J , meaning that the equivalence class of I is greater than the equivalence class of J , if D(I ) ⊃ D(J ). One can easily show that the order defined above extends the order by division. In comparison with the order by division, Toeplitz order makes many more pairs of inner functions comparable while also presenting many new problems. Some of such problems can be shown to be equivalent to well-known problems of analysis and mathematical physics. One of the standard problems of complex analysis is to describe zero sets for functions in various spaces. For functions in K θ this problem translates immediately

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into a problem on Toeplitz dominance sets defined above. A sequence  is the zero set of some f ∈ K θ iff B ∈ D(θ). The problem of completeness of systems of reproducing kernels in K θ discussed above is equivalent to the problem of description of zero sets, since {kλ }λ∈ is incomplete iff  is a zero set. Although this problem has been solved in several basic cases (see the Beurling–Malliavin problem, or some of the results of [5, 6]) solutions for many other natural cases, let alone the general solution, are yet to be found.

6 The General Beurling–Malliavin Problem In its original formulation, the problem asks for what complex sequences  ⊂ C the system of exponentials E = {eiλz }λ∈ is complete in L 2 (−a, a). The Beurling– Malliavin theory created in 1960s to solve this problem is considered one of the deepest parts of the 20th century Harmonic Analysis. Via the Fourier transform the problem translates into the problem of description of zero sets in the Paley–Wiener spaces: E is incomplete in L 2 (−a, a) iff there is f ∈ P Wa , f = 0 on . Since K S 2a = eiaz P Wa , the problem becomes a problem on zero sets in K θ spaces in the particular case θ = S = ei z , i.e., on description of  such that B ∈ D(S). The radius of completeness R() is defined as the supremum of a such that E is complete in L 2 (−a, a). As was shown in [5], in terms of Toeplitz kernels,   R() = sup a : N [ S¯ a B ] = 0 = sup a : N + [ S¯ a B ] = 0 . (Here we assume that the sequence  is in C+ and satisfies the Blaschke condition. For sequences on R, B has to be replaced with an inner function which equals to 1 at . If a sequence has points in C− , they need to be replaced with complex conjugates. Finally, it can be easily shown that sequences which do not satisfy the Blaschke condition have infinite radius of completeness.) It follows that in terms of Toeplitz order R() = inf{a|S a > B } (= inf{a|B ∈ D(S a )}). (Useful observation: I ∈ R(S a ) ⇒ S a ≥ I and S a ≥ I ⇒ I ∈ R(S a+ε ).) We call an inner function in the upper half-plane θ a meromorphic inner function (MIF) if it can be extended meromorphically to the whole plane. One of the main results of [5] can be expressed as follows. Theorem 2 Let U be a MIF with |U  |  x κ , κ ≥ 0, U = eiγ on R. Let J be another MIF, J = eiσ . Then (I) If σ − (1 − ε)γ is κ-almost decreasing, then J ∈ D(U ); (II) If σ − (1 + ε)γ is not κ-almost decreasing, then J ∈ / D(U )

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In terms of Toeplitz order it follows that if (I) holds then U > J . One can show that if (II) holds then U > J .

7 The Gap Problem For a fixed closed set X ⊂ R one needs to find its gap characteristic G(X ) defined as the supremum of the length of the gap in the support of the Fourier transform, μ, ˆ taken over all non-zero finite complex measures μ supported on X . Here we will assume that X is a discrete (but not necessarily separated) sequence on R. It was shown in [13] that the general Gap Problem may be reduced to this case. In this case, G(X ) is the supremum of a ≥ 0 such that there exists a MIF θ such that σ(θ) = {θ = 1} ⊂ X and S a < θ (here and throughout the paper S a (z) = eiaz ). A formula for G(X ) was given in [13]. A simpler version of the formula in the case when X is a separated sequence was earlier found in [8]. Both papers used the approach described above (without referring to Toeplitz order, which was not properly defined yet).

8 The Type Problem Let μ be a finite positive measure on R. Denote by Tμ its type, defined as

Tμ = sup{a | ∃ f ∈ L 2 (μ), f ≡ 0,

f μ = 0 on [0, a]}.

(3)

The type problem asks to find a formula for Tμ in terms of μ. As was shown in [14], the general case can be reduced to the case when μ is a discrete measure. It can be easily seen that μ is discrete if and only if it is a Clark measure for a meromorphic inner function θμ . Via this connection, Tμ is the supremum of a, S a < θμ . This approach was used in [14] to obtain a solution for the Type Problem. An equivalent definition for the type is

Tμ = inf{a|eist , s ∈ [−a, a], are complete in L 2 (μ)}. The connection of the type with Toeplitz order is expressed by the following statement. Theorem 3 Tμ = sup{a|S a ≤ θμ } = sup{a|S a ∈ D(θμ )}.

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9 Toeplitz Order in Comparison with Similar Relations Among Inner Functions Intuitively, when ker TI¯ J = 0 for two inner functions I and J it means that I is ’larger’ than J . This relation between I and J starts to resemble a strict order even more after one recalls that by a lemma of Coburn ker TI¯ J and ker TJ¯ I cannot be nontrivial simultaneously. Formally, however, this relation does not constitute an order due to the lack of transitivity: ker TI¯ J = ker TJ¯ L = 0 does not imply ker TI¯ L = 0. Accordingly, the relation I  J , which can be defined to mean that ker TI¯ J = 0 and ker TJ¯ I = 0, fails to produce a formal equivalence. Another important relation between meromorphic inner functions is the ‘twin’ relation. We call the set where the non-tangential values of an inner function θ are equal to 1, the spectrum of θ and denote it by σ(θ). Following [5], we will call two MIFs I and J twins if σ(I ) = σ(J ). Twin MIFs appear in the study of isospectral operators and isospectral hierarchies. Clearly, the twin relation is an equivalence relation on the set of all MIFs, which is different from the Toeplitz equivalence. It is easy to see that I ∼ J does not imply that I is a twin of J . The opposite implication fails in general as well. However, we do have the following statement. ˆ Then Lemma 4 Let I and J be two MIF twins with the common spectrum σ ⊂ R.   I ∼ J iff |I |  |J | on σ. The condition of comparability for the derivatives of the inner functions appearing in the last statement is worth exploring a bit further. Such conditions appear in applications. For instance, inner functions corresponding to Schrödinger equations with regular potentials, as well as to other similar classes of canonical systems, will satisfy this condition. Let us provide the following description of Toeplitz equivalence pertaining to this case. Lemma 5 Consider two MIFs I and J such that |I  |  |J  | on R. Then I ∼ J iff arg I − arg J has a bounded harmonic conjugate. Another important relation between inner functions, which resembles equivalence, comes from invertibility of the Toeplitz operator with the symbol I¯ J . Intuitively, this condition also tells us that the functions I and J are similar. Due to the work of Hrušˇchev et al. [3], this condition became one of the main tools in the study of basis properties for systems of reproducing kernels, including the classical problem on exponential bases as a particular case. Up to some technical details, a system of reproducing kernels kλn forms a Riesz basis in a model space K I if and only if TI¯ B is invertible, where σ(B) = . It is not difficult to show that Toeplitz equivalence is not the same as the invertibility of the Toeplitz operator. As a matter of fact, unlike Toeplitz equivalence, invertibility relation is not a formal equivalence since, once again, it lacks transitivity. Acknowledgements The author was partially supported by NSF Grant DMS-1954085.

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References 1. D. Clark, One dimensional perturbations of restricted shifts. J. Anal. Math. 25, 69–191 (1972) 2. J. Garnett, Bounded Analytic Functions (Academic Press, New York, 1981) 3. S.V. Hrušˇcev, N.K. Nikol’skii, B.S. Pavlov, Unconditional bases of exponentials and of reproducing kernels, in Complex Analysis and Spectral Theory, ed. by V.P. Havin, N.K. Nikol’skii. Lecture Notes in Mathematics, vol. 864 (Springer, Berlin, Heidelberg, 1981), pp. 214–335 4. P. Koosis, Introduction to H p Spaces (Cambridge University Press, Cambridge, 1980) 5. N. Makarov, A. Poltoratski, Meromorphic inner functions, Toeplitz kernels, and the uncertainty principle, in Perspectives in Analysis, (Springer, Berlin, 2005) pp. 185–252 6. N. Makarov, A. Poltoratski, Beurling–Malliavin theory for Toeplitz kernels. Invent. Math. 180(3), 443–480 (2010) 7. N. Makarov, A. Poltoratski, Etudes in inverse spectral problem. Preprint 8. M. Mitkovski, A. Poltoratski, Polya sequences, Toeplitz kernels and gap theorems. Adv. Math. 224, 1057–1070 (2010) 9. N.K. Nikolskii, Treatise on the Shift Operator (Springer, Berlin, 1986) 10. A. Poltoratski, On the boundary behavior of pseudocontinuable functions. (Russian) Algebra i Analiz. 5(2), 189–210; [translation in St. Petersburg Math. J. 5 (2), 389–406 (1994)] 11. A. Poltoratski, Maximal properties of the normalized Cauchy transform. J. Am. Math. Soc. 16(1), 1–17 (2003) 12. A. Poltoratski, Toeplitz order. J. Funct. Anal. 275(3), 660–697 (2018) 13. A. Poltoratski, Spectral gaps for sets and measures. Acta Math. 208(1), 151–209 (2012) 14. A. Poltoratski, A problem on completeness of exponentials. Ann. Math. 178, 983–1016 (2013) 15. A. Poltoratski, Toeplitz Approach to Problems of the Uncertainty Principle. Book in CBMS series (AMS/NSF, 2015) 16. A. Poltoratski, D. Sarason, Aleksandrov-Clark measures, in Recent advances in operatorrelated function theory, Contemp. Math., 393, Amer. Math. Soc., Providence, RI (2006), pp. 1–14

Schmidt Subspaces of Hankel Operators Patrick Gérard and Alexander Pushnitski

Abstract Let  be a compact Hankel operator acting on the Hardy class H 2 over the unit circle. The purpose of the lectures is to discuss the structure of the Schmidt spaces of  (i.e. the eigenspaces of  ∗ ) as a class of subspaces of H 2 . It turns out that the Schmidt spaces of  are the images of model spaces under the action of isometric multipliers. The action of  on the Schmidt spaces can also be explicitly described. All of these notions will be introduced and discussed in detail in the lectures. If time permits, an inverse spectral problem for  will be briefly described. The lectures are based on recent joint work of the author with Patrick Gérard (Orsay).

1 Introduction 1.1 Motivation The motivation for this topic is the work of Patrick Gérard and Sandrine Grellier in 2010–2014. In [3], they have introduced the cubic Szeg˝o equation i

∂u = P(|u|2 u), u = u(z; t), z ∈ T, t ∈ R, ∂t

as a model for totally non-dispersive evolution equations. Here for each t ∈ R, the function u(·, t) is an element of the Hardy class H 2 = H 2 (T) and P is the the Szeg˝o projection, i.e. the orthogonal projection in L 2 (T) onto the Hardy class H 2 (precise definitions will be given below).

P. Gérard Laboratoire de Mathématiques d’Orsay, Université Paris-Saclay, CNRS, 91405 Orsay, France e-mail: [email protected] A. Pushnitski (B) King’s College London, London, United Kingdom e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. Abakumov et al. (eds.), Extended Abstracts Fall 2019, Trends in Mathematics 12, https://doi.org/10.1007/978-3-030-74417-5_24

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It turned out [3, 4] that this equation is completely integrable and possesses a Lax pair. Indeed, a function u is a solution to the cubic Szeg˝o equation if and only if d Hu = [Bu , Hu ], dt where Hu is a Hankel operator with the symbol u (see Sect. 1.7 for the definition), and Bu is a certain auxiliarly skew-selfadjoint operator. In particular, it follows that if the operator Hu is compact, then its singular values are integrals of motion for the cubic Szeg˝o equation. In order to solve the Cauchy problem for the cubic Szeg˝o equation, one must therefore develop a version of direct and inverse spectral theory for Hu . The spectral data in this problem involves the sequence of singular values of Hu and the sequence of inner functions, parameterizing the Schmidt subspaces of Hu (i.e. the eigenspaces of |Hu |). This was achieved in [5, 6] for u ∈ VMOA(T), which corresponds to compact Hankel operators Hu .

1.2 Summary The important ingredient of the work by Gérard-Grellier was the description of the structure of the Schmidt subspaces of Hu . This description was later made both more precise and more general in [7, 9]. The purpose of this mini-course is to describe this structure, its consequences and some related ideas. To put it briefly, the aim of this mini-course is to state precisely and to prove Main theorem. Every Schmidt subspace of a Hankel operator has the form pK θ , where θ is an inner function, K θ is a model space and p is an isometric multiplier on Kθ. All the underlined terms will be defined and discussed; some ideas of the proof will be given, and some consequences will be mentioned.

1.3 Schmidt Subspaces Let A be a bounded operator in a Hilbert space. We will say that s > 0 is a singular value of A, if the Schmidt subspace E A (s) := Ker(A∗ A − s 2 I ) is non-trivial: E A (s) = {0}. In other words, this means that there exists a non-zero pair (ξ, η) (called the Schmidt pair) of elements in our Hilbert space such that Aξ = sη and A∗ η = sξ.

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It is straightforward to see that A maps the Schmidt subspace E A (s) onto E A∗ (s). Suppose for simplicity of discussion that A is compact. Observe that our Hilbert space can be represented as Ker A ⊕



 E A (s) ,

s

where the orthogonal sum is taken over all singular values of A. Observe that if we know all singular values of A, all Schmidt subspaces E A (s) and we know how A acts from E A (s) to E A∗ (s), then we can reconstruct the operator A from this information.

1.4 Hankel and Toeplitz Matrices The most elementary way of approaching the definition of Hankel and Toeplitz operators is to consider them as infinite matrices in 2 (Z+ ), Z+ = {0, 1, 2, . . . }, with the following structure:  = {a(n + m)}∞ n,m=0 ,

T = {a(n − m)}∞ n,m=0 ,

where {a(n)}∞ n=−∞ is a sequence of complex numbers. Our main focus will be on Hankel operators. We will discuss the boundedness and other analytic properties very soon, but at present let us focus on the following “algebraic” aspect. Observe that the matrix  is symmetric, i.e.   = . It follows that C =  ∗ C, where C is the (anti-linear) operator of complex conjugation in 2 : ∞ C{xn }∞ n=0 = {x n }n=0 .

From here we easily see that C maps E ∗ (s) onto E  (s) and vice versa. Thus, the antilinear operator C maps the subspace E ∗ (s) onto itself; in fact, it is an anti-linear involution on this subspace. Observe also that (C)2 = CC =  ∗ , and so E ∗ (s) = ker((C)2 − s 2 I ).

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Thus, it will be convenient to deal with the anti-linear operator C; our aim will be to describe the Schmidt subspaces ker((C)2 − s 2 I ) and the action of C on these subspaces.

1.5 Hardy Space We use the standard notation for the unit circle and the unit disk in C: T = {z ∈ C : |z| = 1}, D = {z ∈ C : |z| < 1}. We recall basic definitions related to the Hardy space H 2 in the unit disk. This space consists of all functions in L 2 (T) of the form f (z) =

∞ 

 f (n)z n , |z| = 1,

(1)

n=0

  2 where  f 2 = ∞ n=0 | f (n)| < ∞. We will denote by f, g the inner product of f and g in H 2 , and we will denote by 1 the function in H 2 which is identically equal to 1. The Szeg˝o projection P is the orthogonal projection in L 2 (T) onto H 2 , given by P:

∞ 

 f (n)z n →

n=−∞

∞ 

 f (n)z n .

n=0

If f ∈ H 2 , sometimes it is convenient to consider it not as a function on the unit circle T, but as a holomorphic function in the open unit disk D, given by the same formula (1). In this case, the function on the unit circle can be recovered, for example, as the radial (in fact, non-tangential) limit f (eiθ ) = lim f (r eiθ ), r →1−

a.e. θ ∈ (0, 2π).

We refer e.g. to [12, 14] for the theory of boundary behaviour of holomorphic functions in the unit disk. The shift operator S in H 2 is defined by S f (z) = z f (z), and its adjoint S ∗ in H 2 is given by f (z) − f (0) . S ∗ f (z) = z We will also need the Hardy space H ∞ , which can be defined as H 2 ∩ L ∞ (or as the space of all bounded analytic functions in the unit disk).

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1.6 Toeplitz Operators in Hardy Space Let a ∈ L ∞ (T); consider the operator Ta in H 2 , given by Ta f = P(a · f ),

f ∈ H 2.

The function a is called the symbol of Ta . It is straightforward to see that Ta is a (n − m)}∞ bounded and that the matrix of Ta in the standard basis {z n }∞ n=0 is { n,m=0 , i.e. it is a Toeplitz matrix. Conversely, it is not difficult to prove that any bounded Toeplitz matrix T = {t (n − m)}∞ n,m=0

on 2 (Z+ )

is unitarily equivalent to Ta with some a ∈ L ∞ (we will say that T is realized as Ta in the Hardy space). Toeplitz operators satisfy the commutation relation S ∗ Ta S = Ta ; the proof of this is a simple exercise. In fact, if a bounded operator Ta satisfies this commutation relation, then it is necessarily a Toeplitz operator (this is called the Brown-Halmos theorem). We refer, e.g. to [13] for background information on Toeplitz operators.

1.7 Hankel Operators in Hardy Space Consider the symbol u ∈ H ∞ . We define the anti-linear Hankel operator in H 2 by Hu f = P(u · f ),

f ∈ H 2.

u (n + It is straightforward to see that the matrix of Hu in the standard basis is  = { . Remembering the complex conjugation over f , we see that H is unitarily m)}∞ u n,m=0 equivalent to the operator C in 2 . We note that instead of u ∈ H ∞ , we could have taken u ∈ L ∞ . However, Hu depends only on the analytic part of u, and requiring that u is analytic ensures that the symbol u is uniquely defined by the operator Hu . It is an easy exercise to check the commutation relation S ∗ Hu = Hu S. In fact, it is not difficult to show that any bounded anti-linear operator that satisfies this relation, is a Hankel operator.

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Although this is not a focus of our mini-course, we briefly mention the following facts: 1. Condition u ∈ H ∞ is sufficient, but not necessary for the boundedness of Hu . In fact, Hu is bounded if and only if u ∈ BMOA, i.e. u is an analytic BMO function; this fact is often called the Nehari-Fefferman theorem. The class BMOA satisfies H ∞ ⊂ BMOA ⊂ H p , ∀ p < ∞. In particular, BMOA ⊂ H 2 , and so the symbol u of a bounded Hankel operator Hu is in H 2 . This can also be seen directly, because Hu 1 = P(u1) = u. 2. Kronecker’s theorem asserts that Hu is a finite rank operator if and only if u is a rational function with no poles in the closed unit disk. Example Let u α (z) = of H 2 , i.e.

1 , where |α| 1−αz

< 1. Recall that u α is the reproducing kernel

f, u α = f (α),

f ∈ H 2.

It is a simple exercise to check that Hu α f = u α , f u α , i.e. Hu α is a rank one Hankel operator. Using this example, it is not difficult to prove one part of Kronecker’s theorem. If v is a rational function, we can represent it as a sum of elementary fractions u α and their derivatives. Each of them gives rise to a finite rank Hankel operator. We refer, e.g. to [13, 15] for background information on Hankel operators.

2 Inner Functions, Model Spaces and Isometric Multipliers 2.1 Inner Functions A non-constant function θ ∈ H ∞ is called inner, if |θ(z)| = 1 for almost all z in the unit circle. N Example Let N ∈ N and let {z n }n=1 be points in the open unit disk. Define

θ(z) =

N  zn − z 1 − zn z n=1

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N for |z| < 1. Then θ is inner; it is a Blaschke product of degree N with zeros {z n }n=1 .

Example The previous example can be modified to the case of infinitely many zeros. The only new aspect is that one has to take care about the convergence of the infinite product. Let {z n }∞ n=1 be points in the open unit disk, satisfying the condition ∞  (1 − |z n |) < ∞. n=1

Define θ(z) =

∞  zn zn − z |z | 1 − z n z n=1 n

for |z| < 1. (The terms z n /|z n | are inserted in order to make the infinite product converge.) Then θ is inner; it is an infinite Blaschke product. We define the degree of θ to be infinity. Example Let μ ≥ 0 be a finite singular measure on the unit circle T; define

θ(z) = exp −

π −π

eit + z dμ(t) eit − z



for |z| < 1. Then θ is inner; it is a singular inner function; by definition, the degree of θ is infinity. For example, if μ is a point mass at 1 with μ({1}) = c > 0, we have  z+1 θ(z) = exp c . z−1 In fact, every inner function can be represented as a product of a Blaschke product and a singular inner function.

2.2 Model Spaces The following theorem due to A. Beurling (1949) is fundamental to much of analysis. Theorem 2.1 Let M ⊂ H 2 be a closed subspace, M = {0} and M = H 2 . Suppose that M is invariant under the shift operator: S M ⊂ M. Then there exists an inner function θ such that M = θ H 2 := {θ f : f ∈ H 2 }. Of course, the converse is also true: every subspace θ H 2 is invariant for S. Observing that S M ⊂ M if and only if S ∗ M ⊥ ⊂ M ⊥ , we obtain

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Corollary 2.2 Let M ⊂ H 2 be a closed subspace, M = {0} and M = H 2 . Suppose that M is invariant under the backwards shift operator: S ∗ M ⊂ M. Then there exists an inner function θ such that M = K θ := H 2 ∩ (θ H 2 )⊥ . The space K θ is called a model space. Let us rewrite the condition f ∈ K θ in an equivalent way: f ⊥ θ H 2 ⇔ θ f ⊥ H 2 ⇔ zθ f ∈ H 2 . Example Let θ(z) = z N , N ∈ N. Then K z N = {a0 + · · · + a N −1 z N −1 : a0 , . . . , a N −1 ∈ C} is simply the space of all polynomials of degree ≤ N − 1. Example Let N ∈ N and let θ be a Blaschke product of degree N with distinct zeros N {z n }n=1 . It is easy to see that N K θ = span{u zn }n=1 , u α (z) =

1 . 1 − αz

Indeed, recalling that u α is the reproducing kernel, we see that the orthogonal comN is precisely the linear subspace of functions f ∈ H 2 that plement to span{u zn }n=1 N vanish at all points {z n }n=1 . This subspace coincides with θ H 2 . Remark 1. It is easy to see that θ(0) = 0 if and only if 1 ∈ K θ . 2. It is easy to see that the map f → zθ f is an involution on K θ .

2.3 Isometric Multipliers on Model Spaces Let M be a closed subspace in H 2 , and let p be an analytic function in the open unit disk. One says that p is an isometric multiplier on M, if for every f ∈ M, we have p f ∈ H 2 and  p f  =  f . In this case we will denote pM = { p f : f ∈ M}. Clearly, pM is a closed subspace in H 2 . Remark Observe that if 1 ∈ M, then (taking f = 1) we have p ∈ M and  p = 1.

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Exercise Check that if 1 ∈ M, then p is (up to a unimodular complex factor) the normalized projection of 1 onto the space pM. The interest to isometric multipliers on model spaces arose due to the following result by E. Hayashi from 1986: Theorem 2.3 ([10]) Let T be a bounded Toeplitz operator in H 2 with a non-trivial kernel. Then there exists an inner function θ and an isometric multiplier p on K θ such that kerT = pK θ . D. Sarason has characterized all isometric multipliers on a given space K θ . Theorem 2.4 ([17]) Let θ be an inner function with θ(0) = 0, and let p ∈ H 2 be a function of norm one. Then p is an isometric multiplier on K θ if and only if it can be represented as a(z) , |z| < 1, p(z) = 1 − θ(z)b(z) where a, b ∈ H ∞ is a pair of functions such that |a|2 + |b|2 = 1 almost everywhere on the unit circle. Some ideas of the proof We will proof only the easy part of the theorem (the “if” part) in the easy case when |b| ≤ const < 1. Let us write | p|2 on the unit circle. By a simple algebra, we have | p|2 =

θb |a|2 1 − |b|2 θb + = =1+ . 2 2 1 − θb 1 − θb |1 − θb| |1 − θb|

Now let us multiply this by | f |2 and write the result as | p|2 | f |2 = | f |2 +

bf bf θf + θ f. 1 − θb 1 − θb

(2)

Consider the second term in the right hand side. By the assumption on b, the term bf 1 − θb is an element in H 2 . Further, since f ∈ K θ , we have zθ f ∈ H 2 , and so θ f is an element in H 2 which vanishes at the origin. It follows that bf θf 1 − θb

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is a function in H 1 which vanishes at the origin. Thus, its integral over the unit circle vanishes. The same considerations apply to the last term in the right hand side of (2): its integral over the unit circle vanishes. So, integrating (2), we obtain

π −π

2

2

| p(eit )| | f (eit )| dt =



π −π

2

| f (eit )| dt,

which means precisely that  p f  =  f .

2.4 Frostman Shifts Here we address the following question. Let M = pK θ , where p is an isometric multiplier on K θ . Are the parameters p, θ unique in the representation M = pK θ ? It is clear that one can multiply both p and θ by unimodular complex numbers without changing the space pK θ . It turns out that there is another natural family of transformations on p and θ that leaves the space pK θ invariant. To begin, consider the example of the previous theorem with both a and b being constants. Changing notation slightly, we see that for every |α| < 1, the function 1 − |α|2 /(1 − αθ) is an isometric multiplier on K θ . Exercise 1. Check that 1 − |α|2 α−θ K θ ⊂ K θα , θ α = . 1 − αθ 1 − αθ 2. Check that in fact we have the equality 1 − |α|2 K θ = K θα . 1 − αθ Hint: use the fact that (θα )α = θ and

1 − |α|2 1 − αθ = . 1 − αθα 1 − |α|2

We can rewrite the result of this exercise as follows: 1 − αθ K θ = gα K θα , gα = , 1 − |α|2 and gα is the isometric multiplier on K θα . This transformation is called the Frostman shift. From here we see that if p is an isometric multiplier on K θ , then the space pK θ can be equivalently written as

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185

pK θ = pgα K θα , where pgα is an isometric multiplier on K θα . In fact, the converse statement also holds, see [2]. If p K θ , pK θ = p is an isometric multiplier on K θ , then where p is an isometric multiplier on K θ and for some constants |α| < 1, |c1 | = 1, |c2 | = 1 we have

p = c2 pgα . θ = c1 θα , Suppose we have a subspace of the form pK θ . It is often convenient to perform a Frostman shift with α = θ(0). Then θα (0) = 0 and we write our subspace in an θ(0) = 0. equivalent form p K θ with Also, θ(0) = 0 is a convenient normalisation which fixes the choices of θ and p up to unimodular constant factors.

2.5 Nearly Invariant Subspaces The following definition was introduced by D. Hitt in 1988, see [11]. A closed subspace M ⊂ H 2 is called nearly S ∗ -invariant, if S ∗ (M ∩ 1⊥ ) ⊂ M. In other words, we require that if f ∈ M and f (0) = 0, then f (z)/z ∈ M. Observe that if M = {0} is nearly S ∗ -invariant, then M ⊥ 1. Indeed, if M ⊥ 1 and if f ∈ M, then after dividing by z a finite number of times, we must arrive at a function which does not vanish at the origin, which contradicts the assumption M ⊥ 1. Because of this simple observation, the condition that M ⊥ 1 is often included in the definition of nearly S ∗ -invariance. Theorem 2.5 (Hitt, [11]) Every nearly S ∗ -invariant subspace M is of the form M = pN , where S ∗ N ⊂ N and p is an isometric multiplier on N . Thus, we have two possibilities: (i) M = pK θ , where θ is inner and p is an isometric multiplier on K θ ; (ii) M = p H 2 , where p is an inner function. Some ideas of the proof (1) Let p be the normalized projection of 1 onto M. For f ∈ M, write f = c0 p + f 1 , f 1 ⊥ p. Since both f and p are in M, we also have f 1 ∈ M. By orthogonality,  f 2 = |c0 |2 +  f 1 2 .

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Further, f 1 ⊥ p means f 1 ⊥ 1 and so, by the nearly S ∗ -invariance, we have S ∗ f 1 = f 1 /z ∈ M. For f 1 /z we write again f 1 /z = c1 p + f 2 ,

f 2 ⊥ p.

Then we get  f 1 2 = |c1 |2 +  f 2 2 and again S ∗ f 2 ∈ M. Continuing recursively, we get f n /z = cn p + f n+1 ,

f n+1 ⊥ p,

and  f n 2 = |cn |2 +  f n+1 2 . Linking these equations together gives f (z) = (c0 + c1 z + · · · + cn z n ) p(z) + z n f n+1 (z) and  f 2 = |c0 |2 + · · · + |cn |2 +  f n+1 2 ≥ |c0 |2 + · · · + |cn |2 . Inspecting the Taylor series of f / p at zero, we find that f (z)/ p(z) =

∞ 

cn z n

n=0

and  f / p = 2

∞ 

|cn |2 ≤  f 2 .

n=0

So the operator T1/ p : M → H 2 is a contraction. (2) Consider the set M0 = { f ∈ M :  f / p =  f }.

Exercise 1. Using the previous step of the proof, prove that M0 is a linear (linearity is non-trivial!) closed subspace of M. 2. Prove that T1/ p (M0 ) is S ∗ -invariant. (3) Using a separate clever calculation with reproducing kernels, Hitt shows that actually M0 = M. So now we have T1/ p M = N , or M = pN , where p is an isometric multiplier on N . This completes the proof of Theorem 2.5.

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2.6 Toeplitz Kernels As already mentioned, Toeplitz operators satisfy the key commutation relation S ∗ Ta S = Ta .

(3)

Here we determine the structure of Toeplitz kernels. First we make two remarks: (1) Since I is a Toeplitz operator with the symbol 1, we have ker(Ta − λI ) = kerTa−λ1 . Thus, describing the structure of Toeplitz kernels is equivalent to describing the structure of all Toeplitz eigenspaces. (2) A deep theorem by Rosenblum [16] says that if Ta is a bounded self-adjoint Toeplitz operator with a non-constant symbol a, then the spectrum of Ta is purely absolutely continuous. In particular, Ta has no eigenvalues. This shows that the study of kernels of Toeplitz operators is a specifically non-selfadjoint problem. Now let us prove Hayashi’s theorem on the structure of Toeplitz kernels by using Hitt’s theorem. Let Ta be a bounded Toeplitz operator with kerTa = {0}. (1) Let us check that kerTa is nearly S ∗ -invariant. Suppose Ta f = 0 and f ⊥ 1; then f = SS ∗ f . Let us apply the commutation relation (3) to S ∗ f : we get S ∗ Ta SS ∗ f = Ta S ∗ f. The left hand side is S ∗ Ta f = 0, hence S ∗ f ∈ kerTa , as claimed. (2) Since kerTa is a nearly invariant subspace, by Hitt’s theorem there are two possibilities: (i) kerTa = pK θ where p is an isometric multiplier on K θ , and (ii) kerTa = p H 2 , where p is inner. Let us show that the second possibility implies Ta = 0. Let f ∈ H 2 , and Ta ( p f ) = 0, where p is inner. This means P(ap f ) = 0, i.e. ap f ∈ z H 2 . Since p is inner, this can be equivalently rewritten as a f ∈ pz H 2 . Since p H 2 ⊂ H 2 , we obtain f ∈ kerTa . Recall that f was an arbitrary element in H 2 ; this, we get Ta = 0. Example Let θ be an inner function; consider the Toeplitz operator Tθ . It is easy to see that in this case kerTθ = K θ .

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3 Schmidt Subspaces of Hankel Operators 3.1 Preliminaries Recall that Hankel operators satisfy the commutation relation S ∗ Hu = Hu S.

(4)

First we discuss Hankel kernels. If Hu f = 0, then by (4) we also have 0 = S ∗ Hu f = Hu S f. Thus, Hankel kernels are invariant under the shift operator S, and so by Beurling’s theorem they have the form ψ H 2 for some inner function ψ. Taking orthogonal complements, we obtain Ran Hu = K ψ .

(5)

Next, we want to discuss one particular example: Hankel operators with inner symbols. Let θ be inner; consider the Hankel operator Hθ , Hθ f = P(θ f ). It is straightforward to see that in this case we have ker Hθ = zθ H 2 , Ran Hθ = K zθ , and Hθ is an anti-linear involution on K zθ , Hθ f = θ f ,

f ∈ K zθ .

It follows that Hθ2 is the orthogonal projection onto K zθ . In other words, in this case we have only one singular value s = 1, and the corresponding Schmidt subspace E Hθ (1) = K zθ .

3.2 Main Result Theorem 3.1 Let Hu be a bounded Hankel operator on H 2 , and let s > 0 be a singular value of Hu : E Hu (s) = ker(Hu2 − s 2 I ) = {0}. Then there exists an inner function θ and an isometric multiplier p on K θ such that

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189

E Hu (s) = pK θ . Before embarking on the proof, we note that E Hu (s) may or may not be nearly S ∗ -invariant. Indeed, it may happen that E Hu (s) is orthogonal to 1. Example Let 0 < α < 1, and let u(z) =

1 − α2 . 1 − αz 2

Exercise Check that Hu u = u. Thus, 1 is a singular value, and u ∈ E Hu (1). Exercise Applying S ∗ to the identity Hu u = u, check that Hu (zu) = α(zu). Thus, α is a singular value, and zu ∈ E Hu (α). Exercise Check that rank Hu = 2. Deduce that E Hu (1) = span{u},

E Hu (α) = span{zu}.

Summarizing, we see that E Hu (α) ⊥ 1. Proof of Theorem 3.1 in the case E Hu (s) ⊥ 1 (1) Let us establish some identities. Besides (4), we need the rank one identity SS ∗ = I − ·, 1 1, and the obvious identity Hu 1 = u. Using (4), we have S ∗ Hu2 S = Hu SS ∗ Hu = Hu2 − ·, Hu 1 Hu 1 = Hu2 − ·, u u. (Compare this with the identity S ∗ T S = T for Toeplitz operators!) Let us multiply the last identity by S ∗ on the right. After rearranging, we obtain S ∗ Hu2 − Hu2 S ∗ = ·, 1 S ∗ Hu u − ·, Su u.

(6)

(2) We need to establish the existence of an element h ∈ E Hu (s) such that u, g = 0. This follows from the assumption E Hu (s) ⊥ 1. Indeed, let h ∈ E Hu (s) be such that h, 1 = 0; take g = Hu h. Then u, g = ug, 1 = Hu g, 1 = Hu2 h, 1 = s 2 h, 1 = 0. (3) Let us prove that E Hu (s) is nearly S ∗ -invariant. Let f ∈ E Hu (s) ∩ 1⊥ , and let g as above. Let us take the bilinear form of (6) on the elements f , g. For the left hand side, we have

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S ∗ Hu2 f, g − Hu2 S ∗ f, g = s 2 S ∗ f, g − S ∗ f, Hu2 g = s 2 S ∗ f, g − s 2 S ∗ f, g = 0.

For the right hand side, we have f, 1 S ∗ Hu u, g − f, Su u, g ; by assumption f ⊥ 1, and so we obtain f, Su u, g = 0. But u, g = 0, and so we obtain that f, Su = 0. Now let us substitute f back into (6) and use the latter fact; the right hand side vanishes and we have S ∗ Hu2 f − Hu2 S ∗ f = 0. Since Hu2 f = s 2 f , this can be rewritten as (Hu2 − s 2 I )S ∗ f = 0, and so S ∗ f ∈ E Hu (s), as claimed. Thus, E Hu (s) is a nearly S ∗ -invariant subspace. (4) By Hitt’s theorem, either E Hu (s) = pK θ (where p is an isometric multiplier on K θ ) or E Hu (s) = p H 2 (where p is inner). Exercise Show that the second option is not possible. Use the fact that the calculation from the previous step of the proof shows that S ∗ (E Hu (s) ∩ 1⊥ ) ⊂ E Hu (s) ∩ u ⊥ . Compare this with

S ∗ ( p H 2 ∩ 1⊥ ) = p H 2

if p is inner. Bring this to a contradiction. This completes the proof in the case of E Hu (s) ⊥ 1. Proof of Theorem 3.1 in case E Hu (s) ⊥ 1 Let α, |α| < 1, be such that α is not a common zero of all elements of E Hu (s). Let μ be the Moebius map μ(z) =

α−z , 1 − αz

mapping the unit disk onto itself, and let Uμ be the corresponding unitary operator on H 2 : 1 − |α|2 Uμ f (z) = f (μ(z)). 1 − αz

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191

By a direct calculation, Uμ is a unitary involution on H 2 . Exercise Check that Uμ Hu Uμ = Hw with some symbol w. Now consider M = Uμ E Hu (s). Then M = E Hw (s) and z = 0 is not a common zero of all elements of M, i.e. M ⊥ 1. By the previous part of the proof, it follows that M = pK θ with some θ, and p is an isometric multiplier on K θ . Now E Hu (s) = Uμ ( pK θ ). Exercise Check that Uμ ( pK θ ) = ( p ◦ μ)K θ◦μ , and p ◦ μ is an isometric multiplier on K θ◦μ . This completes the proof in the case E Hu (s) ⊥ 1.



In the remainder of this section, we will discuss (mostly without proofs) some consequences of Theorem 3.1 and some related statements.

3.3 The Action of Hu on E Hu (s) In fact, by the same method we obtain not only the formula for the subspace E Hu (s) but also the formula for the action of Hu on this subspace. Assume the hypothesis of Theorem 3.1; we have E Hu (s) = pK θ , where p is an isometric multiplier on K θ . By performing a Frostman shift, we can always make sure that θ(0) = 0. In this case, we will write zθ instead of θ. Theorem 3.2 Assume the hypothesis of Theorem 3.1, and let E Hu (s) = pK zθ , where θ is an inner function and p is an isometric multiplier on K zθ . Then, for some unimodular constant eiγ , the action of Hu on E Hu (s) is given by the following formula: Hu ( p f ) = seiγ p Hθ f = seiγ θ f ,

f ∈ K zθ .

(7)

The constant eiγ depends on the normalisation of p and θ; in particular, we can normalize p and θ so that eiγ = 1, in which case formula (7) becomes particularly simple: Hu ( p f ) = sp Hθ f, f ∈ K zθ .

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One can also write this formula in operator theoretic terms as Hu T p = sT p Hθ ,

on K zθ .

Thus, the operator T p intertwines the action of Hu on E Hu (s) and the action of the standard involution f → θ f on K zθ .

3.4 Decompositions of Model Spaces Let Hu be a finite rank Hankel operator. Recall that by (5), we have Ran Hu = K ψ for some inner function ψ. Suppose that Hu has singular values s1 , . . . , s N , and the corresponding Schmidt subspaces are represented as E Hu (sn ) = pn K zθn , where pn is an isometric multiplier on K zθn . Thus, we arrive at an interesting orthogonal decomposition of the model space K ψ : Kψ =

N 

pn K zθn .

(8)

n=1

3.5 The Adamyan-Arov-Krein Theorem Here we briefly discuss the Adamyan-Arov-Krein (AAK) theorem, which gives additional information about the inner factors of pn in formula (8). The theorem below is essentially due to AAK in [1]; however it was expressed there in a different form, since Theorem 3.1 was not available to AAK at that point. The precise statement below is from [7]. Theorem 3.3 Assume the hypothesis of Theorem 3.1, and let E Hu (s) = pK zθ , where θ is an inner function and p is an isometric multiplier on K zθ . Then the degree of the inner factor of p equals the total multiplicity of the spectrum of Hu2 in the open interval (s 2 , ∞). To illustrate this, let us come back to the decomposition (8) and write pn = qn ϕn , where qn is outer and ϕn is inner:

Schmidt Subspaces of Hankel Operators

Kψ =

193 N 

qn ϕn K zθn .

n=1

Assume that the singular values have been ordered as s1 > s2 > · · · > s N > 0, and let qn ϕn K zθn correspond to the singular value sn . As the dimension of K zθn is deg(zθn ), we have for every n = 1, . . . , N deg ϕn =

n−1 

deg(zθk ).

k=1

In particular, for n = 1 this formula says that p1 is an outer function. Below we prove the theorem in this particular case (following [1]). The general case is much more difficult. Poof of Theorem 3.3 for the top singular value Assume s = Hu ; let us prove that p is outer. (1) First we observe that in this case f ∈ E Hu (s) if and only if Hu f  ≥ s f . (2) Let f = a f 0 ∈ E Hu (s), where a is inner and f 0 ∈ H 2 . We have s f 0  = s f  = Hu f  = P(ua f 0 ) = P(au f 0 ). Observe that Pa(I − P) = 0 and therefore P(au f 0 ) = P(a P(u f 0 )). Thus, P(au f 0 ) = P(a P(u f 0 )) = P(a Hu f 0 ) ≤ Hu f 0 . We conclude that s f 0  ≤ Hu f 0 , and therefore f 0 ∈ E Hu (s). (3) Suppose p = qϕ, where ϕ is inner and q is outer. As p ∈ E Hu (s), by the previous step we have q ∈ E Hu (s). Then q = ph, h ∈ K zθ ; so q = qϕh, and we conclude that ϕ = const.

3.6 Inverse Spectral Problems Finally, without going into details, we would like to show how the parameterization of Schmidt subspaces in terms of model spaces can be used in inverse spectral problems. The following facts are borrowed from [9]. Let N ∈ N, let s1 > s2 > s2 > · · · > s N > sN ≥ 0 s1 > be real numbers, and let ψ1 , . . . , ψ N be any inner functions. For z ∈ D, consider the N × N matrix

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 C(z) =

sk ψ j (z) s j − z 2 s j − sk2

N j,k=1

and the vectors in C N , ⎞ ψ1 (z) ⎟ ⎜  = ⎝ ... ⎠ , ⎛

ψ N (z)

⎛ ⎞ 1 ⎜ .. ⎟ 1 = ⎝.⎠. 1

Then (this is a non-trivial fact!) the matrix C(z) is invertible for all z ∈ D and the function u(z) = C(z)−1 , 1 C N is in H ∞ . Consider the Hankel operator Hu . Then Hu has the singular values {s j } Nj=1 (and no others), and each Schmidt subspace of Hu can be represented as E Hu (s j ) = p j K zψ j

(9)

with some isometric multipliers p j (which can also be written explicitly in terms of s j , ψ j ). Remark 1. The numbers s j are arbitrary parameters in this construction. In fact, they coincide with the singular values of the “associated Hankel operator” HS ∗ u . 2. In this construction, all Schmidt subspaces E Hu (s j ) are nearly S ∗ -invariant, i.e. none of them is orthogonal to 1. It is however possible to modify this construction so that Hu has also some Schmidt subspaces orthogonal to 1. 3. The above construction gives all possible Hankel operators with the singular values {s j } Nj=1 (and no others), satisfying (9) and satisfying the additional requirement that E Hu (s j ) ⊥ 1 for all j. In other words,   N sk }k=1 , {ψ j } Nj=1 {s j } Nj=1 , { is a complete independent set of spectral data for this inverse problem.

Schmidt Subspaces of Hankel Operators

195

References 1. V. M. Adamjan, D. Z. Arov, M. G. Kre˘ın, Analytic properties of the Schmidt pairs of a Hankel operator and the generalized Schur-Takagi problem, Math. USSR-Sb. 15 (1971), 31–73 2. R. B. Crofoot, Multipliers between invariant subspaces of the backwards shift, Pacific J. Math. 166, no. 2 (1994), 225–256 3. P. Gérard, S. Grellier, The cubic Szeg˝o equation, Ann. Scient. Éc. Norm. Sup. 43 (2010), 761–810 4. P. Gérard, S. Grellier, Invariant tori for the cubic Szeg˝o equation, Invent. Math. 187, no. 3 (2012), 707–754 5. P. Gérard, S. Grellier, Inverse spectral problems for compact Hankel operators, J. Inst. Math. Jussieu 13 (2014), 273–301 6. P. Gérard, S. Grellier, The cubic Szeg˝o equation and Hankel operators. Astérisque 389 (2017) 7. P. Gérard, A. Pushnitski, Weighted model spaces and Schmidt subspaces of Hankel operators. Journal of London Math. Soc. 101(1), 271–298 (2020) 8. P. Gérard, A. Pushnitski, Inverse spectral theory for a class of non-compact Hankel operators, Mathematika 65, no. 1 (2019), 132–156 9. P. Gérard, A. Pushnitski, The Structure of Schmidt Subspaces of Hankel Operators: A Short Proof, preprint arXiv:1907.05629, to appear in Studia Mathematica 10. E. Hayashi, The kernel of a Toeplitz operator. Integral Equations Operator Theory 9(4), 588– 591 (1986) 11. D. Hitt, Invariant subspaces of H2 of an annulus. Pacific J. Math. 134(1), 101–120 (1988) 12. P. Koosis, Introduction to H p Spaces (Cambridge University Press, Cambridge, 1998) 13. N.K. Nikolski, Operators, Functions, and Systems: An Easy Reading, Vol. 1: Hardy, Hankel, and Toeplitz (AMS, 2002) 14. N.K. Nikolski, Hardy Spaces (Cambridge University Press, Cambridge, 2019) 15. V.V. Peller, Hankel Operators and Their Applications (Springer, Berlin, 2003) 16. M. Rosenblum, The absolute continuity of Toeplitz’s matrices, Pacific J. Math. 10 (1960), 987–996 17. D. Sarason, Nearly invariant subspaces of the backward shift. Operator Theory: Advances and Applications 35, 481–493 (1988)

Maximum Principle and Comparison of Singular Numbers for Composition Operators Hervé Queffélec

Abstract We compare the decay of singular numbers for composition operators with a given symbol, but acting on different Hilbert spaces of analytic functions.

1 Introduction 1.1 General Setting Let H be a Hilbert space such that H → H (D), the holomorphic functions on D, and ϕ : D → D, analytic (a symbol). The composition operator Cϕ : H → H (D) is formally defined by Cϕ ( f ) = f ◦ ϕ. The questions, listed in ambition order, which one poses concerning Cϕ are the five following ones: 1. Boundedness: H → H . 2. Compactness.  p 3. Membership in a Schatten class S p , p > 0, i.e. ∞ n=1 [an (C ϕ )] < ∞ where an (Cϕ ) is the distance of T to operators of rank < n. 4. Specific decay rate of an (Cϕ ). 5. Form of the best approximating operators, in the style of Adamyan–Arov–Krein’s theory for Hankel operators. The “historical evolution” when H = H 2 shows the difficulty of the problem. 1. Boundedness: it always holds (Littlewood’s subordination principle, 1925). 2. Compactness: it was characterized by B. McCluer (Carleson measures, 1985) and J. Shapiro (Nevanlinna counting function, 1987) independently. H. Queffélec (B) CNRS, Laboratorie Paul Painlevé, UMR 8524, Université de Lille, Cité Scientifique, Bâtiment M2, 59655 Villeneuve D’ascq, Cedex, France e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. Abakumov et al. (eds.), Extended Abstracts Fall 2019, Trends in Mathematics 12, https://doi.org/10.1007/978-3-030-74417-5_25

197

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3. Membership in a Schatten class S p , p > 0. It was characterized by Luecking, then Luecking-Zhu, in the same terms, around 1990. 4. Decay rate of an (Cϕ ). It was undertaken by Li, Queffélec, Rodríguez-Piazza in 2012. 5. Form of the best approximating operators (nobody knows). Given a sequence (βk )k≥0 of positive numbers with lim inf k→∞ (βk )1/k ≥ 1, we define the weighted Hardy space H = H 2 (β) → H (D): H 2 (β) :=



f (z) =



ck z k :

k≥0

∞ 

 |ck |2 βk =:  f 2 < ∞ .

(1.1)

k=0

A good example is given by the norms:   f 2 = | f (0)|2 + | f  (z)|2 ω(1 − |z|2 )d A(z)

(1.2)

D

 where

1

 ω(t)dt < ∞, corresponding to βk = k

0

1. 2. 3. 4. 5.

1

2

(1 − t)k−1 ω(t)dt.

0

If ω(t) = t , βk = Bergman space B . If ω(t) = t, βk = 1, Hardy space H 2 . If ω(t) = 1, βk = k + 1, Dirichlet space D2 . If ω(t) = t γ+2 , βk = (k + 1)−1−γ , weighted Bergman space B2γ , |γ| ≤ 1. If ω(t) = t α , βk = (k + 1)1−α , weighted Dirichlet space Dα2 , −1 < α ≤ 1. 2

1 , k+1

2

Given a symbol ϕ, one tries to compare Cϕ acting on different spaces H 2 (β). Various comparison results in this spirit (compactness, or membership in S p ) when the space H 2 (β) changes were given by J. Shapiro, P. D.Taylor, N. Zorboska, etc. But the proofs strongly use the specific form of composition operators. Here, we argue as in complex interpolation theory, in which the maximum modulus principle is strongly involved (three lines theorem).

1.2 Starting Point We wished since long to compare the singular numbers of a given composition operator acting on two different Hilbert spaces H and H  , typically H = H 2,

H  = D2 .

Specifically, in [5, 6], we were interested by the “lens map” λ = λθ , 0 < θ < 1, and the “cusp map” χ. Lens maps are defined later. The cusp map has an ugly analytic expression, but its image is simpler to describe: it is the intersection of the exterior of both circles C(1 ± i/2, 1/2) and of the interior of the circle C(1/2, 1/2). We obtained the following table.

Maximum Principle and Comparison of Singular Numbers … Spaces H = H 2 = D12 H H H H H

= H 2 = D12 = Dα2 , 0 < α ≤ 1 = Dα2 , 0 < α ≤ 1 = D2 = D02 = D2 = D02

199

Operators Cλ lens map

Singular numbers √ an (Cλ ) ≈ e−b n

Cχ Cλ Cχ Cλ Cχ

an (Cχ ) ≈ e log n an (Cλ ) ≈ ? an (Cχ ) ≈ ? an (Cλ ) ≈ 1 √ an (Cχ ) ≈ e−b n

cusp map lens map cusp map lens map cusp map

−b

n

Cusps on D2 behave like lens on H 2 , what does that mean? We had the feeling to understand nothing! In this talk, we come back to the question (see [7] for more details).

2 V. Katsnelson’s Result, One New Application 2.1 Maximum Principle We recall the maximum principle for subharmonic functions. Theorem 2.1 Let  ⊂ C be open (possibly unbounded), with  = C, and let also u :  → R ∪ {−∞} a function which is u. s. c. on  and subharmonic on . We assume that, for numbers −∞ ≤ m ≤ M < ∞, one has: • u(z) ≤ m throughout ∂; • u(z) ≤ M throughout . Then, u(z) ≤ m throughout .

2.2 The Result In the seventies, the Ukrainian mathematician V. E. Katsnelson found a beautiful application of Theorem 2.1 to operators [3]. The result was used e.g. by Chalendar and Partington [1, 2], notably in terms of Schatten classes when p ≥ 1; here, we go one step beyond, in terms of approximation numbers; then, we apply it to composition operators. Theorem 2.2 (Katsnelson) Let H be a Hilbert space with a ONB (e j ) j≥0 ; let T ∈ L(H ) be lower-triangular and let (d j ) j≥0 a non-decreasing sequence of positive numbers. Let D be the (generally unbounded) diagonal operator “defined” by D(e j ) = d j e j for all j. Then, D −1 T D is bounded on H , and D −1 T D ≤ T . Proof We give it when dim H < ∞. We have assumed

200

H. Queffélec

ti, j := T (e j ), ei  = 0 for i < j. Let  = {z : Re z > 0}. Then we set, for z ∈ :   T (z) = D −z T D z = ti, j (z) with ti, j (z) =

ti, j (d j /di )z if i ≥ j 0 otherwise.

Next, we set u(z) = T(z), where T(z) is holomorphic with values in L(H ); hence u is subharmonic on  and continuous on . By construction, |ti, j (z)| ≤ |ti, j | ≤ T , so that u is bounded on . Now, the argument splits in two: 1. By Theorem 2.1, D −1 T D = u(1) ≤ supz∈∂ u(z). 2. If z ∈ ∂, u(z) = T  because then D z and D −z are unitary. We thus get the result.

2.3 Comments on the Assumptions “Lower triangular” is not as restrictive as it might seem. If H = H 2 (β) with its canonical basis and T = Mw Cϕ , a weighted composition operator with ϕ(0) = 0, then for i < j: T (e j ), ei  = wϕ j , z i  up constants, and this is 0 for i < j, since then one has wϕ j =  to multiplicative k k≥ j ck ( j)z . In the general case, we compose on the left with an involutive automorphism ϕa such that ϕa (0) = a := ϕ(0). This works for example for the weighted Dirichlet spaces Dα2 when α > −1. As for the second assumption ((d j ) increasing), it will be fulfilled in most cases of interest for us.

3 Singular Numbers, Improvement on Katsnelson 3.1 Singular Numbers Let T : H → H be compact, with eigenvalues (λn (T ))n≥1 in decreasing order of modulus. We can write T =

∞  n=1

sn u n ⊗ vn , or else T (x) =

∞  n=1

sn x, vn u n

(3.1)

Maximum Principle and Comparison of Singular Numbers …

201

where (u n ) and (vn ) are two ON sequences, and s1 ≥s2 ≥ · · · sn ≥ · · · with lim sn = 0, are the singular numbers of T , namely sn (T ) = λn (|T |) where |T | =

√

T ∗T .

Fact. sn (T ) = an (T ), the nth approximation number of T . Schatten class. S p = {T ∈ K(H ) : (sn (T ))n≥1 ∈  p },

p > 0.

Theorem 3.1 (H. Weyl) One has the inequality n

|λ j (T )| ≤

j=1

n

s j (T ) for all n ≥ 1.

(3.2)

j=1

This is a fundamental inequality, which motivates the next subsection.

3.2 Subordination and Log-Subordination Let us consider S = {u = (u j ) j≥1 } the set of non-increasing sequences of real numbers. S + = {u = (u j ) j≥1 } the subset of S formed by real positive sequences. If u, v ∈ S, we say that u is subordinate to v, and write u ≺ v, if n 

uj ≤

j=1

n 

v j for all n ≥ 1.

j=1

If u, v ∈ S + , we say that u is log-subordinate to v if log u ≺ log v, or again  if nj=1 u j ≤ nj=1 v j for all n ≥ 1. Weyl’s inequality can be expressed by saying that |λ| is log-subordinate to s. Log-subordination (cf. log-convexity, logsubharmonicity) is much stronger than subordination, and allows pointwise comparisons, like √ log u ≺ log v =⇒ u 2n ≤ v1 vn . Moreover, we have the simple Proposition 3.1 Let u, v ∈ S with values in an interval I , and let h : I → R increasing and convex. Then 1. u ≺ v =⇒ h(u) ≺ h(v). 2. If u, v ∈ S + , then u ≺ v =⇒ u p ≺ v p for all p ≥ 1. 3. If u, v ∈ S + , then log u ≺ log v ⇐⇒ u p ≺ v p for all p > 0.

202

H. Queffélec

3.3 New Theorem Here is an improvement on Katsnelson, and Chalendar-Partington. Theorem 3.2 (Lefèvre-Li-Queffélec-Rodríguez) Same assumptions as in Theorem 2.2, with moreover T compact. Then D −1 T D is compact as well, and moreover n

s j (D −1 T D) ≤

j=1

n

s j (T ) for all n ≥ 1.

j=1

In other terms, s(D −1 T D) is log-subordinate to s(T ). Proof Here is a short “abstract nonsense” type proof, which shows that Theorem 3.2 follows directly from Katsnelson’s result. Let I denote the set of all increasing n-tuples α = (i 1 < i 2 < · · · < i n ) of non-negative integers. Let (u α )α∈I be the orthonormal basis of the antisymmetric tensor product n (H ) defined by u α = ei1 ∧ ei2 ∧ . . . ∧ ein , α ∈ I.

(3.3)

Now, we use the general fact that n

s j (D −1 T D) = n (D −1 T D)

j=1

where n denotes the n-skew product. Moreover  −1 n (D −1 T D) = n (D −1 ) n (T ) n (D) = n (D) n (T ) n (D) where n (D) is the diagonal operator on the basis (u α ) with diagonal elements δα = di1 · · · din if α = (i 1 < i 2 < · · · < i n ). We next show that n (T ) is lower triangular in the following sense, denoting by α = (i 1 < i 2 < · · · < i n ) and β = ( j1 < j2 < · · · < jn ) two elements of I : δα < δβ ⇒ n (T )(u β ), u α  = 0.

(3.4)

And (3.4) is exactly what is needed for Katsnelson’s theorem to apply to n (T ) and n (D)!

Maximum Principle and Comparison of Singular Numbers …

203

4 Application to Composition Operators 4.1 Our Theorem in This Context As an application of the general principles of Sect. 3, we have the following result [7], whose first two items were previously obtained by Chalendar and Partington in [1] and [2]. P designates the set of polynomials. Theorem 4.1 (Lefèvre-Li-Queffélec-Rodríguez) Let H 2 (β) and H 2 (γ) be two weighted Hilbert spaces as in (1.1). Assume that γ is dominated by β in the sense that the sequence (βk /γk ) is increasing, so that H 2 (β) → H 2 (γ). Let T : H (D) → H (D) be a linear operator, with T : P → H 2 (β), whose restriction Tβ to H 2 (β) formally has a lower triangular matrix (ti, j ) on the ONB (ei ) of H 2 (β). Then (with obvious notation): (1) if Tβ is bounded on H 2 (β), Tγ is also bounded on H 2 (γ), and moreover Tγ  ≤ Tβ ; (2) if Tβ is compact, so is Tγ ; (3) with obvious notation the sequence s γ is log-subordinate to the sequence s β , so that: γ β β (a) s2n ≤ s1 sn , for all n ≥ 1; (b) Tβ ∈ S p (H 2 (β)) ⇒ Tγ ∈ S p (H 2 (γ)), for any p > 0. Proof Let us consider the non decreasing sequence

dk =

βk . γk

The spaces H 2 (β) and H 2 (γ) have respective orthonormal bases zk ek = √ , βk

zk f k = √ = dk ek . γk

Let D : H 2 (β) → H 2 (β) be the diagonal operator defined by: D(ek ) = dk ek . The key observation, obvious to check, is Tγ f j , f i γ = D −1 Tβ D(e j ), ei β .

(4.1)

This means that Tγ is unitarily equivalent to D −1 Tβ D, and we are back to the Katsnelson situation.

204

H. Queffélec

Remark This theorem applies perfectly well to a weighted composition operator T = Mw Cϕ when ϕ(0) = 0, and to the weighted Dirichlet spaces Dβ2 and Dγ2 where −1 < β < γ (then, βk γk−1 decreases).

5 “Strong” Points We just mention two examples. 1. The first example is the validity of the Chalendar-Partington comparison results for S p when 0 < p < 1. 2. The second example goes as follows: for ϕ a lens map, we proved that, on the Hardy space H 2 , we have [6]: √ an ≤ a exp(−b n). Recall that the lens map ϕ with parameter θ, 0 < θ < 1, is defined by ϕ(z) =

(1 + z)θ − (1 − z)θ . (1 + z)θ + (1 − z)θ

Our Theorem 4.1 easily gives us the following: Theorem 5.1 Denote respectively by sn and tn the approximation numbers of the on H 2 and B2 . Suppose that, for some constants a, b > 0, we have lens map λ = λθ√ sn ≤ a exp − b n . Then  2b √  n . tn ≤ a exp − 3

6 “Weak” Points We will content ourselves with one result [7], which shows that our general Theorem 4.1 misses some kind of “hypercontractivity”. Recall that H 2 = B2−1 and B2 = B20 . Theorem 6.1 Let p > 0, and ϕ a symbol. Suppose that Cϕ ∈ S p (H 2 ). Then 1. Cϕ ∈ S p/2 (B2 ). 2. The converse holds if ϕ is injective, or boundedly valent. / Sr (B2 ) as soon as r < 3. There are symbols ϕ for which Cϕ ∈ S p (H 2 ), but Cϕ ∈ p/2.

Maximum Principle and Comparison of Singular Numbers …

205

Proof We use a result of Luecking and Zhu [4]: let Nϕ, β , Mϕ, β (β ≥ 1) be the weighted and normalized weighted Nevanlinna counting function, defined as:  

Nϕ, β (w) =

ϕ(z)=w

1 β log , |z|

Mϕ, β (w) =

Nϕ, β (w) (log 1/|w|)β

if w ∈ ϕ(D) \ {ϕ(0)}, Nϕ, β (w) = Mϕ, β (w) = 0 otherwise; and by dν(w) = the hyperbolic (Möbius invariant) measure on D. Then, for γ ≥ −1:

(6.1)

d A(w) (1−|w|2 )2

Theorem 6.2 Cϕ ∈ S p (B2γ ) if and only if Mϕ,γ+2 ∈ L p (ν). 2 , and (Cauchy–Schwarz) Our result follows since Mϕ,2 ≤ Mϕ,1 2 Mϕ,1 ≤

√

s × Mϕ,2 if ϕ is at most s-valent.

And moreover, we were able ([5], Theorems  5.1, 5.4, and [7]) to produce injective symbols, say ϕ, such that Cϕ ∈ S p (H 2 ) \ r < p Sr (H 2 ). The application to Theorem  6.1 is clear: let ϕ be injective with Cϕ ∈ S p (H 2 ) \ r < p Sr (H 2 ). Then, we know that Cϕ ∈ S p/2 (B2 ). But if Cϕ ∈ Sr (B2 ) for some r < p/2, then Cϕ ∈ S2r (H 2 ) with 2r < p, which is a contradiction. 

7 Back to Lens and Cusps We could in [7], make a “homotopic completion of our table”. Spaces H = H 2 = D12 H H H H H

= H 2 = D12 = Dα2 , 0 < α ≤ 1 = Dα2 , 0 < α ≤ 1 = D2 = D02 = D2 = D02

Operators Cλ lens map

Singular numbers √ an (Cλ ) ≈ e−b n

Cχ Cλ Cχ Cλ Cχ

an (Cχ ) ≈ e log n √ √ an (Cλ )  exp(−b α n) an (Cχ )  exp(−bα logn n ) an (Cλ ) ≈ 1 √ an (Cχ ) ≈ e−b n

cusp map lens map cusp map lens map cusp map

−b

n

This is more satisfactory. When α decreases from 1 to 0, we see the decay rate degrading (sub)-linearly, with a drop when α = 0. Lens and cusps have really different decays, which must be considered separately, and the only coincidence seems to be an accident!

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H. Queffélec

References 1. I. Chalendar, J.R. Partington, Norm estimates for weighted composition operators on spaces of holomorphic functions. Complex Anal. Oper. Theory 8(5), 1087–1095 (2014) 2. I. Chalendar, J.R. Partington, Compactness and norm estimates for weighted composition operators on spaces of holomorphic functions, in Harmonic Analysis, Function Theory, Operator Theory, and Their Applications, Theta Ser. Adv. Math. vol. 19 (Theta, Bucharest, 2017), pp. 81–89 3. V.E. Katsnelson, A remark on canonical factorization in certain spaces of analytic functions(Russian), in Investigations on Linear Operators and the Theory of Functions III, ed. by N.K. Nikolskii, Z. Naucn. Sem. Leningrad Otdel. Mat. Inst. Steklov (LOMI) 30 (1972), pp. 163–164. Translation: J. Soviet Math. 4 (1975), no. 2 (1976), 444–445 4. D.H. Luecking, K.H. Zhu, Composition operators belonging to the Schatten ideals. Am. J. Math. 114(2), 1127–1145 (1992) 5. P. Lefèvre, D. Li, H. Queffélec, L. Rodríguez-Piazza, Some examples of compact composition operators on H 2 , J. Funct. Anal. 255, no. 11 (2008), 3098–3124 6. P. Lefèvre, D. Li, H. Queffélec, L. Rodríguez-Piazza, Some new properties of composition operators associated with lens maps, Israel J. Math. 195, no. 2 (2013), 801–824 7. P. Lefèvre, D. Li, H. Queffélec, L. Rodríguez-Piazza, Comparison of singular numbers of composition operators on different Hilbert spaces of analytic functions. J. Funct. Anal. 280(3), 108834 (2021)

Canonical Systems in Classes of Compact Operators Roman Romanov and Harald Woracek

Abstract Spectral properties of two-dimensional canonical systems with locally integrable Hamiltonian are studied. We give a criterion of discreteness of the spectrum of the associated selfadjoint operator, and study asymptotic distribution of this spectrum in terms of symmetrically normed ideals of compact operators. Simultaneously, we answer a 1968 question of Louis de Branges on description of the Hamiltonians which are structure functions of some de Branges spaces. In this talk we present our results from [1] on the discrete spectrum of 2 × 2 canonical systems on an interval with one singular end. Let h 1 , h 2 , h 3 be real functions on the interval (0, b), b ≤ ∞, Lebesgue  summable  h h over any interval (0, c) with c < b. Let the matrix function H = h 1 h 3 be such that 3 2 the set {t ∈ [0, b) : H (t) = 0} has measure 0, and H (t) ≥ 0 for a. e. t ∈ [0, b). Such a function is called the Hamiltonian. Assume that  b trace H (t) dt = ∞. 0

Under a certain technical condition at t = 0, a selfadjoint operator A[H ] can be associated with the two-dimensional canonical system

R. Romanov (B) School of Mathematics and Computer Science, St. Petersburg State University, Universitetskaya nab 7-9, 199034, St. Petersburg, Russia e-mail: [email protected] H. Woracek Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8-10 101, 1040 Wien, Austria e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. Abakumov et al. (eds.), Extended Abstracts Fall 2019, Trends in Mathematics 12, https://doi.org/10.1007/978-3-030-74417-5_26

207

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R. Romanov and H. Woracek

y  (t) = z

  0 −1 H (t)y(t), t ∈ [0, b), 1 0

(1)

and the selfadjoint boundary condition (1, 0)y(0) = 0. We shall assume throughout that  b

h 1 (s) ds < ∞.

0

This assumption is a normalization and does not diminish the generality of the results. Our first result is a discreteness criterion for the spectrum of the operator A[H ] .   h h Theorem 1 Let H = h 1 h 3 be a Hamiltonian on [0, b). Then σ(A[H ] ) is discrete 3 2 if and only if  t  b  h 1 (s) ds · h 2 (s) ds = 0. (2) lim tb

t

0

For diagonal Hamiltonians (h 3 = 0) this result has been established by M. Krein as a discreteness criterion for the spectrum of an inhomogeneous string. In the framework of the de Branges theory of spaces of entire functions any entire function of the Hermite–Biehler class without real zeroes has a Hamiltonian as its structure function. In 1968 de Branges posed the question of characterization of the Hamiltonians which appear in this way. This question is equivalent to the problem of description of Hamiltonians with discrete spectrum solved by Theorem 1. Once it is known that the spectrum is discrete it is natural to ask about its asymptotic distribution. Our next result describes the asymptotics of the discrete spectrum in terms of the summability properties or pointwise estimates with respect to growth functions. Let us define a sequence cn , cn  b, by c0 = 0, 

cn

h 1 (t) dt = 2

−n



b

h 1 (t) dt,

0

cn−1

and a sequence (ωn )n∈N by  ωn :=

cn

cn−1

 h 1 (s)ds ·

cn

h 2 (s)ds, n ∈ N.

(3)

cn−1

Let (ωn∗ )n∈N be the nonincreasing rearrangement of the sequence ωn . Let {λn }, 0 ≤ |λ1 | ≤ |λ2 | ≤ . . . , be the eigenvalues of the operator A[H ] .   h h Theorem 2 Let H = h 1 h 3 be a Hamiltonian on the interval [0, b) such that A[H ] 3 2 has discrete spectrum. Moreover, assume that h 1 does not vanish a.e. on any interval (c, b) with c ∈ (a, b). Let g be a growth function with order ρg > 1. Then

Canonical Systems in Classes of Compact Operators

209

∞ 

∞  1 1  = 0. n→∞

The domain D f is the largest in the sense that if D is a domain with cap(D \ D f ) > 0, then no subsequence of [n/n] f (z) converges in capacity to f (z) everywhere in D. The complement  f := C \ D f has very special structure. Theorem 2 The set  f can be decomposed as  f = E 0 ∪ E 1 ∪  j , where E 0 ⊆ E f , E 1 consists of isolated points to which f (z) has unrestricted continuations from infinity leading to at least two distinct function elements, and  j are open analytic arcs. The set  f possesses the S-property

∂g D f (z, ∞) ∂g D f (z, ∞) = on j, ∂n+ ∂n−

(3)

where ∂/∂n± are the one-sided normal derivatives on  j . Define h D f (z) := ∂z g D f (z), 2∂z := ∂x − i∂ y . The function h 2D f (z) is holomorphic in C \ (E 0 ∪ E 1 ), has a zero of order 2 at infinity, and the arcs  j are orthogonal critical trajectories of the quadratic differential h 2D f (z)dz 2 . For each point e ∈ E 0 ∪ E 1 denote by i(e) the number of different arcs  j incident with e. If E f is finite, then h 2D f (z) =



(z − e)i(e)−2

e∈E 0 ∪E 1



(z − e)2 j (e) ,

(4)

e∈E 2

where E 2 is the set of critical points of g D f (z) with j (e) standing for the order of j j (e)+1 e ∈ E 2 , i.e., ∂z g D f (e) = 0 for j ∈ {1, . . . , j (e)} and ∂z g D f (e) = 0. Informally, Padé approximants converge to f (z) in the complement of a branch cut  f that can be characterized as the one of smallest logarithmic capacity or equivalently the one with the symmetry property (3). The latter property becomes essential in understanding the extension of Stahl’s work to multipoint Padé approximants. Given a positive finite Borel measure ω supported in a domain D, denote by  G D (z, ω) :=

g D (z, w)dω(w)

the Green potential of ω relative to D. The potential G D (z, ω) is superharmonic in D and, if extended by zero to C \ D, is subharmonic in some neighborhood of C \ D. Therefore, it possesses a distributional Laplacian in this neighborhood, i.e.,  ω := G D (·, ω)/(2π ), which is a positive Borel measure supported on ∂ D and is

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called the balayage of ω out of D. The measure  δ (∞), the balayage of the delta mass at infinity, is also known as the logarithmic equilibrium distribution of ∂ D as well as C \ D. Gonchar and Rakhmanov [3] have proposed the following generalization of the symmetry property (3) (more generally, they introduced a notion of a contour symmetric in an external field, which specializes to (5) in the case of multipoint Padé approximants). Let  be a system of finitely many Jordan arcs that does not separate the plane. Assume that almost every point of  belongs to an analytic subarc. It is said that  is symmetric with respect to a positive Borel measure ω supported in D := C \  (has the S-property with respect to ω) if ∂G D (z, ω) ∂G D (z, ω) = a.e. on . ∂n+ ∂n−

(5)

To make use of (5), one shall choose an interpolation scheme that is asymptotic to ω. More precisely, given , D, and ω as in (5), a function f (z) analytic in D, and an interpolation scheme V supported in D, that is, ∩n ∪k≥n Vk ⊂ D, it is said that V is asymptotic to ω if ∗

ωn → ω as n → ∞, ωn :=

2n 1  δ(vn,i ), 2n i=1 ∗

where δ(v) is the Dirac’s  delta distribution supported at v (as usual, ωn → ω as n → ∞ if φdωn → φdω as n → ∞ for every function φ continuous in D and supported on a closed subset of D). The following is an adaptation of [3, Lemmas 1 and 2] to the case of multipoint Padé approximants. Theorem 3 Let , D, and ω be as in (5). If a function f (z) is holomorphic in D and the jump of f (z) across  is non-zero almost everywhere, then the diagonal multipoint Padé approximants [n/n; Vn ] f (z) associated with an interpolation scheme V asymptotic to ω converge to f (z) in logarithmic capacity in D with the rate function exp{−2nG D (z, ω)} in the sense of (2). Moreover, the normalized counting measures ω. of the poles of [n/n; Vn ] f (z) converge weak∗ to  It should be stressed that the above theorem assumes existence of a symmetric contour while Stahl’s theorem proves it but in a very specific case.

3 Szeg˝o-Type Convergence If one wants to strengthen convergence in capacity to uniform convergence, the notion of symmetry needs to be refined. Let us start with a case of a single arc. Let √  be a rectifiable Jordan arc with endpoints ±1 oriented from −1 to 1. Set w(z) := z 2 − 1,

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w(z)/z → 1 as z → ∞, to be the branch holomorphic in C \ . Define (z) := z + w(z), z ∈ D := C \ , which is a non-vanishing univalent holomorphic function in D except for a simple pole at infinity. Observe that + (s)− (s) ≡ 1 for s ∈ . Let v ∈ D. Define (z, v) :=

1 (z) − (v) , |v| < ∞, and (z, ∞) := , z ∈ D. 1 − (v)(z) (z)

Each (z, v) is a holomorphic function in D with a simple zero at v and nonvanishing otherwise. Given an interpolation scheme V in D, let n (z) :=



(z, v), z ∈ D.

v∈Vn

It is holomorphic in D with 2n zeros there and its traces on  satisfy n+ (s) n− (s)≡1. The following definition has been proposed in [2] by Baratchart and the author. It is said that  is symmetric with respect to an interpolation scheme V if the functions n (z) satisfy |n± (s)| = O(1) uniformly on  and n (z) = o(1) locally uniformly in D as n → ∞. This notion has the following connection to (5), see [2, Theorem 1]. Theorem 4 Let  be a rectifiable Jordan arc such that for the endpoints x = ±1 and all s ∈  sufficiently close to x it holds that |s,x | ≤ c|x − s|β , β > 1/2, where |s,x | is the arclength of the subsarc of  joining s and x. Then the following are equivalent: (a) there exists V supported in D such that  is symmetric with respect to V; (b) there exists a probability Borel measure ω supported in D such that (5) holds; (c)  is an analytic Jordan arc, i.e., there exists a univalent function (z) holomorphic in some neighborhood of [−1, 1] such that  = ([−1, 1]). As expected, the proof shows that if  is symmetric with respect to V and ω is a weak∗ limit point of the normalized counting measures of the elements of Vn , then  is symmetric with respect to ω. Moreover, the following result holds, see [2, Theorem 4]. Theorem 5 Let  be an analytic Jordan arc connecting ±1 symmetric with respect to an interpolation scheme V. Let f (z) :=

1 2π i

 

ρ(s) ds , z ∈ D, s − z w+ (s)

(6)

where ρ(s) is a non-vanishing Dini-continuous complex-valued function on . Then f (z) − [n/n; Vn ] f (z) =

1 + o(1) 2 S (z)n (z) w(z) ρ

(7)

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locally uniformly in D, where Sρ (z) = exp of ρ.

M. L. Yattselev

 w(z)  2πi

log ρ(s) ds   w+ (s) s−z is the

Szeg˝o function

The above results can be generalized to more complicated geometries in the following way, see [10]. Let E = {e0 , . . . , e2g+1 } be a set of 2g + 2 distinct points in C and  (8) R := (z, w) : w 2 = (z − e0 ) · · · (z − e2g+1 ), z ∈ C be a hyperelliptic Riemann surface, necessarily of genus g. Define the natural projection π : R → C by π(z, w) = z and denote by E = {e0 , . . . , e2g+1 } the set of ramification points of R. We shall use bold lower case letters z, s, etc. to denote points on R with natural projections z, s, etc. We utilize the symbol ·∗ for the conformal involution on R, that is, z ∗ = (z, −w) if z = (z, w). Given v ∈ R \ E, denote by g(z, v) a function that is harmonic in R \ {v, v ∗ }, normalized so that g(e0 , v) = 0, and such that   log |z − v|, |v| < ∞, log |z − v|, |v| < ∞, g(z, v) + and g(z, v) − − log |z|, v = ∞, − log |z|, v = ∞, are harmonic functions of z around v and v ∗ , respectively. Such a function always exists as it is simply the real part of an integral of the third kind differential with poles at v and v ∗ that have residues −1 and 1, respectively, and whose periods are purely imaginary. Definition 6 Let  be a system of open analytic arcs together with the set E of their 2n endpoints and V, Vn = {vn,i }i=1 , be an interpolation scheme in D := C \ . We say that  is symmetric with respect to (R, V) if (i) R \ ,  := π −1 (), consists of two disjoint domains, say D (0) and D (1) , and no closed  proper subset of  has this property;  (0)  2n are uniformly bounded above and below on  and g z, vn,i (ii) the sums i=0 converge to −∞ locally uniformly in D (1) , where z (i) := π −1 (z) ∩ D (i) , z ∈ D. The first condition above says that  does not separate the plane and serves as a branch cut for w(z) from (8) (w(z)/z g+1 → 1 as z → ∞), which has a non-zero jump across every subarc of . The second one is essentially a non-Hermitian Blaschketype condition. It is in fact true that if  is symmetric with respect to (R, V) and V is asymptotic to a measure ω, then  is symmetric with respect to ω. The following can be said about the existence of such contours, see [10, Theorem 3.2]. Theorem 7 Given R as in (8) and v ∈ C \ E, there always exists a contour v symmetric with respect to (R, Vv ), where Vv consists of sets containing only the point v. Further, let c > 0 be a constant such that L c := {s : g Dv (s, ∞) = c} is a smooth Jordan curve, where Dv := C \ v . If (z) is a univalent function in the interior of L c such that (e) = e for every e ∈ E, then there exists an interpolation scheme V in C \ () such that () is symmetric with respect to (R, V).

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The right-hand side of (7) is obviously defined explicitly, but also can be characterized as a function with a certain zero/pole divisor that solves a particular boundary value problem. We take this second approach to define functions describing the asymptotic behavior of the multipoint Padé approximants for more general contours . Proposition 8 Let  be as in Definition 6 and ρ(s) be a Lipschitz continuous and non-vanishing function on . There exists a sectionally meromorphic in R \  function n (z) whose zeros and poles there are described by the divisor1 (n − g)∞(1) + z n,1 + · · · + z n,g − n∞(0)

(9)

for some set of g points z n,i on R, and whose traces on  are continuous and satisfy   n− (s) = ρ(s)/vn (s) n+ (s), s ∈ ,

(10)

where  is oriented so that D (0) lies to the left of . If two functions (z), ∗ (z) satisfy (9) and (10), then (z)/∗ (z) = R(π(z)) for some rational function R(z) g with at most g/2 poles. In particular, if the set {z n,i }i=1 does not contain involutionsymmetric pairs (z n,i = z ∗n, j for some i = j), then n (z) is unique up to a multiplicative constant. g

The sets {z n,i }i=1 can be independently introduced as solutions of a certain explicitly defined Jacobi inversion problem. The asymptotics of Padé approximants now can be describes as follows [10, Theorem 3.7]. Theorem 9 Given R as in (8), let  be symmetric with respect to (R, V) for some interpolation scheme V supported in D = C \  and f (z) be given by (6), where ρ(s) is a non-vanishing Lipschitz continuous function on . Further, let N∗ ⊆ N be g a subsequence such that the sets {z n,i }i=1 as well as their topological limit points in g g R \ g , the quotient of R by the symmetric group g , contain neither involutionsymmetric pairs nor ∞(0) . Then     vn (z) n z (1) 1 + εn1 (z) + εn2 (z)ϒn z (1)     f (z) − [n/n; Vn ] f (z) = w(z) n z (0) 1 + εn1 (z) + εn2 (z)ϒn z (0) for n ∈ N∗ , where εni (z) = o(1) locally uniformly in D and vanish at infinity and ϒn (z) is a rational function on R that vanishes at ∞(0) and whose divisor of poles is equal to z n,1 + · · · + z n,g + ∞(1) . Moreover, it holds that



 2n 

v (z)  z (1) 

   n

n (0) (1)   ≤ C K exp g z , vn,i = o(1)

w(z) n z (0)

i=1   A meromorphic function (z) has a zero/pole divisor i m i x i − n i yi if (z) has a zero of order m i at x i , a pole of order n i at yi , and otherwise is non-vanishing and finite.

1

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for every closed subset K ⊂ D, where the last estimate follows from Definition 6(ii). It needs to be stressed that this theorem is conditional. Even though Theorem 7 describes some symmetric contours, it is by no means comprehensive. More importantly, Theorem 9 is conditional on existence of a sequence N∗ . Existence of such a sequence is known only for the case of classical Padé approximants, i.e., V = V∞ , see [1, Propositions 2.2 and 2.5] or [10, Theorem 3.6].

References 1. A.I. Aptekarev, M. Yattselev, Padé approximants for functions with branch points – strong asymptotics of Nuttall-Stahl polynomials. Acta Math. 215(2), 217–280 (2015) 2. L. Baratchart, M. Yattselev, Convergent interpolation to Cauchy integrals over analytic arcs. Found. Comput. Math. 9(6), 675–715 (2009) 3. A.A. Gonchar, E.A. Rakhmanov, Equilibrium distributions and the degree of rational approximation of analytic functions. Mat. Sb. 134(176)(3), 306–352 (1987). English transl. in Math. USSR Sbornik 62(2):305–348, 1989 4. T. Ransford, Potential Theory in the Complex Plane. London Mathematical Society Student Texts, vol. 28. (Cambridge University Press, Cambridge, 1995) 5. E.B. Saff, V. Totik, Logarithmic Potentials with External Fields. Grundlehren der Math, vol. 316. (Wissenschaften. Springer, Berlin, 1997) 6. H. Stahl, Extremal domains associated with an analytic function. I, II. Complex Variables Theory Appl. 4, 311–324, 325–338 (1985) 7. H. Stahl, Structure of extremal domains associated with an analytic function. Complex Var. Theory Appl. 4, 339–356 (1985) 8. H. Stahl, Orthogonal polynomials with complex valued weight function. I, II. Constr. Approx. 2(3), 225–240, 241–251 (1986) 9. H. Stahl, The convergence of Padé approximants to functions with branch points. J. Approx. Theory 91, 139–204 (1997) 10. M.L. Yattselev, Symmetric contours and convergent interpolation. J. Approx. Theory 1225, 76–105 (2018)

Special Conformal Mappings and Extremal Problems P. Yuditskii

Abstract In this talk we demonstrate two recent applications of the special conformal mappings on comb domains in getting exact constants and asymptotics in the approximation theory.

1 Kharkov’s Edition of the Classical Chebyshev Theorem Already Chebyshev himself reduced many problems on extremal polynomials to certain functional equations recalling Pell’s equation from number theory. Akhiezer added to this method the apparatus of geometric theory of functions of a complex variable. A quite detailed presentation of these ideas can be found in our review [4]. By “Kharkov’s edition” of the Chebyshev theorem we understand its interpretation by means of special representations of the Chebyshev extremal polynomials. Namely, the structure of Chebyshev polynomials is revealed by their representation via conformal mappings on so called n-regular comb domains. Definition 1 By n-regular comb we mean a half strip with a system of slits  = {z : Re z ∈ π(n − , n + ), Im z > 0} \ {z = πk + i y : y ∈ (0, h k )} where k ∈ Z, k ∈ (n − , n + ), and n = n + − n − . We call h k , h k ≥ 0, the hight of the kth slit, with a possible degeneration h j = 0 for some j. Let θ : C+ →  be a conformal mapping from the upper half plane on an nregular comb such that θ(∞) = ∞. Then Tn (z) = cos θ(z) is a polynomial of degree n. Indeed, the function is well defined in the upper half plane and it is real on the real axis. Therefore it can be extended in the lower half plane P. Yuditskii (B) Abteilung fur Dynamische Systeme und Approximationstheorie, Institut fur Analysis, Johannes Kepler Universitat Linz, 4040 Linz, Austria e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. Abakumov et al. (eds.), Extended Abstracts Fall 2019, Trends in Mathematics 12, https://doi.org/10.1007/978-3-030-74417-5_28

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by the symmetry principle. Thus, Tn (z) is an entire function. From the asymptotic at infinity we get that it is a polynomial of degree n. Assume that h 0 > 0 and let us fix uniquely this conformal mapping by two extra conditions θ(a) = −0, θ(b) = +0, a, b ∈ R. Theorem 1 Let E be a closed subset of R, (a, b) ⊂ R \ E. Assume that E contains the following collection of points: (i) both end points a and b, (ii) at least one of two possibly different points θ−1 (πk ± 0) for all other k ∈ (n − , n + ), (iii) one of two points b0 = θ−1 (πn − ) or a0 = θ−1 (πn + ). Assume that Pn is a polynomial of degree at most n. Then |Pn (x)| ≤ 1 for x ∈ E implies |Pn (x)| ≤ Tn (x), ∀x ∈ (a, b). Vice versa, if (a, b) is a gap in a compact E ⊂ R (containing at least n + 1 points) and Tn (x0 ) = sup{|Pn (x0 )|, |Pn (x)| ≤ 1, x ∈ E}, x0 ∈ (a, b), then Tn (x) = cos θ(z) with a suitable comb . Remark 1 The following degenerations are allowed h n − +1 = ∞, or alternatively, h n + −1 = ∞. In both cases the extremal polynomial appears to be of degree n − 1. For a proof of the theorem in a general case of entire functions of a fixed exponential type, see [4, Basic Theorem].

2 Asymptotics of Chebyshev Polynomials on Cantor Sets The following problem was solved in [1]. Problem 1 Let E be a Cantor type set of a positive Lebesgue measure. Find asymptotics of Chebyshev polynomials Tn (z) = Tn (z, E, ∞) = Tn (z, E, x0 ), x0 ∈ R \ [b0 , a0 ], as n → ∞. By a Cantor type set of a positive measure we mean the set, which remains in the standard iterative construction. We start with the interval [−1, 1]. On the nth step from each of 2n−1 intervals we remove  (from the middle) the κn th part. So that the remaining set has the length |E| = 2 n≥1 (1 − κn ) > 0. To comment our solution we introduce the notion of a general comb. We normalize the base of the half strip to the interval [0, π]. Assume that it contains a countably many slits with bases ωk and heights h k , moreover limk→∞ h k = 0.

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Proposition 1 Let θ be a conformal mapping on a regular comb, θ(∞) = ∞, θ(a0 ) = π, θ(b0 ) = 0. Let  = C \ E, E = θ−1 ([0, π]). Then  is regular in the sense of potential theory. Moreover, if G(z) = G(z, ∞) is the Green function of this domain with respect to infinity, then Im θ(z) = G(z) in the upper half plane. For this reason (z) = eiθ(z) is called the complex Green function. It has a simple zero at infinity. Note that this is a multivalued function in  with a single valued modulus, |(z)| = e−G(z) . Let π1 () be the fundamental group of  with the standard generators γk ; the contour γk is a closed loop going through the gap (ak , bk ) ⊂ R \ E and the negative half axis below b0 . Then the analytic extension of the function (z) along γk possesses the property (γk (z)) = e2iωk (z), z ∈ , where the gap (ak , bk ) corresponds to the slit in the comb with the base ωk . By π1 ()∗ we denote the group of characters: α ∈ π1 ()∗ if α(γ) ∈ R/Z and α(γγ ) = α(γ) + α(γ ) for γ, γ ∈ π1 ().

Definition 2 Let F(z) be analytic multivalued function in , s.t. F(γ(z)) = e2πiα(γ) F(z), ∀γ ∈ π1 (). We say that F is character automorphic with the character α. A complex Green function z0 (z), |z0 (z)| = e−G(z,z0 ) , can be associated with an arbitrary point z 0 ∈ . These functions are character automorphic. We will be particularly interested in complex Green functions related to points x j in the gaps (a j , b j ). Note that such functions are also related to conformal mappings θx j on comb domains with the condition θx j (x j ) = ∞, x j (z) = eiθx j (z) , x j (∞) > 0. The characters generated by these functions are denoted by βx j , x j (γ(z)) = e2πiβx j (γ) x j (z). In particular, we denote β := β∞ . Our main result is based on the following modification of the classical AbelJacobi inversion problem. Each closed gap [a j , b j ] is endowed with the topology of a circle; we identify the end points in this case, a j ≡ b j . By X (E) we denote the direct product of these circles with the standard product topology, X (E) = {X = {x j } j≥1 : x j ∈ [a j , b j ], a j ≡ b j }. The modified Abel map A : X (E) → π1 ()∗ is given by

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A(X ) =



βx j .

j≥1

 In other words, A(X ) is the character of the infinite product j≥1 x j (z). Note that this definition agrees with the identification of the end points of gaps since lim x j →a j x j (z) = lim x j →b j x j (z) = 1, z ∈ . Theorem 2 Let E be a Cantor type set s.t. |E| > 0. Then A is a well defined homeomorphism. Our result generalizes the Widom construction [6]. Theorem 3 Let E be a Cantor type set, |E| > 0. Let Tn (z) be the Chebyshev polynomials associated to E. Then (on compact subsets in ) n (z)Tn (z) −

1 x (n) (z) → 0, n → ∞, 2 j≥1 j

(1)

where X n = {x j (n)} = A−1 (nβ). We illustrate an essential hint in the direction of proving of this theorem by Figs. 1 and 2. In Fig. 1 we sketch the graph of the nth Chebyshev polynomial. Let E n = Tn−1 ([−1, 1]) ⊃ E. Comparably to E, all initial gaps (except for a finite number of them) are closed in E n . However, some of them are only partially closed. The typical example is shown in Fig. 1 as the gap (a j , b j ) ⊂ R \ E. This gap contains an additional interval I j = I j (n) ⊂ E n . The comb n associated to Tn , according to Theorem 1, is shown as the left comb in Fig. 2. Recall that it is enough that the preimage of only one of base points of the slit belongs to the set E, see case (ii). Thus (2) the preimage of the gap (a j , b j ) in the shown case consists of two slits S (1) j , S j , and an interval on the base (of the length π) between them. The last one is the preimage of I j (n) for θn : n → C+ . Now we normalize such combs, i.e., we consider the sequence of domains n /n and using Caratheodory theorem consider all convergent subsequences. The limit comb

Fig. 1 Chebyshev polynomial Tn (z) = cos θn (z)

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Fig. 2 Comb n and the limit comb  = limn k →∞ n k /n k

({n k }) = lim

k→∞

1 n nk k

is already associated to the initial set E. Therefore the corresponding conformal mapping is related to the Green function of the domain  = C \ E. Thus, the limit does not depend on the subsequence, and we get ({n k }) = . Practically, in this way we get the first term in asymptotics (1), which can be rewritten into the form  1 θx j (n) (z) + o(1), Im z > 0. log 2Tn (z) = −nθ(z) + i j≥1 The second term in this asymptotic deals with a much more delicate problem. Note that it represents an almost periodic sequence in n, i.e., the limit  lim

n→∞



1 log 2Tn (z) + nθ(z) i

(2)

does not exists: it is already different along certain subsequences. Nevertheless, looking at Fig. 2, we can explain appearance of different limits in terms of the collections {x j } j≥1 as the limit points in convergent subsequences limk→∞ x j (n k ) = x j . Indeed, in the shown example, for a fixed degree n, we associate to a given gap (2) (a j , b j ) two slits, say S (1) j (n) and S j (n), which can be different in size. Recall that after normalization, the distance between their bases is π/n. Thus, in the limit procedure (along a certain subsequence) such couples of slits would form a unique slit in the comb . However, certain parts of the limit points are the limit points of the different slits in the couples. In Fig. 2 (right) these points are separated by the point denoted as (x j ): the left edge of the slit above (x j ) and its right edge are produced by the limit points of the slits S (2) j (n k ), k → ∞ (in the Caratheodory sense). In other words, in terms of z-variable for a fixed n k we get an additional interval I j (n k ) in the gap (a j , b j ). In the limit procedure these additional intervals shrink to a point, I j (n k ) → x j ∈ (a j , b j ) as k → ∞, and (x j ) denotes the image of this point in θ-plane.

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To summarize, we get different limits along different subsequences in (2). These different subsequences {n k } are marked by different limit characters α = α({n k }) = lim n k β k→∞

so that α = A(X ),

X = {x j }, x j = lim x j (n k ). k→∞

3 Remez Problem for Trigonometric Polynomials Another application was given recently in [5]. Theorem 4 Let an algebraic polynomial Pn (ζ) of degree n be such that |Pn (ζ)| ≤ 1 for ζ ∈ E ⊂ T and |E| ≥ 2π − s. Then  s , sup |Pn (ζ)| ≤ Tn sec 4 ζ∈T where Tn is the classical Chebyshev polynomial of degree n. The equality holds if and only if   s z − c0 , c0 , c1 ∈ R. Pn (ei z ) = ei(nz/2+c1 ) Tn sec cos 4 2 Note that such an estimate was conjectured by Erdélyi [2] for trigonometric polynomials (this corresponds to even n in our theorem). The main ingredient in the proof is the representation of the extremal polynomial by means of the conformal mapping θ(z) on an n-regular periodic comb n n n Tn (ζ) = e 2 i z cos θ(z), ζ = ei z , 2 where

g

n = C+ \ ∪k=0 ∪m∈Z {z = ωk + 2πm + i y (0 < y ≤ h k )} and ωk = 2π njk , 0 ≤ jk < n. Due to this representation the problem is reduced to the standard principle of monotonicity of harmonic measure [3, Chap. 3]. Acknowledgements Supported by the Austrian Science Fund FWF, project no: P29363-N32.

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References 1. J.S. Christiansen, B. Simon, P. Yuditskii, M. Zinchenko, Asymptotics of Chebyshev polynomials, II. DCT subsets of R. Duke Math. J. 168(2), 325–349 (2019) 2. T. Erdélyi, Remez-type inequalities and their applications. J. Comput. Appl. Math. 47, 167–210 (1993) 3. R. Nevanlinna, Analytic Functions (Springer, Berlin, 1970) 4. M. Sodin, P. Yuditskii, Functions that deviate least from zero on closed subsets of the real axis, Algebra i Analiz 4, 1–61; English translation in St. Petersburg. Math. J. 4(1993), 201–249 (1992) 5. S. Tikhonov, P. Yuditskii, Sharp Remez inequality. Constr. Approx. (2019). https://doi.org/10. 1007/s00365-019-09473-2 6. H. Widom, Extremal polynomials associated with a system of curves in the complex plane. Adv. Math. 3, 127–232 (1969)