Sampling, Approximation, and Signal Analysis: Harmonic Analysis in the Spirit of J. Rowland Higgins (Applied and Numerical Harmonic Analysis) 3031411293, 9783031411298

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Applied and Numerical Harmonic Analysis

Stephen D. Casey M. Maurice Dodson Paulo J. S. G. Ferreira Ahmed Zayed Editors

Sampling, Approximation, and Signal Analysis Harmonic Analysis in the Spirit of J. Rowland Higgins

Applied and Numerical Harmonic Analysis Series Editors John J. Benedetto, University of Maryland, College Park, MD, USA Wojciech Czaja, Mathematics, University of Maryland, College Park, College Park, MD, USA Kasso Okoudjou, Dept of Mathematics, Tufts University, Medford, MA, USA Editorial Board Members Akram Aldroubi, Vanderbilt University, Nashville, TN, USA Peter Casazza, Math Department, University of Missouri, Columbia, USA Douglas Cochran, Arizona State University, Phoenix, AZ, USA Hans G. Feichtinger, University of Vienna, Vienna, Austria Anna C. Gilbert, Dept of Statistics and Data Science, Yale University, New Haven, CT, USA Christopher Heil, Georgia Institute of Technology, Atlanta, GA, USA Stéphane Jaffard, University of Paris XII, Paris, France Gitta Kutyniok, Ludwig Maximilian University of Munich, München, Bayern, Germany Mauro Maggioni, Johns Hopkins University, Baltimore, MD, USA Ursula Molter, University of Buenos Aires, Buenos Aires, Argentina Zuowei Shen, National University of Singapore, Singapore, Singapore Thomas Strohmer, University of California, Davis, CA, USA Michael Unser, Laboratoire d’imagerie biomédicale, École Polytechnique Fédérale de Lausa, Lausanne, Switzerland Yang Wang, Hong Kong University of Science & Technology, Kowloon, Hong Kong

Applied and Numerical Harmonic Analysis (ANHA) publishes works ranging from abstract harmonic analysis to engineering and scientific subjects having significant applicable harmonic analysis components. The interface between mathematics, science, and engineering is the overriding theme. Harmonic analysis is a wellspring of ideas and applicability in mathematics, engineering, and the sciences that has flourished, evolved, and deepened with continued research and exploration. The ANHA series reflects the intricate and fundamental relationship between harmonic analysis and general disciplines such as signal and image processing, partial differential equations, machine learning, and data science. This series provides a means of disseminating important, current information along with computational tools for harmonic analysis. The following topics are covered: * Analytic Number theory * Antenna Theory * Artificial Intelligence * Biomedical Signal Processing * Classical Fourier Analysis * Coding Theory * Communications Theory * Compressed Sensing * Crystallography and Quasi-Crystals * Data Mining * Data Science * Deep Learning * Digital Signal Processing * Dimension Reduction and Classification * Fast Algorithms * Frame Theory and Applications * Gabor Theory and Applications * Geophysics * Image Processing * Machine Learning * Manifold Learning * Numerical Partial Differential Equations * Neural Networks * Phaseless Reconstruction * Prediction Theory * Quantum Information Theory * Radar Applications * Sampling Theory (Uniform and Non-uniform) and Applications * Spectral Estimation * Speech Processing * Statistical Signal Processing * Super-resolution * Time Series * Time-Frequency and Time-Scale Analysis * Tomography * Turbulence * Uncertainty Principles * Waveform design * Wavelet Theory and Applications. The series includes professional monographs, advanced textbooks, and cohesive and carefully edited contributed works.

Stephen D. Casey • M. Maurice Dodson • Paulo J. S. G. Ferreira • Ahmed Zayed Editors

Sampling, Approximation, and Signal Analysis Harmonic Analysis in the Spirit of J. Rowland Higgins

Editors Stephen D. Casey Department of Mathematics and Statistics American University Washington, DC, USA

M. Maurice Dodson Department of Mathematics University of York York, UK

Paulo J. S. G. Ferreira IEETA University of Aveiro Aveiro, Portugal

Ahmed Zayed Department of Mathematical Sciences DePaul University Chicago, IL, USA

ISSN 2296-5009 ISSN 2296-5017 (electronic) Applied and Numerical Harmonic Analysis ISBN 978-3-031-41129-8 ISBN 978-3-031-41130-4 (eBook) https://doi.org/10.1007/978-3-031-41130-4 Mathematics Subject Classification: 94A20, 41-XX, 42-XX, 43-XX © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 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 Paper in this product is recyclable.

ANHA Series Preface

The Applied and Numerical Harmonic Analysis (ANHA) book series aims to provide the engineering, mathematical, and scientific communities with significant developments in harmonic analysis, ranging from abstract harmonic analysis to basic applications. The title of the series reflects the importance of applications and numerical implementation, but richness and relevance of applications and implementation depend fundamentally on the structure and depth of theoretical underpinnings. Thus, from our point of view, the interleaving of theory and applications and their creative symbiotic evolution is axiomatic. Harmonic analysis is a wellspring of ideas and applicability that has flourished, developed, and deepened over time within many disciplines and by means of creative cross-fertilization with diverse areas. The intricate and fundamental relationship between harmonic analysis and fields such as signal processing, partial differential equations (PDEs), and image processing is reflected in our state-of-theart ANHA series. Our vision of modern harmonic analysis includes a broad array of mathematical areas, e.g., wavelet theory, Banach algebras, classical Fourier analysis, timefrequency analysis, deep learning, and fractal geometry, as well as the diverse topics that impinge on them. For example, wavelet theory can be considered an appropriate tool to deal with some basic problems in digital signal processing, speech and image processing, geophysics, pattern recognition, biomedical engineering, and turbulence. These areas implement the latest technology from sampling methods on surfaces to fast algorithms and computer vision methods. The underlying mathematics of wavelet theory depends not only on classical Fourier analysis, but also on ideas from abstract harmonic analysis, including von Neumann algebras and the affine group. This leads to a study of the Heisenberg group and its relationship to Gabor systems, and of the metaplectic group for a meaningful interaction of signal decomposition methods. The unifying influence of wavelet theory in the aforementioned topics illustrates the justification for providing a means for centralizing and disseminating information from the broader, but still focused, area of harmonic analysis. This will be a key

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role of ANHA. We intend to publish with the scope and interaction that such a host of issues demands. Along with our commitment to publish mathematically significant works at the frontiers of harmonic analysis, we have a comparably strong commitment to publish major advances in the following applicable topics in which harmonic analysis plays a substantial role: *Analytic Number Theory * Antenna Theory * Artificial Intelligence * Biomedical Signal Processing * Classical Fourier Analysis * Coding Theory * Communications Theory * Compressed Sensing * Crystallography and Quasi-Crystals * Data Mining * Data Science * Deep Learning * Digital Signal Processing * Dimension Reduction and Classification * Fast Algorithms * Frame Theory and Applications * Gabor Theory and Applications * Geophysics * Image Processing * Machine Learning * Manifold Learning * Numerical Partial Differential Equations * Neural Networks * Phaseless Reconstruction * Prediction Theory * Quantum Information Theory * Radar Applications * Sampling Theory (Uniform and Non-uniform) and Applications * Spectral Estimation * Speech Processing * Statistical Signal Processing * Super-resolution * Time Series * Time-Frequency and Time-Scale Analysis * Tomography * Turbulence * Uncertainty Principles *Waveform design * Wavelet Theory and Applications The above point of view for the ANHA book series is inspired by the history of Fourier analysis itself, whose tentacles reach into so many fields. In the last two centuries Fourier analysis has had a major impact on the development of mathematics, on the understanding of many engineering and scientific phenomena, and on the solution of some of the most important problems in mathematics and the sciences. Historically, Fourier series were developed in the analysis of some of the classical PDEs of mathematical physics; these series were used to solve such equations. In order to understand Fourier series and the kinds of solutions they could represent, some of the most basic notions of analysis were defined, e.g., the concept of “function." Since the coefficients of Fourier series are integrals, it is no surprise that Riemann integrals were conceived to deal with uniqueness properties of trigonometric series. Cantor’s set theory was also developed because of such uniqueness questions. A basic problem in Fourier analysis is to show how complicated phenomena, such as sound waves, can be described in terms of elementary harmonics. There are two aspects of this problem: first, to find, or even define properly, the harmonics or spectrum of a given phenomenon, e.g., the spectroscopy problem in optics; second, to determine which phenomena can be constructed from given classes of harmonics, as done, for example, by the mechanical synthesizers in tidal analysis. Fourier analysis is also the natural setting for many other problems in engineering, mathematics, and the sciences. For example, Wiener’s Tauberian theorem in Fourier analysis not only characterizes the behavior of the prime numbers, but is a fundamental tool for analyzing the ideal structures of Banach algebras. It also provides the proper notion of spectrum for phenomena such as white light. This latter process leads to the Fourier analysis associated with correlation functions in

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filtering and prediction problems. These problems, in turn, deal naturally with Hardy spaces in complex analysis, as well as inspiring Wiener to consider communications engineering in terms of feedback and stability, his cybernetics. This latter theory develops concepts to understand complex systems such as learning and cognition and neural networks; and it is arguably a precursor of deep learning and its spectacular interactions with data science and AI. Nowadays, some of the theory of PDEs has given way to the study of Fourier integral operators. Problems in antenna theory are studied in terms of unimodular trigonometric polynomials. Applications of Fourier analysis abound in signal processing, whether with the fast Fourier transform (FFT), or filter design, or the adaptive modeling inherent in time-frequency-scale methods such as wavelet theory. The coherent states of mathematical physics are translated and modulated Fourier transforms, and these are used, in conjunction with the uncertainty principle, for dealing with signal reconstruction in communications theory. We are back to the raison d’etre of the ANHA series! College Park, MD, USA Boston, MA, USA

John Benedetto Wojciech Czaja Kasso Okoudjou

John Rowland Higgins

Biography John Rowland Higgins was born on 17 August 1935 in Hitchin, a market town 40 miles north of London. During the Battle of Britain in 1940, bombers and fighters frequently flew overhead to or from London. By the age of 5, Rowland, always known by his second name, far from being frightened, could identify the planes and would stand in the garden calmly naming them. He had also become interested in bird watching; his interest in flying, be it of planes or birds, remained with him all his life. His father, who had volunteered for the Royal Navy, died in 1940 and life became harder for the family. Rowland and his older sister Nancy both attended a local infants school. When 8, Rowland left and went to a nearby boys preparatory boarding school. From there, at 13, he went on to Aldenham, an independent secondary boarding school for boys. After passing School Certificate (the forerunner of the current General Certificate of Secondary Education), Rowland left school, having just turned 16, and following the family tradition and his interest in planes, joined the Fleet Air Arm, the naval arm of the Royal Air Force, to be a pilot. He completed his cadet training for the Royal Navy and then did his flying training in the United States. However, after four years in the Navy, Rowland made a remarkable career change. He decided that mathematics was more worthwhile than flying jet fighters off and onto aircraft carriers and, with characteristic determination, left the Fleet Air Arm and began a completely different career in mathematics by preparing for university as a mature student at Sir John Cass College, which was associated with the University of London. Having qualified for university entrance, he went on to graduate in 1961 with a BSc in Mathematics from the University of London. The next step was graduate school in the United States, with a Teaching Assistantship at the University of Minnesota Institute of Technology at Minneapolis while working for a master’s degree. He stayed on there to study for a PhD in Mathematics. He showed independence in choosing his topic and was described as “going against the flow.” Characteristically, he continued in the direction he had chosen, was successful, and xi

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was awarded a PhD in 1970. While at the University he met Nancy Gustafson, a librarian at the University Library. They were married in Minneapolis, Minnesota and moved to England in 1970. By now in his mid-30s, Rowland took up an academic post in the Mathematics and Statistics Division at the Cambridgeshire College of Arts and Technology. He and Nan bought a house with a large vegetable garden in Bourn, a village outside Cambridge, where they brought up their two daughters. At that time, the College provided technical and academic qualifications for professions and universities. For Rowland it had the huge advantage of being in the same city as the Cambridge University Mathematics Department and the famous University Library, an invaluable facility for academic and scholarly research. Through his research, Rowland made and maintained contact with mathematicians and engineers in the UK and abroad, often travelling with Nan when visiting them. In 1995 he and Nan went to an international conference/workshop on Signal Theory at the University of Riga, where the participants included Professor P. L. Butzer, Professor H. Feichtinger and Professor A. J. Jerri. This was a seminal meeting, as the participants appreciated the important role of sampling both in signal processing and in mathematics, with the prospect of exciting advances arising from the overlap between mathematical and engineering research. The idea of regular international conference on sampling theory and its applications for mathematicians and engineers with interdisciplinary interests to be held all over the world, combined with establishing an international journal, was clearly desirable, so the Sampling Theory and Applications (SAMPTA) Committee was established to organize them. This was a very successful venture, starting from a relatively small workshop/conference to the present fully-fledged biennial conferences held all over the world and attracting an unusually even mix of mathematicians and engineers as well as publishing an international journal. Friendly, modest and approachable, an advocate of collegiality and collaboration, with sound judgement and an encyclopaedic knowledge of sampling that he was always happy to share, Rowland played an active part in the development of SAMPTA from its inception at Riga. From serving on the original Steering Committee of the SAMPTA meetings, he became a valued mainstay of the SAMPTA Community, encouraging and advising younger members and, drawing on his extensive mathematical knowledge, pointing out new interconnections in the theory. His university was pleased by the international recognition of his research and had provided him with considerable financial support. The latter half of the twentieth century was a period of university expansion in the UK and in 1992 the College merged with other regional academic institutions to form part of the Anglia Polytechnic University, which in 2005 was renamed Anglia Ruskin University. However, in this process, Mathematics was absorbed into Computing and Statistics, an all too common fate for small departments. After nearly 30 years of service, Rowland felt it time to retire and left in 1999 as a Professor Emeritus, in recognition of his outstanding contribution to mathematics. On 31 December 1999, Rowland, who was also a bell-ringer at the Bourn village church, rang in the new Millennium until 3 am, 1 January 2000. On the first day

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of the new Millennium, he and Nan drove to France, moving to the village of Montclar, near Carcassonne, for a last adventure and lived happily there for 20 years. They enjoyed living in France, exploring the Languedoc and trying the local cuisine and wines. They renovated their house and organised the garden, which had been part of an old vineyard. Hospitable and welcoming company, they had many visits from family and friends. In addition, Rowland, who had remained a keen birdwatcher, continued his birdwatching in the surrounding countryside, often with Paulo Ferreira, a SAMPTA friend from Portugal. Rowland never lost his early interest in aviation and for his 70th birthday was taken for a spin over Carcassonne in a small 2-seater, briefly taking over the controls. Mathematically, he had a fruitful two decades of remarkably varied and collaborative research, from meticulously tracing the sometimes obscure origins of key developments in sampling (including an account of interpolation by Babylonian mathematicians to solve a compound interest problem 4000 years ago!) to important advances in the current theory. He continued to do mathematics, producing a paper almost annually, attending conferences and lecturing and maintaining his close links with SAMPTA. They returned to Cambridge early in 2020 to be closer to their children. Sadly Rowland died on 24 February 2020, leaving his widow Nan, two daughters Jenny and Emily, each with a grandson, Solomon and Joe, respectively, and his sister Mrs. Nancy Hughes. We are grateful to Mrs. Nan Higgins and Mrs. Nancy Hughes for information and help with preparing this brief account of his life.

Publications Higgins’ paper Five short stories about the cardinal series, Bulletin of the AMS, Volume 12, Number 1, pp. 45–89 (1985) is one of the most referenced articles about Sampling Theory. It is “required reading” for all in the field, and has been frequently passed from teacher to student. The paper begins with a short table of contents. CONTENTS INTRODUCTION STORY ONE. HISTORICAL NOTES STORY TWO. SOME METHODS FOR DERIVING THE CARDINAL AND ALLIED SERIES

STORY THREE. .L2 AND .Lp THEORY STORY FOUR. THE CARDINAL SERIES AND LCA GROUPS STORY FIVE. EXTENSIONS TO HIGHER DIMENSIONS CONCLUSION

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Many of Higgins’ papers, appropriately, fit into the categories established by these “short stories.” • An interpolation series associated with the Bessel-Hankel transform. J. London Math. Soc., 5, (2), 707–714, (1972). • A sampling theorem for irregularly spaced sample points. IEEE Trans. Inform. Theory, IT- 22, (5), 621–622, (1976). • Completeness and basis properties of sets of special functions. Cambridge Tracts in Mathematics, No. 72. Cambridge University Press, Cambridge-New York-Melbourne, (1977). • Some orthogonal and complete sets of Bessel functions associated with the vibrating plate. Math. Proc. Cambridge Philos. Soc., 91, (3), 503–515, (1982). • Five short stories about the cardinal series. Bull. AMS, 12, (1), 45–89, (1985). • A fresh approach to the derivative sampling theorem. Mathematics in signal processing, Inst. Math. Appl. Conf. Ser. New Ser. Bath, 12, Oxford Sci. Pub., (1985), 25–31, Oxford Univ. Press, New York, (1987). • Sampling theorems and the contour integral method. Applied Analysis., 41, (1–4), 155–169, (1991). • Sampling and aliasing for functions band-limited to a thin shell. Numer. Function Analysis Optimization, 12, (3–4), 327–337, (1991). • with Beaty, M. G., Dodson, M. M., Approximating Paley-Wiener functions by smoothed step functions. J. Approx. Theory 78, (no. 3), 433–445, (1994). • Sampling theory for Paley-Wiener spaces in the Riesz basis setting. Proc. Royal Irish Acadamy, 94, (2) Sect. A, 219–236, (1994). • with Beaty, M. G., Aliasing and Poisson summation in the sampling theory of Paley-Wiener spaces. J. Fourier Anal. Appl., 1, no. 1, 67–85, (1994). • Sampling for multi-band functions. Mathematical analysis, wavelets, and signal processing (Cairo, 1994), Contemp. Math., (190), Amer. Math. Soc., Providence, RI, 165–170, (1995). • An appreciation of Paul Butzer’s work in signal theory. Results Math., 34, (1–2), 3–19, (1998). • Sampling Theory in Fourier and Signal Analysis: Foundations, Clarendon Press, Oxford, (1996).

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• with Stens, R. L. (Eds.), Sampling Theory in Fourier and Signal analysis: Advanced Topics, Oxford University Press, Oxford, (1999). • with Schmeisser, G., Voss, J. J., The sampling theorem and several equivalent results in analysis. J. Comput. Anal. Appl., 2, no. 4, 333–371, (2000). • with Butzer, P. L., Stens, R. L., Sampling theory of signal analysis. Development of mathematics 1950-2000, 193–234, Birkhäuser, Basel, (2000). • Sampling theorems from the iteration of low order differential operators. Modern sampling theory, 219–227, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, (2001). • A sampling principle associated with Saitoh’s fundamental theory of linear transformations. Analytic extension formulas and their applications (Fukuoka, 1999/Kyoto, 2000), 73–86, Int. Soc. Anal. Appl. Comput., 9, Kluwer Acad. Publ., Dordrecht, (2001). • The ’Riesz basis method’ for deriving sampling series: an overview and some applications. Trends in industrial and applied mathematics (Amritsar, 2001), 63–76, Appl. Optim., 72, Kluwer Acad. Publ., Dordrecht, (2002). • Historical origins of interpolation and sampling, up to 1841. Sampl. Theory Signal Image Process, 2, no. 2, 117–128, (2003). • with Butzer, P. L., Stens, R. L., Classical and approximate sampling theorems; studies in the .Lp (R) and the uniform norm. J. Approx. Theory, 137, no. 2, 250–263, (2005). • Some groupings of equivalent results in analysis that include sampling principles. Sampl. Theory Signal Image Process., 4, no. 1, 19–31, (2005). • Integral transforms and sampling theorems. Integral Transforms Spec. Funct., 17, no. 1, 45–52, (2006). • Linear interpolation and a clay tablet of the Old Babylonian period. Sampl. Theory Signal Image Process, 6, no. 3, 243–247, (2007). • The Riemann zeta function and the sampling theorem. Sampl. Theory Signal Image Process., 8, no. 1, 1–12, (2009). • with Beaty, M. G., Dodson, M. M., Eveson, S. P., On the approximate form of Kluvánek’s theorem. J. Approx. Theory, 160, no. 1–2, 281–303, (2009).

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• with Butzer, P. L., Nashed, M. Z., Abdul Jerri—an appreciation on his seventy-seventh birthday. Sampling Theory Signal Image Processes, 8, no. 3, 209–223, (2009). • Paley-Wiener spaces and their reproducing formulae. Progress in analysis and its applications, World Sci. Publ., Hackensack, NJ, 273– 279, (2010). • with Butzer, P. L., Ferreira, P. J. S. G., Schmeisser, G., Stens, R. L., Interpolation and sampling: E. T. Whittaker, K. Ogura and their followers. J. Fourier Anal. Appl., 17, no. 2, 320–354, (2011). • with Butzer, P. L., Ferreira, P. J. S. G., Schmeisser, G., Stens, R. L., The sampling theorem, Poisson’s summation formula, general Parseval formula, reproducing kernel formula and the Paley-Wiener theorem for bandlimited signals—their interconnections. Applied Analysis, 90, no. 3–4, 431–461, (2011). • with Butzer, P. L., Dodson, M. M., Ferreira, P. J. S. G., Lange, O., Stens, R. L., Multiplex signal transmission and the development of sampling techniques: the work of Herbert Raabe in contrast to that of Claude Shannon. Appl. Anal., 90, no. 3–4, 643–688, (2011). • with Ferreira, P. J. S. G., The Establishment of Sampling as a Scientific Principle—A Striking Case of Multiple Discovery. Not. Am Math Soc, 58, (10), 1448–1450 (2011). • Sampling in reproducing kernel Hilbert space. New perspectives on approximation and sampling theory, 23–38, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, Cham, (2014). • Converse sampling and interpolation. Sampl. Theory Signal Image Process, 14, no. 2, 145–152, (2015). • with Dodson, M. M., Sampling theory in a Fourier algebra setting. Sampling: Theory and Applications, 51–91, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, Cham, (2020). • Claude Shannon: American genius. Sampling: theory and applications, 1–7, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, Cham, (2020). Rowland was a gift to our community. Through his deep interest in the rich mathematics of sampling theory and his wide knowledge of its fascinating history, he provided for all of us an informed perspective and a sense of the history of the subject. We were able to see the deep roots and interconnections of sampling and appreciate the tremendous potential of this theory, as it branched out to exciting new areas, even beyond the genius of Shannon.

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However, even more brilliant than the excellence of Rowland as a mathematician and lecturer, was his role to our community as a friend and mentor. With an encyclopaedic knowledge of sampling which he wore lightly, he was in a real sense a scholar and a gentleman. His kindness, generosity and courtesy brought out, in many ways, our best. And for that gift we will always be grateful. Washington, DC, USA York, UK

Stephen D. Casey M. Maurice Dodson

Sampling: Theory and Applications – A History of the SAMPTA Meetings

Abstract SAMPTA (Sampling : Theory and Applications) is a biennial interdisciplinary international conference for mathematicians, engineers, and applied scientists. The main purpose of SampTA is to exchange recent advances in sampling theory and to explore new trends and directions in the related areas of application. The SampTA conferences are a bridge between the mathematical and engineering signal processing communities. There have been 13 official meetings and a preliminary “pre-meeting,” at which the concept of the conference was developed. The conferences included talks and papers on signal and image processing, compressed sensing, frames, geometry, wavelets, non-uniform and weighted sampling, finite rate of innovation, universal sampling, time-frequency analysis, operator theory, and, of course, traditional sampling from both a mathematical and engineering perspective (A-to-D conversion). The purpose of this chapter is to give the reader an overview of SampTA and its contribution to the mathematical and engineering communities. Professor J. R. Higgins, to whom this volume is dedicated, was one of the founding members of SampTA and whose work inspired many researchers in the field.

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Introduction: Overview of SAMPTA The genesis of the SAMPTA meetings occurred at the International Conference on Mathematical Analysis and Signal Processing, held at Cairo University in Egypt, January 3–9, 1994. The conference was organized by Profs. A. Zayed, M. Ismail, Z. Nahed, and A. Ghaleb. Participants included Prof. J. R. Higgins, Prof. P. L. Butzer, Prof. Z. Nashed, Prof. A. Jerri, and Prof. G. Walter. At the Cairo conference, a meeting was planned—the first SAMPTA meeting. The prospect of exciting advances arising from the overlap between mathematical and engineering research led to the idea of regular international conference on sampling theory and its applications for mathematicians and engineers with interdisciplinary interests to be held all over the world. The first official SAMPTA meeting occurred during 1995 in Riga, Latvia, an international conference/workshop on Signal Theory at the Institute of Electronics and Computer Science in Riga, Latvia. Participants included Prof. J. Rowland Higgins, Prof. P. L. Butzer, Prof. H. Feichtinger, Prof. A. Jerri, Prof. F. Marvasti, and Prof. A. Zayed, the core group that organized and sustained the SAMPTA meetings. The Riga meeting was particularly fruitful. A Sampling Theory and Applications (SAMPTA) Committee was established to organize the meetings, combined with establishing an international journal—Sampling Theory in Signal and Image Processing (STSIP). In a spirit true to the SAMPTA meetings, STSIP was proposed during a walk along the coast by some conference attendees. STSIP continues today as Sampling Theory, Signal Processing, and Data Analysis, and focuses on the mathematics relating to sampling theory, signal processing, data analysis, and associated recovery problems from partial or indirect information. This journal continues the SAMPTA mission of inducing interactions leading to crossdisciplinary advances. The second SAMPTA meeting was two years later, in the University of Aveiro in Portugal, and was chaired by Prof. Paulo J. S. G. Ferreira. These first two meetings established the SAMPTA meetings as a regular event in the harmonic analysis/signal processing communities. The participants appreciated the important role played by sampling both in engineering and in mathematics. This role was envisioned in Claude Shannon’s seminal papers. Shannon placed what we commonly refer to as the Classical (a.k.a., Whittaker-Kotel’nikov-Shannon (WKS)) Sampling Theorem as a cornerstone of information theory in his paper Communications in the presence of noise.1 SAMPTA is a very successful venture, starting from a relatively small workshop/conference to the present fully-fledged biennial conferences held all over the world and attracting an unusually balanced mix of mathematicians and engineers and publishing an international journal. The conference is now a biennial inter-

1 Shannon himself was careful to note that the Sampling Theorem did not originate with him. The history of the theorem is rich, and includes the names Cauchy, Raabe, and Ogura, among others. Prof. Higgins wrote extensively on this.

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disciplinary international conference for mathematicians, engineers, and applied scientists. The main purpose of SAMPTA is to exchange recent advances in sampling theory and to explore new trends and directions in the related areas of application. SAMPTA has focused on such fields as signal processing and image processing, coding theory, control theory, real analysis and complex analysis, harmonic analysis, and the theory of differential equations. The conference has always featured plenary talks by prominent speakers, special sessions on selected topics reflecting the current trends in sampling theory and its applications to the engineering sciences, as well as regular sessions about traditional topics in sampling theory, and poster sessions. • The intellectual merit of SAMPTA: The SAMPTA conferences are a bridge between the mathematical and engineering signal processing communities. The mix between mathematicians and engineers is unique, and leads to extremely useful and constructive dialog between the two communities. For example, SAMPTA conferences have had sessions on theory—compressed sensing, frames, geometry, wavelets, non-uniform and weighted sampling, finite rate of innovation, universal sampling, time-frequency analysis, operator theory, and application—A-to-D conversion, computational neuroscience, mobile sampling issues, and biomedical applications. The interaction at SAMPTA pushed the envelopes on these topics forward in both communities. The meetings bring together world-renowned mathematicians and engineers to work on these subjects.

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• The broader impacts resulting from SAMPTA : The SAMPTA conferences have and will continue to serve as a meeting ground for harmonic analysts and electrical engineers, and give graduate students and junior investigators a chance to learn about the developments of the subjects. SAMPTA gives the community an opportunity to interact with some of the leaders in the field in a relaxed and yet very constructive environment. The plenary talks are delivered by top researchers in the fields. The conference papers were reviewed and presented via the EDAS system, and then uploaded into IEEE Xplore. Longer papers were published as journal papers in Sampling Theory in Signal and Image Processing (STSIP) and contributed chapters in Springer-Birkhäuser books in the Applied and Numerical Harmonic Analysis Series.

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SAMPTA Meetings • • • • • • • • • • •

SAMPTA 2019—University of Bordeaux, Bordeaux, France, July 8–12, 2019 SAMPTA 2017—Tallinn University, Tallinn, Estonia, July 3–7, 2017 SAMPTA 2015—American University, Washington, DC, USA, May 25–29, 2015 SAMPTA 2013—Jacobs University, Bremen, Germany, July 1–5, 2013 SAMPTA 2011—Nanyang Technical University, Singapore, May 2–6, 2011 SAMPTA 2009—CIRM, Marseilles, France, May 18–22, 2009 SAMPTA 2007—Aristotle University, Thessaloniki, Greece, June 1–5, 2007 SAMPTA 2005—Samsun, Turkey, July 10–15, 2005 SAMPTA 2003—Strobl, Austria, May 26–30, 2003 SAMPTA 2001—U. of Central Florida, Orlando FL, USA, May 13–17, 2001 SAMPTA 1999—Loen, Norway, August 11–14, 1999

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• SAMPTA 1997—University of Aveiro, Aveiro, Portugal, July 16–19, 1997 • SAMPTA 1995—Riga Technical Institute, Riga, Latvia, September 20–22, 1995 We also mention “SAMPTA Zero,” the International Conference on Mathematical Analysis and Signal Processing, held at Cairo University in Egypt, January 3–9, 1994.

Plenary Talks at SAMPTA Conferences The SAMPTA conferences have and will serve as a meeting ground for mathematicians and engineers. They have also been a strong mix of junior and senior scientists, providing graduate students and junior investigators a chance to learn about the latest developments of the subjects. SAMPTA gives the community an opportunity to interact with some of the leaders in the field in a relaxed and yet very constructive environment. The plenary talks were delivered by top researchers in the fields.

Plenary Speakers for SAMPTA 2019 – Bordeaux Name Alexandre d’Aspremont Bubacarr Bah Marcin Bownik Massimo Fornasier Anna Gilbert Rémi Gribonval Urbashi Mitra Ursula Molter Pierre Vandergheynst

Affiliation ENS Paris, France AIMS, South Africa University of Oregon, USA T.U. Munich, Germany University of Michigan, USA INRIA Rennes, France University of South California, USA University of Buenos Aires, Argentina EPF Lausanne, Switzerland

Sampling: Theory and Applications – A History of the SAMPTA Meetings

Plenary Speakers for SAMPTA 2017 – Tallinn Name David Gross Stéphane Jaffard José Príncipe Justin Romberg Amit Singer Rene Vidal Laura Waller Rachel Ward

Affiliation University of Cologne, Germany Université Paris-Est, France University of Florida, USA Georgia Institute of Technology, USA Princeton University, USA Johns Hopkins University, USA University of California, Berkeley, USA University of Texas at Austin, USA

Plenary Speakers for SAMPTA 2015 – Washington, DC Name Richard G. Baraniuk Robert Calderbank Laurent Demanet Yonina Eldar Pascal Frossard Stanley Osher Thomas Strohmer Alexander Ulanovskii

Affiliation Rice University, USA Duke University, USA Massachusetts Institute of Technology, USA Technion, Israel EPF Lausanne, Switzerland University of California Los Angeles, USA University of California Davis, USA University of Stavanger, Norway

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Sampling: Theory and Applications – A History of the SAMPTA Meetings

Plenary Speakers for SAMPTA 2013 – Bremen Name Emmanuel Candès Wolfgang Dahmen Hans Feichtinger Piotr Indyk Michal Irani Nikolai Nikolski Jan Rabey Yannis Tsividi Roman Vershynin

Affiliation Stanford University, USA RWTH Aachen, Germany University of Vienna, Austria Massachusetts Institute of Technology, USA Weizmann Institute, Israel Université Bordeaux, France UC Berkeley, USA Columbia University, USA University of Michigan, USA

Plenary Speakers for SAMPTA 2011 – Singapore Name Martin Vetterli Stéphane Mallat John Benedetto Graham Goodwin Albert Cohen Shen Zouwei Willy Sansen Steven Smale

Affiliation EPF Lausanne, Switzerland École Polytechnique, France University of Maryland, USA University of Newcastle, Australia Université Pierre et Marie Curie, France Nanyang Technical University, Singapore Katholieke Universiteit Leuven, Belgium University of California, Berkeley, USA

Plenary Speakers for SAMPTA 2009 – Marseilles Name Yurii Lyubarskii Mauro Maggioni Ron DeVore Jean-Luc Starck

Affiliation NTNU, Trondheim, Norway Johns Hopkins University, USA Texas A&M University, USA CEA-Saclay, France

Sampling: Theory and Applications – A History of the SAMPTA Meetings

Plenary Speakers for SAMPTA 2007 – Thessaloniki Name Michael Unser Sergei Avdoni Antonios Melas Sinan Güntürk Martin Vetterli Ognyan Kounchev Joel Tropp Ioannis Pitas

Affiliation EPF Lausanne, Switzerland University of Alaska, USA University of Athens, Greece NYU-Courant, USA EPF Lausanne, Switzerland Bulgarian Academy of Sciences, Bulgaria California Institute of Technology, USA Aristotle University, Thessaloniki, Greece

Plenary Speakers for SAMPTA 2005 – Samsun Name Yoram Bresler Ingrid Daubechies M. Maurice Dodson Hans Feichtinger Paulo Ferreira Karlheinz Gröchenig Abdul Jerri Yurii Lyubarskii Steven Smale Jared Tanner Ahmed Tewfik

Affiliation University of Illinois, USA Princeton University, USA University of York, UK University of Vienna, Austria Universidade de Aveiro, Portugal University of Connecticut, USA Clarkson University, USA NTNU, Trondheim, Norway University of Chicago, USA Stanford University, USA University Of Minnesota, USA

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Sampling: Theory and Applications – A History of the SAMPTA Meetings

Plenary Speakers for SAMPTA 2003 – Strobl Name Vadim Olshevsky Michael Lacey J. Rowland Higgins Amir Averbuch Jeremy Levesley

Affiliation University of Connecticut, USA University of Texas, USA Anglia Ruskin, UK Tel Aviv, Israel University of Leicester, UK

Plenary Speakers for SAMPTA 2001 – Orlando, FL Name Stéphane Jaffard N. K. Bose John Benedetto Michael Unser P. P. Vaidyanathan Alberto Grunbaum Dennis M. Healy Wasfy Mikhael

Affiliation University of Paris, France Pennsylvania State University, USA University of Maryland, USA EPF Lausanne, Switzerland California Institute of Technology, USA University of California- Berkeley, USA University of Maryland, USA University of Central Florida, USA

Plenary Speakers for SAMPTA 1999 – Loen Name Alexander Petrovskii William J. Fitzgerald

Affiliation Belarusian State University, Belarus University of Cambridge, UK

Sampling: Theory and Applications – A History of the SAMPTA Meetings

Plenary Speakers for SAMPTA 1997 – Aveiro Name Abdul Jerri Michael Unser John Benedetto

Affiliation Clarkson University, USA EPF Lausanne, Switzerland University of Maryland, USA

Plenary Speakers for SAMPTA 1995 – Riga Name Hans Feichtinger Paul Butzer Ahmed Zayed Ivars Bilinskis Farokh Marvasti

Affiliation University of Vienna, Austria RWTH Aachen, Germany University of Central Florida, USA Riga Technical Institute, Latvia King’s College London, UK

Plenary Speakers for “SAMPTA ZERO” – Cairo Name Paul Butzer Zuhair Nashed Carlos Kenig W. N. Everitt Charles Chui Victor Wickerhauser

Affiliation RWTH, Germany University of Delaware, USA University of Chicago, USA University of Birmingham, UK Texas A&M, USA Washington University, USA

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Sampling: Theory and Applications – A History of the SAMPTA Meetings

A Sampling of Special Session Topics at SAMPTA Conferences

Special Sessions for SAMPTA Compressed sensing and low rank matrix recovery Frame theory Time-frequency analysis Deep learning Graph signal processing Quantization Mathematical data processing with optimization Bilinear inverse problems Mathematics of deep learning Mathematical theory of quantization Low rank matrix recovery and phase retrieval Shannon sampling 100 years after his birth Time-frequency analysis Dynamical sampling Sampling and geometry Frames, nonlinear approximations and the Hilbert transform Spectral estimation and acoustics Frame Theory Dynamic Mobile and Nonlinear Sampling Sampling in Non-Euclidean Spaces Low Rank Matrix and Tensor Recovery Universal Sampling, Fourier Frames and Riesz Bases of Exponentials Compressed Sensing and Sparsity Based Regularizations Phase Retrieval A to D Algorithms and Chip Design Sampling Signals with Finite Rate of Innovation in Biomedical Applications Sampling and Stochastic Processes

F. Krahmer, R. Kueng J. Jasper, D. Mixon A. Haimi, J. Romero M. Belkin, M. Soltanolkotab K. Gröchenig, I. Pesenson S. Dirksen, R. Saab A. Bandeira, D. Mixon, D.Needell F. Krahmer, M. Soltanolkotabi H. Boelcskei, P. Grohs, M. Rodrigues S. Dirksen, R. Saab R. Balan, D. Gross, X. Li G. Pfander, A. Zayed K. Gröchenig, J. Romero A. Aldroubi, J. Bouchot S. Casey, G. Olafsson, J. Christensen A. Aldroubi, S. Kaushik, S. Sharma L. Abreu, P. Balasz G. Kutyniok, G. Pfander R. Aceksa, J. Romero, Q. Sun G. Olafsson H. Rauhut J. Antezana, J. Marzo B. Adcock, F. Krahmer B. Bodmann L. Fesquet, S. Hoyos, B. Sadler P. Marziliano M. Unser

Sampling: Theory and Applications – A History of the SAMPTA Meetings

SAMPTA Organizing Committees

Steering Committee for SAMPTA Name Ahmed Zayed Akram Aldroubi John Benedetto Yonina Eldar Paulo Ferreira Gitta Kutyniok Farokh Marvasti Götz Pfander Bruno Torrésani Michael Unser

Affiliation DePaul University, USA Vanderbilt University, USA University of Maryland, USA Weizmann, Israel Universidade de Aveiro, Portugal TU Berlin, Germany Sharif University of Technology, Iran Katholische Universität Eichstätt, Germany Aix-Marseille Université, France EPF Lausanne, Switzerland

Founding Members for SAMPTA Name Paul Butzer Hans Feichtinger Karlheinz Gröchenig Rowland Higgins Abdul Jerri Andi Kivinuuk Yurii Lyubarskii Farokh Marvasti Gerhard Schmeisser Ahmed Zayed

Affiliation RWTH Aachen, Germany University of Vienna, Austria University of Vienna, Austria Anglia Polytechnic University, Cambridge, England Clarkson University, USA Tallinn University, Estonia NTNU, Trondheim, Norway Sharif University of Technology, Iran Erlangen-Nürnberg University, Germany DePaul University, USA

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Sampling: Theory and Applications – A History of the SAMPTA Meetings

Chairs of SAMPTA Meetings Meeting SAMPTA 2019 SAMPTA 2017 SAMPTA 2015 SAMPTA 2013 SAMPTA 2011 SAMPTA 2009 SAMPTA 2007 SAMPTA 2005 SAMPTA 2003 SAMPTA 2001 SAMPTA 1999 SAMPTA 1997 SAMPTA 1995

Name Phillipe Jamming Andi Kivanuuk Stephen Casey Götz Pfander Pina Marziliano Bruno Torresani Nikos Atreas A.Turan Gürkanlı Hans Feichtinger Ahmed Zayed Yurii Lyubarskii Paulo Ferreira Ivars Bilinskis

Affiliation University of Bordeaux Tallinn University American University Jacobs University Nanyang Technical University Aix-Marseille Université, France Aristotle University Samsun, Turkey University of Vienna, Austria University of Central Florida NTNU, Norway Universidade de Aveiro, Portugal Riga Technical Institute, Latvia

We again also mention “SAMPTA Zero,” the International Conference on Mathematical Analysis and Signal Processing, held at Cairo University in Egypt, January 3–9, 1994. The conference was organized by Prof. A. Zayed et al.

Publications Each presenter at the conference submitted a brief, yet informative paper outlining their main results and justifying their inclusion in the conference. To aid attendees in selecting appropriate talks, the conference papers were available at first by a volume of printed papers, and then later, electronically, at registration. Following a longstanding tradition in the engineering literature, each conference paper for SAMPTA were generally four to five pages. This length was sufficient to contain a careful explanation of the problem to be addressed in the presenter’s talk and some of the technical highlights of the solution, without going into extensive detail. The conference papers were peer-reviewed for technical accuracy and topical relevance by the session organizers. Following on its effective use during SAMPTA 2013, the conference papers were reviewed and published via the EDAS system. Starting with SAMPTA 2015, accepted papers were published in IEEE Xplore. Longer papers were published in special issues of Sampling Theory in Signal and Image Processing (STSIP), and plenary and special sessions speakers were invited to contribute chapters in books published in the Springer-Birkhäuser Applied and Numerical Harmonic Analysis Series.

Sampling: Theory and Applications – A History of the SAMPTA Meetings

Proceedings of SAMPTA Conferences Meeting affiliation SAMPTA 2019—University of Bordeaux, Bordeaux, France SAMPTA 2017—Tallinn University, Tallinn, Estonia SAMPTA 2015—American University, Washington, DC, USA SAMPTA 2013—Jacobs University, Bremen, Germany SAMPTA 2011—Nanyang Technical University, Singapore SAMPTA 2009—CIRM, Marseilles, France SAMPTA 2007—Aristotle University, Thessaloniki, Greece SAMPTA 2005—Samsun, Turkey SAMPTA 2003—Strobl, Austria SAMPTA 2001—UCF, Orlando Florida, USA SAMPTA 1999—Loen, Norway SAMPTA 1997—University of Aveiro, Aveiro, Portugal SAMPTA 1995—Riga, Latvia

Number of papers 133 144 145 148 102 108 85 86 91 86 80 86 60

Special Issues in STSIP for SAMPTA Meetings Volume number Volumes 17 Volumes 15 and 16 Volumes 13 and 14 Volumes 11 and 12 Volume 10 Volume 8 Volume 6 Volume 3

Meeting affiliation SAMPTA 2017 SAMPTA 2015 SAMPTA 2013 SAMPTA 2011 SAMPTA 2009 SAMPTA 2007 SAMPTA 2005 SAMPTA 2003

Number of papers 8 15 20 10 10 11 12 14

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Sampling: Theory and Applications – A History of the SAMPTA Meetings

Books Generated by Previous SAMPTA Meetings Title Sampling: Theory and Applications Sampling Theory: A Renaissance New Perspectives on Approximation and Sampling Proceedings of SAMPTA 2007 Sampling, Wavelets, and Tomography Modern Sampling Theory

Editors Casey, Okoudjou Robinson, Sadler Pfander Zayed and Schmeisser

Meeting SAMPTA 2015

Atreas and Karanikas Benedetto and Zayed Benedetto and Ferreira

SAMPTA 2007 SAMPTA 2001 SAMPTA 1997

SAMPTA 2013 SAMPTA 2013

SAMPTA 2023 The SAMPTA conferences have and will continue to serve as a meeting ground for harmonic analysts and electrical engineers, and will give graduate students and junior investigators a chance to learn about the developments of the subjects. SAMPTA gives the community an opportunity to interact with some of the leaders in the field in a relaxed and yet very constructive environment. The plenary talks are delivered by top researchers in the fields. The conference papers are reviewed and cataloged. The biennial scheduling of the meetings allows for an opportunity to establish new milestones in the harmonic analysis and signal processing communities. SAMPTA 2023 was organized while this volume was written, and was held at Yale University, New Haven, Connecticut during the week of July 10–14. Washington, DC, USA Aveiro, Portugal Chicago, IL, USA

Stephen D. Casey Paulo J. S. G. Ferreira Ahmed Zayed

Contents

Part I Classical Sampling Classical and Approximate Exponential Sampling Formula: Their Interconnections in Uniform and Mellin–Lebesgue Norms . . . . . . . . . . C. Bardaro, P. L. Butzer, I. Mantellini, G. Schmeisser, and R. L. Stens

3

Asymptotic Theorems for Durrmeyer Sampling Operators with Respect to the .Lp -Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Danilo Costarelli and Gianluca Vinti

23

On Generalized Shannon Sampling Operators in the Cosine Operator Function Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Andi Kivinukk and Gert Tamberg

39

Bernstein Spaces, Sampling, and Riesz-Boas Interpolation Formulas in Mellin Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Isaac Z. Pesenson

69

The Behavior of Frequency Bandlimited Cardinal Interpolants . . . . . . . . . . . W. R. Madych

89

The Balian-Low Theorem for .(Cq )-Systems in Shift-Invariant Spaces . . . . 113 Alexander M. Powell Whittaker-Type Derivative Sampling and .(p, q)-Order Weighted Differential Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Tibor K. Pogány Shannon Sampling Theorem via Poisson, Cauchy, Jacobi, and Levin . . . . . 139 Stephen D. Casey

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Contents

Part II Theoretical Extensions Schoenberg’s Theory of Totally Positive Functions and the Riemann Zeta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 Karlheinz Gröchenig Sampling via the Banach Gelfand Triple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Hans G. Feichtinger Part III Frame Theory A Survey of Fusion Frames in Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 L. Köhldorfer, P. Balazs, P. Casazza, S. Heineken, C. Hollomey, P. Morillas, and M. Shamsabadi Frames of Iterations and Vector-Valued Model Spaces . . . . . . . . . . . . . . . . . . . . . . 329 Carlos Cabrelli, Ursula Molter, and Daniel Suárez A Survey on Frame Representations and Operator Orbits . . . . . . . . . . . . . . . . . 349 Ole Christensen and Marzieh Hasannasab Three Proofs of the Benedetto–Fickus Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 Dustin G. Mixon, Tom Needham, Clayton Shonkwiler, and Soledad Villar Clifford Prolate Spheroidal Wavefunctions and Associated Shift Frames . 393 Hamed Baghal Ghaffari, Jeffrey A. Hogan, and Joseph D. Lakey Part IV Applications Power-Aware Analog to Digital Converters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 Satish Mulleti, Ayush Bhandari, and Yonina C. Eldar Quaternionic Coupled Fractional Fourier Transform on Boehmians . . . . . . 453 R. Kamalakkannan, R. Roopkumar, and A. Zayed Multiscale Tree Sampling Regularization of Inverse Spherical Pseudodifferential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469 Willi Freeden, M. Zuhair Nashed, and Michael Schreiner Acceleration Algorithms for Iterative Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521 Mahdi Shamsi and Farokh Marvasti Applied and Numerical Harmonic Analysis (100 volumes) . . . . . . . . . . . . . . . . . 553

Part I

Classical Sampling

Classical and Approximate Exponential Sampling Formula: Their Interconnections in Uniform and Mellin–Lebesgue Norms C. Bardaro, P. L. Butzer, I. Mantellini, G. Schmeisser, and R. L. Stens

Dedicated to the memory of Rowland Higgins (1935–2020) a friend, model, and mentor of us all

AMS Subject Classification 41A58, 44A15, 94A20, 41A35, 41A80, 47B37

1 Introduction The exponential sampling formula is one of the most useful mathematical tools for studying phenomena in optical physics and in engineering. It was first formally introduced in [9] and [20] as a series representation of solutions of integral equations that are modeling several practical problems related to light scattering, diffraction, and radioastronomy besides other applications. A rigorous mathematical formulation was first introduced by P.L. Butzer and S. Jansche in [13, 14] and later on in [3]. It turns out that the most suitable setting for a treatment of the exponential sampling theorem is the theory of Mellin transform and the consequent Mellin analysis, first developed in [12] and widely exposed in the forthcoming book [1]. The exponential sampling theorem gives an exact series representation of a function with a compactly supported Mellin transform, using exponentially spaced samples on the positive real line. This representation is indeed the Mellin counterpart of the classical Whittaker–Kotel’nikov–Shannon (WKS) sampling theorem C. Bardaro · I. Mantellini Department of Mathematics and Computer Sciences, University of Perugia, Perugia, Italy e-mail: [email protected]; [email protected] P. L. Butzer · R. L. Stens Lehrstuhl A für Mathematik, RWTH Aachen, Aachen, Germany e-mail: [email protected]; [email protected] G. Schmeisser () Department Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Erlangen, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. D. Casey et al. (eds.), Sampling, Approximation, and Signal Analysis, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-031-41130-4_1

3

4

C. Bardaro et al.

of Fourier analysis, which reconstructs a Fourier bandlimited function by a series involving uniformly spaced samples on the whole real line. This was one of the most important areas of Rowland Higgins’ scientific work to which he gave fundamental contributions both theoretical and practical. Although we can formally obtain the exponential sampling theorem from the WKS theorem by suitable substitutions, the mathematical background of the two theorems is not the same. For example, the structure of the Fourier bandlimited functions, characterized by the famous Paley–Wiener theorem (see, e.g., [10, Section 6.8], [18]), is very different from that of the Mellin bandlimited functions. The class of these functions was recently characterized by a Mellin version of the Paley–Wiener theorem (see [5, 7]). When the function f is not bandlimited but has a summable Fourier transform, the representation in terms of the WKS sampling series is no longer exact. In this case, J.L. Brown [11] established an approximate version which includes the WKS theorem as a particular case. In [16], the authors proved that the classical WKS theorem is indeed equivalent to its approximate version in the sense that each can be deduced from the other. As regards the approximate version, the results in [16] include convergence in the uniform norm as well as in the norm of .Lp (R)-spaces. In the latter case, the authors required that the function f belongs to a suitable subspace of .Lp (R), which was first introduced in [2]. Now turning to the exponential sampling theorem, we point out that an approximate version as a counterpart of Brown’s result was studied in [13] (also see [3] and [6]). In the present chapter, we prove that the exponential sampling theorem implies its approximate version, in both, the uniform norm and the norm of the Mellin– p Lebesgue spaces .Xc defined in (1) and (2) below. These spaces represent the Mellin counterpart of the Lebesgue spaces. Here, we consider the case .1 ≤ p ≤ 2 only since, as far as we know, a version of the exponential sampling theorem is not known for .p > 2. p As it happens in Fourier analysis, concerning the convergence in the .Xc -norm, p p we have to restrict ourselves to a suitable subspace .c of .Xc , which was introduced in [8]. A fundamental tool is the notion of a mixed Mellin–Hilbert transform here introduced as the counterpart of the mixed Hilbert transform employed in [19] (also p see [2]). This transform acts on sequence spaces .c which are Mellin counterparts of the Lebesgue sequence spaces (see Sect. 4.2). We prove that this transform is a p p bounded linear operator between the spaces .c and .Xc . Our extension of the results in [16] is obtained independent of the Fourier theory by employing tools from Mellin analysis.

2 Notations and Basic Definitions Let .Z, .N, .R, and .R+ be the sets of all integers, positive integers, real numbers, and positive real numbers, respectively. By .C(R) and .C(R+ ), we denote the spaces of all the uniformly continuous functions on .R and .R+ , respectively.

Classical and Approximate Exponential Formula

5

For .1 ≤ p < ∞, the Lebesgue space .Lp (R) comprises all measurable functions such that .|f |p is Lebesgue integrable on .R. It is endowed with the norm  f p :=

1/p

∞

|f (x)| dx p

.

−∞

.

The Lebesgue space .L∞ (R) comprises all essentially bounded measurable functions on .R and is endowed with the norm .f ∞ := ess supx∈R |f (x)|. Analogous definitions hold for .R+ . For .1 ≤ p < ∞, we denote by .p the sequence space comprising all the sequences .x = {xn }n∈Z such that  xp :=



.

1/p |xn |

< ∞,

p

n∈Z

while .∞ will be the space of all bounded sequences. p For .1 ≤ p < ∞ and .c ∈ R, the Mellin–Lebesgue space .Xc is defined by Xc := {f : R+ → C : f (·) (·)c−1/p ∈ Lp (R+ )}

.

p

(1)

and endowed with the norm  f 

p Xc

.

:=

∞

p cp dx

|f (x)| x

0

1/p .

x

(2)

The Mellin–Lebesgue space .Xc∞ comprises all measurable functions f on .R+ such that .f Xc∞ := ess supx∈R+ x c |f (x)| < ∞. For .p = 1, we simply write .Xc instead of .Xc1 . For a differentiable function .f : R+ → C, we define the Mellin derivative .c f as (c f )(x) = xf  (x) + cf (x)

.

(see [1, 12]). Higher-order Mellin derivatives are defined recursively. r,p For .1 ≤ p < ∞, the Mellin–Sobolev space .Wc (R+ ) is defined as the space of p all functions .f ∈ Xc such that .f = ϕ a.e., where .ϕ is .(r − 1)-times differentiable p and .ϕ (r−1) is locally absolutely continuous with .rc ϕ ∈ Xc . p For .1 ≤ p < ∞, we denote by .c the Mellin–Lebesgue sequence space comprising all sequences .x = {xn }n∈Z such that  xpc :=



.

n∈Z

1/p |xn | e

p ncp

0).

(4)

3 The Exponential Sampling Formula For any .c ∈ R, we introduce the .lin function as linc (x) := x −c sinc(log x)

.

(x ∈ R+ ),

where the .sinc function is defined by sinc(x) :=

.

sin π x πx

(x ∈ R)

with the continuous extensions .linc (1) = sinc(0) = 1. The following two theorems are to be found in [13, 14] (also see [6]). 2 Theorem A (Exponential Sampling Formula) Let .T > 0 and .f ∈ Bc,π T . Then

f (x) = (ST f )(x) :=



.

f (ek/T ) linc/T (e−k x T )

(x > 0),

k∈Z

the series being absolutely and uniformly convergent on compact subsets of .R+ . Theorem B (Approximate Exponential Sampling Formula) Let .f ∈ M2c . Then for .T > 0,   x −c ∧ k/T −k T . f (x) − [f ] 2 (c + it) dt. ≤ f (e ) lin (e x ) c/T M c π |t|≥π T k∈Z

In particular, f (x) = lim

.

T →∞



f (ek/T ) linc/T (e−k x T )

k∈Z

uniformly on compact subsets of .R+ . Obviously, Theorem A can be deduced from Theorem B. In fact, Theorem A is just a special case of Theorem B. The inverse implication, that Theorem B can be deduced from Theorem A, is not so trivial and takes up the remaining part of this section. We start with the following observation.

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C. Bardaro et al. p

Lemma 1 Let .c ∈ R and .1 < p < 2. Then .Mc ⊂ M2c . p

Proof Let .f ∈ Mc for .p ∈]1, 2[. Then one has by (4) that .x c f (x) is a bounded function. Thus, for a constant .C > 0, |f (x)|2 x 2c−1 = |f (x)|p x pc−1 |f (x)|2−p x (2−p)c ≤ C 2−p |f (x)|p x pc−1 ,

.

 

and the assertion follows. The main result of this section now reads as: p

Theorem 1 Let .f ∈ Mc , .1 < p ≤ 2. Then Theorem A implies Theorem B. Proof In view of Lemma 1, it suffices to consider the case .p = 2 only. Noting that [f ]∧ ∈ L1 (c + iR) ∩ L2 (c + iR), we can apply the inversion formula (4) and split Mc2 f for any .T > 0 as

.

f (x) = f1 (x) + f2 (x),

.

where f1 (x) :=

.



x −c 2π

πT

−π T

[f ]∧ (c + it)x −it dt M2 c

and f2 (x) :=

.

x −c 2π

 |t|≥π T

[f ]∧ (c + it)x −it dt. M2 c

Since the Mellin transform is a bijective operator from .M2c onto .L2 (c + iR), it 2 2 follows that .f1 ∈ Bc,π T and .f2 ∈ Xc , and by Theorem A, we have f1 (x) =



.

f1 (ek/T ) linc/T (e−k x T ).

(5)

k∈Z

As to the exponential sampling series .(ST f2 )(x), we show that it is convergent for all .x > 0 and deduce an integral representation. To this end, fix .x > 0 and define .gx (t) as the .2π T -periodic extension of the function .t → x −it = e−it log x from .] − π T , π T ] to the whole real axis, that is, gx (t) = ei(t−2π nT ) log x

.

 t ∈ ](2n − 1)π T , (2n + 1)π T ]; n ∈ Z .

Since the Fourier coefficients of the function .gx are given by 1 .(gx )k = 2π T



πT −π T

e−it log x e−ikt/T dt = sinc(T log x + k)

(k ∈ Z)

Classical and Approximate Exponential Formula

9

and the function .gx is of bounded variation, we have gx (t) =



.

sinc(T log x + k)eikt/T =

k∈Z



sinc(T log x − k)e−ikt/T

(6)

k∈Z

at each continuity point of .gx , i.e., for all .t = (2n + 1)π T , .n ∈ Z. Moreover, the symmetric partial sums are uniformly bounded (see [22, page 28]). Now it follows by the definition of .f2 that N 

f2 (ek/T )linc/T (e−k x T )

.

k=−N

=

 N  −kc/T  

 e −ikt/T [f ]∧ (c + it) e dt ekc/T x −c sinc log(e−k x T ) 2 M c 2π |t|≥π T

k=−N

=



x −c 2π

|t|≥π T

N 

[f ]∧ (c + it) M2 c

e−ikt/T sinc(T log x − k)dt.

k=−N

Letting .N → ∞, yields by (6),

.

N 

lim

N →∞

f2 (ek/T )linc/T (e−k x T )

k=−N

x −c = 2π =

x −c 2π

 |t|≥π T

 |t|≥π T

[f ]∧ (c + it) M2 c

∞ 

e−ikt/T sinc(T log x − k)dt

k=−∞

[f ]∧ (c + it)gx (t)dt, M2 c

the interchange of the limit and the integral being justified by the uniform boundedness of the partial sums of (6) and Lebesgue’s dominated convergence theorem. This shows the convergence of the sampling series .(ST f2 )(x) and gives the desired integral representation. Moreover, we have (ST f2 )(x) =

.

=

x −c 2π

 |t|≥π T

 

[f ]∧ (c + it)gx (t)dt M2 c

(2n+1)π T

n∈Z\{0} (2n−1)π T

[f ]∧ (c + it)x −i(t−2nπ T ) dt. M2 c

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C. Bardaro et al.

Thus, we can write f (x) = f1 (x) + f2 (x) = (ST f1 )(x) + f2 (x) = (ST f )(x) + (RT f )(x),

.

where (RT f )(x) = f (x) − (ST f )(x) = f2 (x) − (ST f2 )(x)

.

=

  (2n+1π T x −c 

1 − x 2inπ T [f ]∧ (c + it)x −it dt. Mc2 2π (2n−1)π T n∈Z

From this representation, we obtain the error estimate of Theorem B.

 

p

4 Studies in Xc -Norm In this section, we will investigate the convergence of the exponential sampling p series and its approximate version in .Xc . Since the exponential sampling series is p not a bounded linear operator on the whole of .Xc , we have to introduce suitable subspaces which guarantee the boundedness.

p

4.1 Subspaces of Xc

A partition . = {xj }j ∈Z of .R+ is called admissible, if



 0 < := inf log xj − log xj −1 ≤ sup log xj − log xj −1 =: < ∞.

.

j ∈Z

j ∈Z

(7) It follows that .limj →−∞ log xj = −∞, and so .limj →−∞ xj = 0; similarly, limj →∞ xj = ∞. For .c ∈ R, .p ≥ 1 and any admissible partition . = {xj }j ∈Z of .R+ , we denote p by .c ( ) the space of all functions .f : R+ → C such that

.



.

|f |pc ( )

 f (xk )x c p (log xk − log xk−1 ) := k

1/p < ∞.

k∈Z

In view of (7), we see that for an admissible partition . = {xj }j ∈Z , we have .f ∈ p p c ( ) if and only if . k∈Z f (xk )xkc < ∞.

Classical and Approximate Exponential Formula

11

p

For .c ∈ R and .p ≥ 1, we denote by .c the space of all measurable and bounded functions .f : R+ → C such that .|f |pc ( ) < ∞ for every admissible partition . of + .R . For such spaces, see also [8]. For .c ∈ R, we associate with .f : R+ → C the function .fc : R+ → C defined p p by .fc (t) := f (t)t c for .t ∈ R+ . Obviously, .fc ∈ 0 if and only if .f ∈ c . With an admissible partition . = {xj }j ∈Z of .R+ and a bounded function .g : ∗ : R+ → C piecewise defined by + R → C, we associate the function .g ∗ g (t) :=

sup

.

u∈[xj −1 ,xj ]

|g(u)|

for

t ∈ ]xj −1 , xj ]; j ∈ Z.

Lemma 2 Let .c ∈ R and .p ≥ 1. The following statements are equivalent: p

(i) .f ∈ c . p (ii) .(fc )∗ ∈ 0 for each admissible partition . of .R+ . (iii) For each admissible partition . = {xj }j ∈Z of .R+ and any choice of .ξj ∈ [xj −1 , xj ], .j ∈ Z, we have .



 fc (ξj ) p log xj − log xj −1 < ∞. j ∈Z

Proof (i) .⇒ (ii): Let . = {yj }j ∈Z . We first show that .(fc )∗ ∈ 0 ( ) by proceeding as follows. For each .k ∈ Z, there exists a point .tk ∈ [yk−1 , yk ] such that p

.

(fc )∗ (yk ) ≤ |fc (tk )| +

1 . 1 + k2

By Minkowski’s inequality, this implies that   .

1/p (fc )∗

(yk )

≤

p

k∈Z

 



1/p |fc (tk )|

+

p

k∈Z

 k∈Z

1 1 + k2

p 1/p .

(8)

Clearly, the second term on the right-hand side is bounded. By statement (i), the first term would also be bounded if .{tj }j ∈Z were an admissible partition of .R+ . Unfortunately, this cannot be guaranteed since .infj ∈Z (log tj − log tj −1 ) may be zero. However, it is easily verified that . 0 := {t2j }j ∈Z and . 1 := {t2j +1 }j ∈Z are admissible partitions of .R+ with . m ≥ and . m ≤ 3 for .m ∈ {0, 1}. Thus, as a consequence of (i), 



.

k∈Z

1/p |fc (tk )|

p

≤

 



1/p |fc (t2k )|

+

p

k∈Z

 k∈Z

Combined with (8), we find that .(fc )∗ ∈ 0 ( ). p

1/p |fc (t2k+1 )|

p

< ∞.

12

C. Bardaro et al.

Now let .  = {xj }j ∈Z be an arbitrary admissible partition of .R+ . Then  .

(fc )∗ (xj )p =





(fc )∗ (xj )p

k∈Z xj ∈]yk−1 ,yk ]

j ∈Z

=





(fc )∗ (yk )p

1

xj ∈]yk−1 ,yk ]

k∈Z



≤ 1+







(fc )∗ (yk )p .

k∈Z

Thus, .(fc )∗ ∈ 0 ( ) implies that .(fc )∗ ∈ 0 (  ) for any admissible partition .  p of .R+ , and so we have proved that .(fc )∗ ∈ 0 . (ii) .⇒ (iii): Let . = {xj }j ∈Z be an admissible partition of .R+ and .ξj ∈ [xj −1 , xj ]. Then p

.

p

  fc (ξj ) p (log xj − log xj −1 ) ≤ (fc )∗ (xj )p (log xj − log xj −1 ), j ∈Z

j ∈Z

which shows that (iii) is a consequence of (ii). (iii) .⇒ (i): Under the validity of (iii), we may choose .ξj = xj , which gives (i) immediately.   In the next lemma, we denote by .Rcomp the set of all functions .g : R+ → C which are Riemann integrable on compact intervals of .R+ . Lemma 3 Let .c ∈ R and .p ≥ 1. The following assertions hold: p

p

(a) . c is a proper linear subspace of .Xc . p (ν) (b) Let .f ∈ c ∩ Rcomp , and for each .ν > 0, let . ν := {xj }j ∈Z be an admissible partition of .R+ such that . ν tends to zero as .ν → ∞. Suppose that .ξj

(ν)

arbitrary point

.

(ν) (ν) in .[xj −1 , xj ]

lim |f |pc ( ν )

ν→∞

(ν)

for each .j ∈ Z. Then

⎧ ⎫1/p ⎨ ⎬ (ν) p (ν) = lim = f Xcp , fc (ξj ) j ν→∞ ⎩ ⎭

(ν)

is an

(9)

j ∈Z

(ν)

where . j = log xj − log xj −1 . Proof p

(a) Obviously, .c is a linear manifold, and . 0 :=

 j e j ∈Z is an admissible

partition of .R+ . Therefore, if .f ∈ c , then by Lemma 2, p

Classical and Approximate Exponential Formula



 f (x)x c p dx = x

∞

.

0



ej

j −1 j ∈Z e

13

dx  ≤ (fc )∗ 0 (ej )p < ∞. x

|fc (x)|p

j ∈Z

p

p

This shows that .f ∈ c implies .f ∈ Xc . Since the function .g : R+ → C, defined by  g(x) =

.

p

e−cj if x = ej , j ∈ Z, 0 otherwise,

p

p

p

belongs to .Xc but not to .c ( 0 ), we see that .c is a proper subspace of .Xc . p (b) Let .ε > 0 be given. Since .f ∈ Xc , there exists an .N0 ∈ N such that  .

R+ \[e−N ,eN ]

|fc (x)|p

ε dx < x 3

(10)

|fc (x)|p d log x.

(11)

for all integers .N ≥ N0 . We note that  .

[e−N ,eN ]

|fc (x)|p

dx = x



eN e−N

Since .f ∈ Rcomp , the integral on the left-hand side exists as a Riemann integral. This allows us to conclude that the integral on the right-hand side exists as a Riemann–Stieltjes integral. We want to show that this integral can be approxi (ν) p (ν) mated arbitrarily close by a truncation of the series . j ∈Z fc (ξj ) j that (ν)

comprises those terms for which .ξj

∈ ]e−N , eN ].

First we consider the complementary part of the truncation called the remainder. By the assumption on the partitions . ν , there exists a .ν0 > 0 such that . ν ≤ 1 for all .ν ≥ ν0 . With . 0 as in part (a) of the proof, we can estimate the remainder for .ν ≥ ν0 as follows:  .

(ν) p (ν) fc (ξj ) j =





(ν) p (ν) fc (ξj ) j

k∈Z\]−N,N ] ξ (ν) ∈]ek−1 ,ek ]

(ν)

ξj ∈]e−N ,eN ]

j



≤

p (fc )∗ 0 (ek )

k∈Z\]−N,N ]

 (ν)

ξj ∈]ek−1 ,ek ]





≤ 1 + 2 ν

p (fc )∗ 0 (ek )

k∈Z\]−N,N ]

≤3

 k∈Z\]−N,N ]

p (fc )∗ 0 (ek ) .

(ν)

j

14

C. Bardaro et al.

By Lemma 2, the series in the last line converges. Hence, there exists an integer N1 ≥ N0 such that

.

 . (ν) ξj ∈]e−N ,eN ]

ε (ν) p (ν) fc (ξj ) j < 3

(12)

for .ν ≥ ν0 and all integers .N ≥ N1 . Now, for a fixed integer .N ≥ N1 , we denote by .mν the smallest integer j such that .ξj(ν) > e−N and by .nν the largest integer j such that .ξj(ν) ≤ eN . Then we can write  (ν) p (ν) . fc (ξj ) j = Sν + Tν , (ν)

ξj ∈]e−N ,eN ]

where p   (ν) Sν := fc (ξm(ν)ν ) log xm + N ν

.

n ν −1

+

j =mν +1

p   (ν) p (ν) (ν) N − log x ) fc (ξj ) j + fc (ξn(ν) nν −1 ν

and p  p    (ν) (ν) log x Tν := − fc (ξm(ν)ν ) N + log xmν −1 + fc (ξn(ν) ) − N . n ν ν

.

It is easily verified that (ν) 0 ≤ log xm + N < 2 ν ν

and

.

0 ≤ N − log xn(ν) ≤ 2 ν . ν −1

Since . ν → 0 as .ν → ∞, we see that .Sν is a Riemann–Stieltjes sum such that  .

lim Sν =

ν→∞

eN e−N

|fc (x)|p d log x.

As regards .Tν , we note that .

(ν) N + log xmν −1 ≤ ν

and

− N log xn(ν) ≤ ν , ν

which allows us to conclude that .limν→∞ Tν = 0. Thus, there exists a .ν1 ≥ ν0 such that

Classical and Approximate Exponential Formula

15

 N e  ε (ν) p (ν) p |fc (x)| d log x − . fc (ξj ) j < e−N 3 (ν) ξ ∈]e−N ,eN ]

(13)

j

for all .ν ≥ ν1 . Combining (10)–(13), we find that  ∞ p  dx (ν) (ν) p |fc (x)| − . fc (ξj ) j x 0 j ∈Z  ≤

R+ \[e−N ,eN ]

|fc (x)|p

dx + x

(ν) p (ν) fc (ξj ) j

 (ν)

ξj ∈]e−N ,eN ]

 N e p  (ν) (ν) |fc (x)|p d log x − + fc (ξj ) j e−N (ν) ξ ∈]e−N ,eN ] j

0, .p ∈]1, 2] and each .r ∈ N, we have p

r,p

Bc,σ ⊂ Wc

.

p

⊂ c ,

(14)

where the inclusions are strict. Proof For the first inclusion, see [6]. As to the second inclusion, it suffices to prove 1,p it for .r = 1. Let .f ∈ Wc , and let . = {xj }j ∈Z be any admissible partition of .R+ . Putting . j := log xj − log xj −1 , we have to show that

|f |pc ( ) =

.

⎧ ⎨ ⎩

j ∈Z

|fc (xj )|p j

⎫1/p ⎬ ⎭

< ∞.

Since f is continuous and .1/x is positive on .R+ , we see with the help of the meanvalue theorem for integrals that for each .j ∈ Z, there exists .ξj ∈ [xj −1 , xj ] such that

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C. Bardaro et al.



xj

|fc (x)|p

.

xj −1

dx = |fc (ξj )|p j x

and so f Xcp = fc Xp =

.

0

⎧ ⎨ ⎩

|fc (ξj )|p j

j ∈Z

⎫1/p ⎬ .

⎭

Thus, by Minkowski’s inequality, ⎧ ⎨

|f |pc ( ) ≤ f Xcp +

.

⎩

|fc (xj ) − fc (ξj )|p j

j ∈Z

⎫1/p ⎬ ⎭

(15)

.

1,p

Next we estimate the second term on the right-hand side. Since .f ∈ Wc , we conclude that the Mellin derivative .c f exists almost everywhere and 

xj

fc (xj ) − fc (ξj ) =

.

x c (c f )(x)

ξj

dx . x

Now by Hölder’s inequality, we find that  |fc (xj ) − fc (ξj )| ≤

xj

|x c (c f )(x)|

.

xj −1



xj

≤

dx x

|x (c f )(x)| c

xj −1



xj

=

|x (c f )(x)| c

xj −1

p dx

1/p 

x p dx

xj

xj −1

1/p

dx x

1/p

1/p

j

x

with .p being the conjugate exponent of .p. This shows that  |fc (xj ) − fc (ξj )| j ≤ p

.

xj xj −1

|x c (c f )(x)|p

dx 1+p/p

. x p

Combining this estimate with (15) and noting that .c f ∈ Xc , we obtain |f |pc ( ) ≤ f Xcp + c f Xcp < ∞.

.

1,p

This completes the proof of .Wc

p

⊂ c .

Classical and Approximate Exponential Formula

17

Concerning the proof that the inclusions (14) are strict, one may take .f (x) = 1 for .x ∈ I , and .f (x) = 0 elsewhere, I being a compact subinterval of .R+ . Then, p 1,p f obviously belongs to .c but not to .Wc , which shows that the second inclusion is strict. As to the first inclusion, we simply observe that a function belonging to p r,p .Bc,σ is infinitely differentiable (see [5]), whereas functions in .Wc need only be differentiable r-times.  

4.2 Hilbert Transforms The discrete Hilbert transform of a sequence .z = {zk }k∈Z is a sequence .H z = {(H z)j }j ∈Z defined by 

(H z)j :=

.

k∈Z\{j }

zk j −k

(j ∈ Z).

It is well known that this transform is a bounded linear operator in the sequence space .p for .1 < p < ∞ (see, e.g., [15, page 332] and [19, 21]). Thus, H zp ≤ Czp

(16)

.

with a constant .C > 0. We also need the related transformation .H : z → {(Hz)j }j ∈Z defined by 

(Hz)j :=

.

k∈Z\{j }

zk (j − k)2

(j ∈ Z).

It is a bounded linear operator in .p for .1 ≤ p ≤ ∞. Indeed, by a shift of the index k, we may write (Hz)j =

.

 zj +n n2

(j ∈ Z).

n∈Z\{0}

Thus, .Hz = n∈Z\{0} n−2 z(n) , where .z(n) := {zj +n }j ∈Z is a shifted form of the sequence .z. Clearly, .z(n) p = zp for all .n ∈ Z. Hence, by Minkowski’s inequality for infinite summations (see [17, Theorem 165]), we have Hzp ≤



.

n∈Z\{0}

∞

 1 1 (n) zp , z p = 2 2 n2 n n=1

18

C. Bardaro et al.

and so Hzp ≤

.

π3 zp . 3

(17)

Now we turn to the main objective of this section. For .c ∈ R and a given sequence y = {yk }k∈Z , we define

.

(HM y)(x) := x −c



.

k=k(x)

 ekc yk ekc yk = x −c −k log(xe ) log x − k

(x > 0),

k=k(x)

where .k(x) is the unique integer such that .x ∈ [ek(x)−1/2 , ek(x)+1/2 [ or, equivalently, .log x ∈ [k(x) − 1/2, k(x) + 1/2[. We call .HM y the mixed Mellin–Hilbert transform of the sequence y. Lemma 4 Let .1 < p < ∞. The mixed Mellin–Hilbert transform is a bounded p p linear operator from .c to .Xc . p

Proof Let .y = {yk }k∈Z be a sequence in .c . We can write .

1 1 1 = + log(xe−k ) k(x) − k (k(x) − k)2



 log(ek(x) x −1 )(k(x) − k) . log(xe−k )

Now, taking into account of the definition of .k(x), we note that the term in square brackets is bounded. Hence, there exists a constant .C > 0 such that    kc kc ekc yk −c e yk −c e yk −c ≤ + C x x x log(xe−k ) k(x) − k (k(x) − k)2

.

k=k(x)

k=k(x)

≤x

k=k(x)

  ekc |yk | ekc yk −c + Cx . k(x) − k (k(x) − k)2

−c

k=k(x)

k=k(x)

Taking the norm in .Xc and noting that .k(x) = j when .x ∈ [ej −1/2 , ej +1/2 [, we have p

⎧ ⎨ .

⎩

⎫ p ⎬1/p  kc y dx e k log(xe−k ) x ⎭

∞ 0

k=k(x)

≤

⎧ ⎨ ⎩

∞ 0



 k=k(x)

⎫ p ⎬1/p  ekc yk ekc |yk | dx +C k(x) − k x ⎭ (k(x) − k)2 k=k(x)

Classical and Approximate Exponential Formula

=

⎧ ∞  ⎨  ⎩

19

⎫1/p

 kc |y | p dx ⎬ kc y  e e k k +C k(x) − k x ⎭ (k(x) − k)2

ej +1/2

j −1/2 j =−∞ e

k=k(x)

k=k(x)

⎧ ⎫1/p ∞  kc ⎨  kc |y | p ⎬  e e y k k +C . = ⎩ j − k (j − k)2 ⎭ j =−∞

k=j

k=j

Now, by Minkowski’s inequality, ⎧ ⎨ .

⎩

⎫ p ⎬1/p  kc y e dx k log(xe−k ) x ⎭

∞ 0

k=k(x)

    ekc yk  ≤ j −k

   

j ∈Z p

k=j

    ekc |yk |  + C (j − k)2 k=j

   . p

j ∈Z 

Setting .z = {zk }k∈Z with .zk := ekc yk , .k ∈ Z, we have .zp = ypc , and so p .z ∈  . Using the boundedness of the discrete Hilbert transform (16) and (17), we finally obtain HM yXcp ≤ C1 ypc

.

 

with a constant .C1 > 0, as was to be shown.

Using Lemma 4, we can prove the boundedness of certain exponential sampling operators acting on Mellin–Lebesgue sequence spaces. For any sequence .{yk }k∈Z , .c > 0 and .T > 0, we formally define the exponential series  .(ST y)(x) := yk linc/T (e−k x T ) (x > 0). k∈Z

We have the following result. Theorem 2 Let .c > 0, .T > 0, and .1 < p < ∞. Then .ST is a bounded linear p p operator from .c/T to .Xc , the bound being of the form .C/T 1/p , where C is a positive constant not depending on T . Proof We first prove the theorem for .T = 1. Let .y = {yk }k∈Z be a sequence of p complex numbers belonging to the space .c . Then (S1 y)(x) :=



.

k∈Z

yk linc (e−k x) = x −c

 k∈Z

ekc yk sinc(log x − k).

20

C. Bardaro et al.

For any .x ∈ R+ , let .k(x) be the unique integer such that .log x ∈ [k(x)−1/2, k(x)+ 1/2[ or .x ∈ [ek(x)−1/2 , ek(x)+1/2 [. With this notation, we find that |(S1 y)(x)|

.

 sin(π(log x − k(x))) sin(π log x) −c k kc ck(x) +e =x yk(x) (−1) yk e π(log x − k(x)) π(log x − k) k=k(x)  y k −c k kc + x −c eck(x) |yk(x) | =: I1 + I2 . ≤x (−1) e π(log x − k) k=k(x) p

By Lemma 4, the .Xc -norm of the first term .I1 may be estimated as I1 Xcp ≤ C1 ypc

.

with a constant .C1 > 0. For the second term, we have p .I2  p Xc

=



ej +1/2

j −1/2 j ∈Z e

p ck(x)p dx

|yk(x) | e

x

=



ej +1/2

j −1/2 j ∈Z e

|yj |p ecpj

dx p = yp . c x

Thus, we obtain S1 yXcp ≤ Cypc ,

(18)

.

where .C = C1 + 1. This is the assertion for .T = 1. When .T > 0 but .T = 1, a suitable substitution allows us to deduce from (18) that ST yXcp ≤

.

C T 1/p

yp . c/T

 

This completes the proof. As a consequence of Theorem 2, we obtain: p

Corollary 1 Let .c > 0, .T > 0, and .1 < p < ∞. Suppose that .f ∈ c . Then ST f Xcp ≤ C|f |pc ( T )

.

(19)

with a constant .C > 0 independent of T and . T := {ek/T }k∈Z . Proof The result follows by applying Theorem 2 to the sequence .y = {yk }k∈Z = {f (ek/T )}k∈Z .  

Classical and Approximate Exponential Formula

21

Now we are ready for the main result of this section. p

Theorem 3 Let .1 < p ≤ 2 and .c ∈ R. Suppose that .f ∈ c ∩ Rcomp . Then Theorem A implies .

lim ST f − f Xcp = 0.

(20)

T →∞

Proof For .c ∈ R and .ρ > 0, let .Fρc be the Mellin–Fejér kernel, defined by c .Fρ (x)

 x −c x iρ/2 − x −iρ/2 2 := − 2πρ log x

(x ∈ R+ \ {1})

with the continuous extension .Fρc (1) = ρ/(2π ). The function .Fρc belongs to the 1 (for details, see [12]). Furthermore, for .f ∈ X p , Mellin–Paley–Wiener space .Bc, c ρ p the Mellin convolution .f ∗ Fρc belongs to .Bc, ρ and .

lim f ∗ Fρc − f Xcp = 0

ρ→∞

p

(see the corresponding chapters in [1]). Hence, for a fixed .f ∈ c ∩ Rcomp and p c .ε > 0, there exist a .T0 > 0 and a function .gT ∈ B c,π T , in fact .gT := f ∗ Fπ T , such that .f − gT Xcp < ε for all .T ≥ T0 . Moreover, by Lemma 3, there exists a .T1 ≥ T0 such that |f − gT |pc ( T ) < f − gT Xcp + ε

.

for all .T ≥ T1 , where here . T is the partition specified in Corollary 1. Since .gT ∈ p p Bc,π T , Proposition 1 implies that .gT ∈ c . Thus, by Corollary 1 and Theorem A, we have ST f − f Xcp ≤ ST (f − gT )Xcp + ST gT − gT Xcp + gT − f Xcp

.

≤ C|f − gT |pc ( T ) + gT − f Xcp ≤ (2C + 1)ε for all .T ≥ T1 , and so the assertion follows.

 

Acknowledgments Carlo Bardaro and Ilaria Mantellini have been partially supported by the “Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA)” of the “Istituto Nazionale di Alta Matematica (INDAM)” as well as by the project “Ricerca di Base 2019 of the University of Perugia (title: Integrazione, Approssimazione Analisi non Lineare e loro Applicazioni).”

22

C. Bardaro et al.

References 1. C. Bardaro, P.L. Butzer, I. Mantellini, G. Schmeisser, An Introduction to Mellin Analysis and Its Applications. Book in preparation 2. C. Bardaro, P.L. Butzer, R.L. Stens, G. Vinti, Approximation error of the Whittaker cardinal series in term of an averaged modulus of smoothness covering discontinous signals. J. Math. Anal. Appl. 316, 269–306 (2006) 3. C. Bardaro, P.L. Butzer, I. Mantellini, The exponential sampling theorem of signal analysis and the reproducing kernel formula in the Mellin transform setting. Sampl. Theory Signal Image Process. 13(1), 35–66 (2014) 4. C. Bardaro, P.L. Butzer, I. Mantellini, The Mellin-Parseval formula and its interconnections with the exponential sampling theorem of optical physics. Integral Transforms Spec. Funct. 27(1), 17–29 (2016) 5. C. Bardaro, P.L. Butzer, I. Mantellini, G. Schmeisser, On the Paley-Wiener theorem in the Mellin transform setting. J. Approx. Theory 207, 60–75 (2016) 6. C. Bardaro, P.L. Butzer, I. Mantellini, G. Schmeisser, Mellin analysis and its basic associated metric–applications to sampling theory. Analysis Math. 42(4), 297–321 (2016) 7. C. Bardaro, P.L. Butzer, I. Mantellini, G. Schmeisser, A fresh approach to the Paley-Wiener theorem for Mellin transforms and the Mellin-Hardy spaces. Math. Nachr. 290, 2759–2774 (2017) 8. C. Bardaro, I. Mantellini, G. Schmeisser, Exponential sampling series: convergence in MellinLebesgue spaces. Results Math. 74, 119 (2019) 9. M. Bertero, E.R. Pike, Exponential sampling method for Laplace and other dilationally invariant transforms: I. Singular-system analysis. II. Examples in photon correlation spectroscopy and Fraunhofer diffraction. Inverse Probl. 7, 1–20, 21–41 (1991) 10. R.P. Boas Jr., Entire Functions (Academic, New York, 1954) 11. J.L. Brown Jr., On the error in reconstructing a non-bandlimited function by means of the bandpass sampling theorem. J. Math. Anal. Appl. 18, 75–84 (1967); Erratum: J. Math. Anal. Appl. 21, 699 (1968) 12. P.L. Butzer, S. Jansche, A direct approach to the Mellin transform. J. Fourier Anal. Appl. 3, 325–375 (1997) 13. P.L. Butzer, S. Jansche, The exponential sampling theorem of signal analysis. Atti Sem. Mat. Fis. Univ. Modena Suppl. 46, 99–122 (1998); Special issue dedicated to Professor Calogero Vinti 14. P.L. Butzer, S. Jansche, A self-contained approach to Mellin transform analysis for square integrable functions; applications. Integr. Transf. Spec. Funct. 8(3–4), 175–198 (1999) 15. P.L. Butzer, R.J. Nessel, Fourier Analysis and Approximation, vol. I (Academic, New York, 1971) 16. P.L. Butzer, J.R. Higgins, R.L. Stens, Classical and approximate sampling theorems; studies in the Lp (R) and in the uniform norm. J. Approx. Theory 137, 250–263 (2005) 17. G.H. Hardy, J.E. Littlewood, G. Pólya, Inequalities (Cambridge University Press, Cambridge, 1934) 18. J.R. Higgins, Sampling Theory in Fourier and Signal Analysis: Foundations (Clarendon Press, Oxford, 1996) 19. M.J. Marsden, F.B. Richards, S.D. Riemenschneider, Cardinal spline interpolation operators on p data. Indiana Univ. Math. J. 24(7), 677–689 (1975) 20. N. Ostrowsky, D. Sornette, P. Parker, E.R. Pike, Exponential sampling method for light scattering polydispersity analysis. Opt. Acta 28, 1059–1070 (1994) 21. M. Riesz, Sur les fonctions conjugées. Math. Z. 27, 218–244 (1927) 22. A. Zygmund, Trigonometrical Series (Dover Publs, New York, 1955)

Asymptotic Theorems for Durrmeyer Sampling Operators with Respect to the Lp -Norm .

Danilo Costarelli and Gianluca Vinti

To Prof. J. Rowland Higgins in memory of his profound scientific influence and great humanity

1 Introduction It is well-known that the general theory of the cardinal sampling series arises from the classical sampling theorem of Wittaker-Kotel’nikov-Shannon (see, e.g., [21]), originally established in the first half of the 1900s. The above topic has covered a central role in signal analysis during the second half of the 1900s; such period has seen the publication of many contributions by several important authors. One of the more impressive has been J. Rowland Higgins, who gave deep results on the foundations of the sampling theory; among his contributions to the theory, we can find [8, 27–29]. Other important contributions to the above subject can also be found in [2, 32]. One of the possible generalizations of the theory of cardinal series is given by the so-called approximate sampling, which include the study of both the generalized and Kantorovich versions of sampling-type operators; see, e.g., [7, 14, 16, 30, 31, 34]. Recently, the above reconstruction tools have been successfully applied in concrete application problems involving image reconstruction and enhancement [18]. A more recent approach on the study of sampling-type operators resides in the Durrmeyer sampling-type series, in which the reconstruction power of the singular integrals and of the generalized sampling series [33, 35] are merged in a new constructive approximation algorithm, which is suitable for the reconstruction of both continuous and not necessarily continuous signals. On the other hand, Durrmeyer sampling operators represent also a generalization of both the wellknown Kantorovich and generalized sampling series.

D. Costarelli · G. Vinti () University of Perugia, Perugia, Italy e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. D. Casey et al. (eds.), Sampling, Approximation, and Signal Analysis, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-031-41130-4_2

23

24

D. Costarelli and G. Vinti

The Durrmeyer approach resides to the 1967,   where it was applied to Bernstein polynomials replacing the sample value .f nk by a convolution-type integral in which the same generating polynomial kernels .pn,k appear, namely:

.

(Dn f ) (x) := (n + 1)

n  k=0



1

pn,k (x)

pn,k (u)f (u) du, x ∈ [0, 1].

0

The literature about this family of operators and its generalizations is very wide; we quote here, e.g., [9, 22–24, 26]. From the sampling theory point of view, the Durrmeyer sampling operators have been considered in a couple of papers [3, 4], in which Voronovskaja-type theorems have been established for both functions of one and several variables, with respect to usual .sup norm. Actually, the above operators and some related convergence theorems (proposed in a more abstract form) can also be deduced from some results established in [1, 39–42]. Recently, a modular approximation theorem for the above family of Durrmeyer sampling operators has been proved in the general context of Orlicz spaces [19]; here, the .Lp -convergence (and other) can be also deduced as a particular case. Motivated by such results, in the present chapter, we establish high-order asymptotic theorems for the Durrmeyer sampling series with respect to the usual p p .L -norm in certain subspaces of .L , .1 ≤ p ≤ +∞. First, a Voronovskajap type theorem with respect to the .L -norm, .1 ≤ p ≤ +∞, is proved in case of functions belonging to the space of r-th differentiable functions with compact support. The key role in the proof of the above theorem is played by the well-known Vitali convergence theorem. Further, assuming some additional assumptions on the kernels of the above sampling-type operators, the previous result is extended to the case of functions belonging to Sobolev spaces.

2 Preliminaries and Auxiliary Results Let .Lp (R), .1 ≤ p < +∞, be the space of all Lebesgue measurable functions ∞ .f : R → R, for which the usual norm .‖f ‖p is finite; .L (R) is the space of the measurable functions with finite essential sup norm .‖ · ‖∞ . Here, by .C(R), we refer to the space of all uniformly continuous and bounded functions on .R endowed with the usual sup norm .‖·‖∞ .1 Moreover, we denote by .C r (R), .r ∈ N+ , the subspace of (i) , .i = 1, . . . , r, exist and belong to .C(R); .C r (R) .C(R) such that the derivatives .f c r is the subset of .C (R) of functions with compact support.

1 Notice

that we use the same notation to denote the sup and the essential sup norms, as usual.

Asymptotic Theorems for Durrmeyer Sampling Operators with Respect to the .Lp -Norm

25

Further, we also recall the definition of Sobolev spaces: W r,p (R) :=



.

 f ∈ Lp (R) : f (r−1) ∈ AC(R), and f (r) ∈ Lp (R) , 1

≤ p ≤ +∞, where .AC(R) denotes the space of absolutely continuous functions .f : R → R and f (j ) is the j -th derivative of f , .j = 1, . . . , r. Moreover, we denote by:

.

f(v) :=



.

R

f (u) e−i u v du,

(1)

v ∈ R, the Fourier transform of .f ∈ L1 (R). Now, in order to recall the definition of the families of sampling-type operators that will be studied in this chapter, and in order to establish the main approximation theorems, we introduce the following notation. Note that we here summarize all the required assumptions; when in what follows we refer to the case .p = +∞, the assumptions assumed deal with functions belonging to .C(R) or .L∞ (R) (or their subspaces) using the .‖ · ‖∞ norm, while when we refer to the case .1 ≤ p < +∞, we are in the case of functions belonging to .Lp (R) (or their subspaces) using the .‖ · ‖p norm. From now on, we will say that a function .χ : R → R is a discrete kernel, if it satisfies the following assumptions:

.

(χ 1) .(χ 2) .

χ ∈ L1 (R) is bounded on .R. The discrete algebraic moment of order 0:

.



m0 (χ , u) :=

.

χ (u − k) = 1,

u ∈ R.

(2)

k∈Z

(χ 3)

.

For a positive integer r, there exists .β > r + 1, if .p = +∞, or .β > rp + 1, if .1 ≤ p < +∞, such that: χ (u) = O(|u|−β ),

.

as

|u| → +∞.

Remark 1 Note that assumption .(χ 3) implies that (see Lemma 2.1 of [15]): Mν (χ ) := sup



.

|u − k|ν |χ (u − k)| < +∞,

(3)

u∈R k∈Z

for every .0 ≤ ν < β − 1. Moreover, .(χ 3) also implies that, for every .γ > 0: .

lim

w→+∞

 |wx−k|>wγ

|wx − k|ν · |χ (wx − k)| = 0,

(4)

26

D. Costarelli and G. Vinti

uniformly with respect to .x ∈ R, where .0 ≤ ν < β −1. Obviously, if .χ has compact support, it turns out that assumption .(χ 3) is satisfied for every .β > 0 and then (3) and (4) hold for every .ν ≥ 0. Now, we also introduce the following notations that will be useful later. We define, for any function .g : R → R, the so-called continuous algebraic and absolute continuous moments of order .ν ∈ N, respectively, by the following integral:  m ν (g) :=

.

R

uν g(u) du,

ν (g) := M

 R

|u|ν |g(u)| du

ν (g) < Note that, if g satisfies condition .(χ 3), it turns out that .m ν (g) < +∞, .M +∞, for every .0 ≤ ν < β − 1, with .ν ∈ N. Now, we also introduce a new family of kernel functions. From now on, we will say that a function .Ф : R → R is a continuous kernel, if it satisfies the following assumptions:2 Ф ∈ L1 (R).

.m 0 (Ф) = R Ф(u) du = 1. For a positive integer r, there exists .β > r + 1, such that:

(Ф1) .(Ф2) .(Ф3) .

.

Ф(u) = O(|u|−β ),

.

as

|u| → +∞.

Now, given a discrete kernel .χ and a continuous kernel .Ф, we are able to recall the definition of the so-called Durrmeyer sampling operators, which are the following: Ф,χ (Dw f )(x) :=

.

  w f (y) Ф(wy − k) dy χ (wx − k), R

k∈Z

x ∈ R,

w > 0. Note that, if we recall the definition of the generalized sampling operators, namely:

.

(Gχw f )(x) :=



.

k∈Z

f

k w

 χ (wx − k),

x ∈ R,

w > 0,

where .f : R → R is any bounded function (see, e.g., [7, 35]), and we also recall the notion of singular integral (see, e.g., [5, 36]):

2 We stress that .Ф is not necessarily a continuous function. The notion of continuous kernel refers only to functions .Ф satisfying conditions .(Ф1), .(Ф2), and .(Ф3).

Asymptotic Theorems for Durrmeyer Sampling Operators with Respect to the .Lp -Norm

27

 (IwФ f )(x) := (f ∗ Фw )(x) = w

.

 = w

R

R

f (x − y) Ф(wy) dy

f (u) Ф(w[x − u]) du,

x ∈ R, .w > 0, where .f ∈ Lp (R), .1 ≤ p ≤ +∞, .Фw (u) := w Ф(wu), .u ∈ R, we get:

.

Ф,χ (Dw f )(x) = [Gχw (IwФ(−·) f )](x),

x ∈ R.

.

We recall that the above symbol “.∗” refers to the usual convolution product. Remark 2 Note that, if we give a distributional interpretation of the operators Ф,χ Dw , assuming .Ф(u) = δ(u), .u ∈ R, is the well-known Dirac delta distribution, we can get:

.

δ,χ Dw f = Gχw f,

.

w > 0,

for suitable functions f (see, e.g., [19]). Furthermore, if we consider .Ф(u) = 1[0,1] (u), where .1[0,1] denotes the characteristic function of the interval .[0, 1], the 1 ,χ operator .Dw[0,1] f reduces to: 1[0,1] ,χ

(Dw

.

f )(x) :=



  w

k∈Z

(k+1)/w

 f (u) du χ (wx − k),

x ∈ R,

w > 0,

k/w

that are known in the literature as the sampling Kantorovich operators; see, e.g., [14, 16]. Now, in order to establish asymptotic type theorems in the .Lp -setting, we need the following condition which links a discrete kernel .χ with a continuous kernel .Ф. From now on, we always consider a pair of discrete and continuous kernels .χ and .Ф, respectively, satisfying conditions .(χ 3) and .(Ф3) with the same parameter + .r ∈ N and such that:  χ . (k − x)j χ (x − k) =: Aj ∈ R, x ∈ R, j = 1, 2, . . . , r, (5) k∈Z

and j   j .

𝓁=0

𝓁

χ

m 𝓁 (Ф) Aj −𝓁 = 0,

j = 1, 2, . . . , r − 1,

r   r 𝓁=0

𝓁

χ

m 𝓁 (Ф) Ar−𝓁 /= 0. (6)

28

D. Costarelli and G. Vinti

Now, we recall the following lemma, from which we can deduce some equivalent conditions for (2) of .(χ 2) and .(5), and that can be proved as a consequence of the well-known Poisson summation formula (see [5], p. 202). Lemma 1 Let .χ : R → R be a given continuous function satisfying .(χ 1). Assuming in addition that the function .g(u) := −(i u)j χ (u), .j ∈ N (.u ∈ R and i denotes the complex unit) belongs to .L1 (R), we have that:  χ (k − x)j χ (x − k) = Aj ∈ R,

.

x ∈ R,

k∈Z

if and only if  χ  (2π k) =

.

(j )

Aj , 0,

k = 0, k ∈ Z \ {0} ,

(7) χ

where .χ (j ) denotes the j -th derivative of .χ . It turns out that .Aj = (−i)j Aj . Clearly, (2) of .(χ 2) and .(5) are equivalent to the conditions given in Lemma 1 for j = 0 with .A0 = 1 and .j = 1, . . . , r with suitable constants .Aj , respectively. Condition (7) is known as Strang-Fix-type condition; see [38]. In order to construct examples of discrete and integral kernels satisfying conditions (5) and (6), one can consider suitable combinations (for some examples, see [17]) of the kernels recalled below. Examples of (discrete and continuous) kernels satisfying the above assumptions (for the details of the proofs, see [15]) are, e.g., the central B-splines of order N (see, e.g., [6]), defined by:

.

 N −1 N  N 1 i N +x−i , (−1) .BN (x) := i 2 (N − 1)! +

x ∈ R,

i=0

where .(·)+ denotes the positive part. It is well-known that .BN ∈ CcN −2 (R), i.e., they are examples of duration-limited kernels with .supp BN ⊂ [−N/2, N/2]; hence, condition .(χ 3) is satisfied for every .β > 0. Further, we also recall the Jackson-type kernels [9], defined by: JN (x) := cN sinc2N



.

x  , 2N π α

x ∈ R,

with .N ∈ N, .α ≥ 1, and .cN is a non-zero normalization coefficient, given by:  cN :=

.

R

sinc

2N



−1 u  du . 2N π α

Asymptotic Theorems for Durrmeyer Sampling Operators with Respect to the .Lp -Norm

29

Here, the sinc function .sinc(x) is defined as .sin(π x)/π x, if .x /= 0, and 1 if .x = 0. Finally, we can mention the Bochner-Riesz kernels [37], defined by: 2η bη (x) := √ 𝚪(η + 1) |x|−η−1/2 Jη+1/2 (|x|), 2π

.

x ∈ R,

with .η > 0, and where .Jλ is the Bessel function of order .λ [10] and .𝚪 is the usual Euler gamma function. The functions .JN and .bη are examples of band-limited (hence belonging to .C ∞ (R)) kernels. Other examples of kernels can be found, e.g., in [5, 11, 12].

3 Asymptotic Type Theorems In this section, high-order asymptotic type theorems with respect to the .Lp -norm, .1 ≤ p ≤ +∞, are established by Durrmeyer sampling operators. We can prove what follows. Theorem 1 Let .χ and .Ф be a pair of discrete and continuous kernels, respectively (assumed as in Sect. 2). Then, for any .f ∈ C r (R), we have:  r    Ф,χ  f (r)  r χ Dw f − f − . lim ‖w m 𝓁 (Ф) Ar−𝓁 ‖∞ = 0. w→+∞ r! 𝓁 r

𝓁=0

Further, assuming in addition that .Ф has compact support, for .f ∈ Ccr (R) and .1 ≤ p < +∞, we also have:  r    f (r)   Ф,χ r χ . lim ‖w Dw f − f − m 𝓁 (Ф) Ar−𝓁 ‖p = 0. w→+∞ r! 𝓁 r

𝓁=0

Proof Using the Taylor formula with Lagrange remainder until the order r: f (u) =

r−1 (j )  f (x)

.

j =0

j!

(u−x)j +

1 (r) f (θu,x ) (u−x)r =: Tr−1 (u; x) + Rr (u; x), r! (8)

u, .x ∈ R, where .θu,x is a suitable value between u and x. By conditions .(χ 2) and (Ф2), the binomial Newton formula, the change of variable .y = wu − k, the Fubini theorem, and (6), one can obtain:

.

30

D. Costarelli and G. Vinti

   Ф,χ  w Ф(wu − k) [Tr−1 (u; x) + Rr (u; x)] du χ (wx − k) . Dw f (x) = k∈Z

=

R

r−1 (j )  f (x) k∈Z j =0

j! j −𝓁

⎡ ⎣w

 R

Ф(wu − k)

 j   j k 𝓁 u− 𝓁 w 𝓁=0



k du χ (wx − k) −x w   w Ф(wu − k) Rr (u; x) du χ (wx − k) + ×

k∈Z

=

R

  j  r−1 (j )  f (x)  j  𝓁 w Ф(wu − k) − k) du (wu 𝓁 wj j ! R j =0

𝓁=0

k∈Z

j −𝓁

χ (wx − k) × (k − wx)    + w Ф(wu − k) Rr (u; x) du χ (wx − k) k∈Z

R

⎧ ⎫ j  r−1 (j ) ⎬  f (x) ⎨ j χ m = f (x) + (Ф)A 𝓁 j −𝓁 ⎭ j !w j ⎩ 𝓁 j =1

𝓁=0

  w Ф(wu − k)Rr (u; x)du χ (wx − k) + k∈Z

R

  1  (r) r w Ф(wu − k) f (θu,x ) (u − x) du = f (x) + r! R k∈Z

× χ (wx − k). On the other hand, using .(Ф2) and .(χ 2) again, together with the Fubini theorem, we have:   r  f (r) (x)  r χ m 𝓁 (Ф) Ar−𝓁 . 𝓁 r! 𝓁=0

   r   f (r) (x)  r Ф(z) z𝓁 dz (k − wx)r−𝓁 χ (wx − k) . = 𝓁 r! R 𝓁=0

k∈Z

  r  f (r) (x)  r  Ф(z) z𝓁 (k − wx)r−𝓁 dz χ (wx − k) . = 𝓁 r! R 

𝓁=0

k∈Z

Asymptotic Theorems for Durrmeyer Sampling Operators with Respect to the .Lp -Norm

31

    r   f (r) (x)  r 𝓁 r−𝓁 . = Ф(z) dz χ (wx − k) ; z (k − wx) r! 𝓁 R 𝓁=0

k∈Z

thus, by the change of variables .z = wu − k, and the binomial theorem, we easily obtain:  r   f (r) (x)  r χ . m 𝓁 (Ф) Ar−𝓁 r! 𝓁 𝓁=0

  f (r) (x)  . = wФ(wu − k) r! R k∈Z

  r   r 𝓁 r−𝓁 × (wu − k) (k − wx) du χ (wx − k) 𝓁 𝓁=0

   f (r) (x)  r = w Ф(wu − k) (wu − wx) du χ (wx − k) r! R

.

k∈Z

    f (r) (x)  r = w Ф(wu − k) (u − x) du χ (wx − k) . w r! R r

.

(9)

k∈Z

Then, we finally have:  r    Ф,χ  f (r) (x)  r χ w (D. w f )(x) − f (x) − m 𝓁 (Ф) Ar−𝓁 r! 𝓁 r

𝓁=0

    wr  (r) (r) r . = w Ф(wu − k) f (θu,x ) − f (x) (u − x) du χ (wx − k). r! R k∈Z (10) All the above computations show how the “limit term” arise. Now, if .p = +∞ and using (10), the proof immediately follows by the results established in [4]. Note that, here in order to establish the proof, assumption .(Ф3) must be used. Now let .1 ≤ p < +∞ be fixed, .f ∈ Ccr (R), and suppose in addition that .Ф has compact support. We want to prove that:  .

lim

w→+∞ R

 r    Ф,χ  f (r) (x)  r χ m 𝓁 (Ф) Ar−𝓁 w (Dw f )(x) − f (x) − r! 𝓁 r

𝓁=0

p

32

D. Costarelli and G. Vinti

dx = 0, exploiting the Vitali convergence theorem. By the first part of this theorem, it immediately follows that:   r   Ф,χ  f (r)  r p χ m 𝓁 (Ф) Ar−𝓁 ‖∞ = 0. Dw f − f − . lim ‖w w→+∞ r! 𝓁 r

𝓁=0

Now let .ε > 0 be fixed and .γ > 0 such that .supp f ⊂ [−γ , +γ ]. For any .M > γ + 1, and using Jensen inequality (see, e.g., [13]), we can write what follows:   r   Ф,χ  f (r) (x)  r χ w (Dw f )(x) − f (x) − m 𝓁 (Ф) Ar−𝓁 𝓁 r!

 JM :=

r

.

|x|>M

dx

𝓁=0

 .

=

 .

p

≤ M0 (χ )p−1

|x|>M

w rp

Ф,χ w r (Dw f )(x)

|x|>M



 w

γ

−γ

k∈Z

p

dx

|f (u)|p |Ф(wu − k)| du |χ (wx − k)| dx.

Now let .T > 0 such that .supp Ф ⊂ [−T , T ]. Hence, for every .w > T and the change of variable .t = wx − k:  JM ≤ M0 (χ )p−1

|x|>M

.

.



w rp

.

 w

|k|≤w(γ +1) p

≤ M0 (χ )p−1 ‖f ‖∞ ‖Ф‖1

γ −γ



|f (u)|p |Ф(wu−k)|du |χ (wx −k)|dx  w rp |χ (wx − k)| dx

|k|≤w(γ +1) |x|>M



p

≤ 2 M0 (χ )p−1 ‖f ‖∞ ‖Ф‖1 [w(γ + 1) + 1] w rp−1

|t|>w(M−γ −1)

|χ (t)| dt < ε, (11)

for sufficiently large .w > T , since  .

|t|>w(M−γ −1)

|χ (t)| dt = o(w −rp ),

as

w → +∞,

by condition .(χ 3) with .β > rp+1. Therefore, we just proved that in correspondence of .ε, there exists the interval .Eε := [−M, M] such that, for any measurable set F , with .F ∩ Eε = ∅, inequality (11) holds. Now, for any fixed measurable set .B ⊂ R, with:

Asymptotic Theorems for Durrmeyer Sampling Operators with Respect to the .Lp -Norm



 2rp ‖f (r) ‖∞ 0 (Ф) Mr (χ ) p .|B| < ε Mr (Ф) M0 (χ ) + M p (r!)

33

−1 ,

using conditions .(χ 2) and .(Ф2), and recalling (10), we finally get:  w

.

r

B



Ф,χ (Dw f )(x) − f (x)



 r   f (r) (x)  r χ − m 𝓁 (Ф) Ar−𝓁 r! 𝓁

p

dx

𝓁=0

    wr  = w Ф(wu − k) f (r) (θu,x ) − f (r) (x) B r! k∈Z R p ×(u − x)r du χ (wx − k) dx 

≤

2 ‖f (r) ‖∞ (r!)p

  w |Ф(wu − k)||wu − k + k − wx|r du |



R

B k∈Z

×χ (wx − k)||p dx      2rp ‖f (r) ‖∞ r r ≤ du |Ф(wu − k)| |wu − k| + |k − wx| w (r!)p B R k∈Z

p

×|χ (wx − k)| dx 2rp ‖f (r) ‖∞ ≤ (r!)p



  r w |Ф(wu − k)| |wu − k| du |χ (wx − k)|

B

k∈Z

R

  w |Ф(wu − k)| du |k − wx|r |χ (wx − k)| + R

k∈Z

2rp ‖f (r) ‖∞ ≤ (r!)p =



0 (Ф) Mr (χ ) r (Ф) M0 (χ ) + M M

p



p

dx

dx

B

 2rp ‖f (r) ‖∞ 0 (Ф) Mr (χ ) p |B| < ε, Mr (Ф) M0 (χ ) + M (r!)p

for every .w > 0 sufficiently large, by Remark 1. This shows that the integrals  w

.

(·)

r



[Gχw



IwФ f

 r     f (r) (x)  r χ ](x) − f (x) − m 𝓁 (Ф) Ar−𝓁 r! 𝓁

p

dx

𝓁=0

are equi-absolutely continuous. This completes the proof.

⨆ ⨅

34

D. Costarelli and G. Vinti

Asymptotic expansions can also be proved for functions in Sobolev spaces, assuming that .χ has compact support. Theorem 2 Let .χ and .Ф be a pair of discrete and continuous kernels, respectively. We assume in addition that both .χ and .Ф have compact support. Then, for every .f ∈ W r,p (R), .1 ≤ p ≤ +∞ and .r ∈ N+ being the parameter of both conditions .(5) and .(6), there holds:   r   Ф,χ  f (r)  r χ m 𝓁 (Ф) Ar−𝓁 ‖p = 0. Dw f − f − . lim ‖w w→+∞ 𝓁 r! r

𝓁=0

Proof Let .f ∈ W r,p (R), .1 ≤ p ≤ +∞, be fixed. It is well-known that f can be

u (r) (t) (u − written by the Taylor formula (8) with integral remainder .Rr (u; x) = x f(r−1)! r−1 dt (see [20] p. 37). Hence, proceeding as in the proof of Theorem 1, i.e., t) Ф,χ replacing the above Taylor formula in the definition of the operators .Dw f , and noting that, by (9):

.

  r  f (r) (x)  r χ m 𝓁 (Ф) Ar−𝓁 𝓁 r! 𝓁=0

.

   u ! f (r) (x) r  w Ф(wu − k) w (u − t)r−1 dt du χ (wx − k) (r − 1)! R x

=

k∈Z

we immediately obtain:   r  (r) (x)     r f χ r Ф,χ m 𝓁 (Ф) Ar−𝓁 Dw .Jw (x) := w f (x) − f (x) − 𝓁 r! 𝓁=0

   u " ! # wr  (r) (r) r−1 w Ф(wu − k) f (t) − f (x) (u − t) dt du . = (r − 1)! R x k∈Z

× χ (wx − k),

(12)

where the above series can be considered only for .k ∈ Z such that .|wx − k| ≤ T , since .supp χ ⊂ [−T , T ], .T > 0. Now, recalling that also .supp Ф ⊂ [−γ , γ ], for a suitable .γ > 0, for every fixed integer k with .|wx − k| ≤ T , the above convolutiontype integrals can be estimated as follows: 

 w r+1

.

R

|Ф(wu − k)| x

u

|f (r) (t) − f (r) (x)| |u − t|r−1 dt du

Asymptotic Theorems for Durrmeyer Sampling Operators with Respect to the .Lp -Norm

 .

≤w



r+1 R

|Ф(wu − k)| |u − x|

= w r+1

 |wu−k|≤γ

≤ wr+1

≤ w r+1

|f (r) (z + x) − f (r) (x)|dz du

0

 |wu−k|≤γ

|Ф(wu−k)||u−x|r−1 du

γ |z|≤ w + wk −x

 .

u−x

|Ф(wu − k)||u − x|r−1

 .

|f (r) (t) − f (r) (x)| dt du

x

 .

u

r−1

35

|f (r) (z+x)−f (r) (x)|dz



|wu−k|≤γ

|Ф(wu − k)| |u − x|r−1 du

+γ ) |z|≤ (T w

|f (r) (z + x) − f (r) (x)|dz.

Thus, for any .w > 0, we have:    1 (r) (r) . |Jw (x)| ≤ |f (z + x) − f (x)| dz × w +γ ) (r − 1)! |z|≤ (T w 



|Ф(wu − k)||u − x|r−1 du |χ (wx − k)|

 wr

.

|wu−k|≤γ

|wx−k|≤T

   1 . ≤ |f (r) (z + x) − f (r) (x)| dz × w +γ ) (r − 1)! |z|≤ (T w   r−1 du |χ (wx − k)| w |Ф(wu − k)||wu − k + k − wx|

 .

R

|wx−k|≤T

   2r−2 (r) (r) . ≤ |f (z + x) − f (x)| dz × w +γ ) (r − 1)! |z|≤ (T w  .

  w |Ф(wu − k)||wu − k|r−1 du R

|wx−k|≤T

 .

.

+|k − wx|r−1 w

≤

R

|Ф(wu − k)|du |χ (wx − k)|

% 2r−2 $ r−1 (Ф) + Mr−1 (χ )M 0 (Ф) × M0 (χ )M (r − 1)!

  . w

 +γ ) |z|≤ (T w

|f

(r)

(z + x) − f

(r)

(x)|dz < +∞,

36

D. Costarelli and G. Vinti

by Remark 1. Now, recalling the generalized Minkowski-type inequality (see, e.g., [25], p. 148), we have: ‖Jw ‖p ≤

.

% 2r−2 $ r−1 (Ф) + Mr−1 (χ )M 0 (Ф) × M0 (χ ) M (r − 1)!

  . w



+γ ) |z|≤ (T w

.

‖f

(r)

(z + ·) − f

(r)

(·)‖p dz

% 2r−1 $ r−1 (Ф) + Mr−1 (χ )M 0 (Ф) × M0 (χ )M (r − 1)!

≤

(T + γ )

.

sup

|z|≤(T +γ )/w

‖f (r) (z + ·) − f (r) (·)‖p .

Now, since .f (r) ∈ Lp (R), we know that, for every .ε > 0, there exists .δ > 0 such that, for every .|z| ≤ δ, we have .‖f (r) (· + z) − f (r) (·)‖p < ε. Then for .w > 0 sufficiently large, we finally obtain .‖Jw ‖p < ε. This completes the proof. ⨆ ⨅ Acknowledgments The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM), of the network RITA (Research ITalian network on Approximation), and of the UMI group T.A.A. (Teoria dell’Approssimazione e Applicazioni). The authors have been partially supported within the (1) 2022 GNAMPA-INdAM Project “Enhancement e segmentazione di immagini mediante operatori di tipo campionamento e metodi variazionali per lo studio di applicazioni biomediche,” while the second author has been partially funded within the projects (1) Ricerca di Base 2019 dell’Università degli Studi di Perugia— “Integrazione, Approssimazione, Analisi Nonlineare e loro Applicazioni”; (2) “Metodiche di Imaging non invasivo mediante angiografia OCT sequenziale per lo studio delle Retinopatie degenerative dell’Anziano (M.I.R.A.),” funded by FCRP, 2019; and (3) “CARE: A regional information system for Heart Failure and Vascular Disorder,” PRJ Project – 1507 Action 2.3.1 POR FESR 2014-2020, 2020.

References 1. T. Acar, D. Costarelli, G. Vinti, Linear prediction and simultaneous approximation by m-th order Kantorovich type sampling series. Banach J. Math. Anal. 14(4), 1481–1508 (2020) 2. B.A. Bailey, W.R. Madych, Classical sampling series of band limited functions: oversampling and non-existence. Samp. Theory Signal Image Proc. 15, 131–138 (2016) 3. C. Bardaro, I. Mantellini, Asymptotic formulae for linear combinations of generalized sampling operators. Zeitschrift Anal. Anwend. 32(3), 279–298 (2013) 4. C. Bardaro, L. Faina, I. Mantellini, Quantitative Voronovskaja formulae for generalized Durrmeyer sampling type series. Math. Nachr. 289(14–15), 1702–1720 (2016) 5. P.L. Butzer, R.J. Nessel, Fourier Analysis and Approximation I (Academic, New York, 1971)

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6. P.L. Butzer, R.L. Stens, Reconstruction of signals in Lp (R)-space by generalized sampling series based on linear combinations of B-splines. Integral Trans. Spec. Funct. 19(1), 35–58 (2008) 7. P.L. Butzer, W. Splettsto¨ ßer, R.L. Stens, The sampling theorem and linear prediction in signal analysis, Jahresber. Deutsch. Math.-Verein. 90, 1–70 (1988) 8. P.L. Butzer, R.J. Higgins, R.L. Stens, Classical and approximate sampling theorems; studies in the Lp (R) and the uniform norm. J. Approx. Theory 137(2), 250–263 (2005) 9. J.D. Cao, H.H. Gonska, Approximation by boolean sums of positive linear operators III: estimates for some numerical approximation schemes. Num. Funct. Anal. Opt. 10(7–8), 643– 672 (1989) 10. D. Constales, H. De Bie, P. Lian, A new construction of the Clifford-Fourier kernel. J. Fourier Anal. Appl. 23(2), 462–483 (2017) 11. L. Coroianu, S.G. Gal, Approximation by max-product sampling operators based on sinc-type Kernels. Samp. Theory Sign. Image Proc. 10(3), 211–230 (2011) 12. L. Coroianu, S.G. Gal, Lp - approximation by truncated max-product sampling operators of Kantorovich-type based on Fejer kernel. J. Integr. Eq. Appl. 29(2), 349–364 (2017) 13. D. Costarelli, R. Spigler, How sharp is the Jensen inequality? J. Inequal. Appl. 2015(69), 1–10 (2015) 14. D. Costarelli, G. Vinti, An inverse result of approximation by sampling Kantorovich series. Proc. Edinburgh Math. Soc. 62(1), 265–280 (2019) 15. D. Costarelli, G. Vinti, Inverse results of approximation and the saturation order for the sampling Kantorovich series. J. Approx. Theory 242, 64–82 (2019) 16. D. Costarelli, G. Vinti, Saturation by the Fourier transform method for the sampling Kantorovich series based on bandlimited kernels. Anal. Math. Physics 9, 2263–2280 (2019) 17. D. Costarelli, G. Vinti, Approximation properties of the sampling Kantorovich operators: regularization, saturation, inverse results and Favard classes in Lp -spaces. J. Fourier Analy. Appl. 28, Article Number: 49 (2022) 18. D. Costarelli, M. Seracini, G. Vinti, A comparison between the sampling Kantorovich algorithm for digital image processing with some interpolation and quasi-interpolation methods. Appl. Math. Comput. 374, 125046 (2020) 19. D. Costarelli, M. Piconi, G. Vinti, On the convergence properties of Durrmeyer-Sampling type operators in Orlicz spaces. Math. Nachr. 296, 588–609 (2023) 20. R.A. DeVore, G.G. Lorentz, Constructive Approximation, vol. 303 (Springer Science & Business Media, Cham, 1993) 21. M.N. Do, Y.M. Lu, A theory for sampling signals from a union of subspaces. IEEE Trans. Signal Proc. 56(6), 2334–2345 (2008) 22. J.L. Durrmeyer, Une firmule d’inversion de la transformée de Laplace: applications á la théorie des moments, Thése de 3e cycle, Université de Paris, 1967 23. H. Gonska, M. Heilmann, I. Rasa, Convergence of iterates of genuine and ultraspherical Durrmeyer operators to the limiting semigroup: C 2 -estimates. J. Approx. Theory 160(1–2), 243–255 (2009) 24. H. Gonska, D. Kacso, I. Rasa, The genuine Bernstein-Durrmeyer operators revisited. Res. Math. 62(3–4), 295–310 (2012) 25. G.H. Hardy, J.E. Littlewood, G. Pólya, Inequalities (Cambridge University Press, Cambridge, 1952, (xii), 1934) 26. M. Heilmann, I. Rasa, A nice representation for a link between Baskakov- and Szász-MirakjanDurrmeyer operators and their Kantorovich variants. Res. Math. 74, Article number: 9 (2019) 27. R.J. Higgins, Five short stories about the cardinal series. Bull. Amer. Math. Soc. 12, 45–89 (1985) 28. R.J. Higgins, Sampling Theory in Fourier and Signal Analysis: Foundations (Clarendon Press, Oxford, 1996) 29. R.J. Higgins, R.L. Stens, Sampling Theory in Fourier and Signal Analysis: Advanced Topics (Oxford University Press, Oxford, 1999)

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30. H. Karsli, On Urysohn type generalized sampling operators. Dolomites Res. Notes Approx. 14, 58–67 (2021) 31. A. Kivinukk, G. Tamberg, On window methods in generalized shannon sampling operators, in New Perspectives on Approximation and Sampling Theory. Applied and Numerical Harmonic Analysis (Birkhäuser, Cham, 2014), pp. 63–85 32. W.R. Madych, The limiting behavior of certain sampling series and cardinal splines, J. Approx. Theory 249, Art. Numb. 105298 (2020). 33. M. Menekse Yilmaz, G. Uysal, E. Ibikli, A note on rate of convergence of double singular integral operators. Adv. Differ. Eq. 2014, Article number: 287 (2014). 34. O. Orlova, G. Tamberg, On approximation properties of generalized Kantorovich-type sampling operators. J. Approx. Theory 201, 73–86 (2016) 35. S. Ries, R.L. Stens, Approximation by generalized sampling series, in Constructive Theory of Functions’84, Sofia (1984), pp. 746–756 36. M. Rosenthal, H.J. Schmeisser, On the boundedness of singular integrals in Morrey spaces and its preduals. J. Fourier Anal. Appl. 22(2), 462–490 (2016) 37. K. Runovski, H.J. Schmeisser, On approximation methods generated by Bochner-Riesz Kernels. J. Fourier Anal. Appl. 14, 16–38 (2008) 38. G. Strang, G. Fix, A Fourier analysis of the finite element variational method, in Constructive Aspects of Functional Analysis (Springer, Berlin, 1971), pp. 793–840 39. G. Vinti, A general approximation result for nonlinear integral operators and applications to signal processing. Appl. Anal. 79, 217–238 (2001) 40. G. Vinti, L. Zampogni, A unifying approach to convergence of linear sampling type operators in Orlicz spaces. Adv. Differ. Eq. 16, 573–600 (2011) 41. G. Vinti, L. Zampogni, A unified approach for the convergence of linear Kantorovich-type operators. Adv. Nonlinear Stud. 14, 991–1012 (2014) 42. G. Vinti, L. Zampogni, A general approximation approach for the simultaneous treatment of integral and discrete operators. Adv. Nonlinear Stud. 18(4), 705–724 (2018)

On Generalized Shannon Sampling Operators in the Cosine Operator Function Framework Andi Kivinukk and Gert Tamberg

1 Introduction We shall study the windowed Shannon sampling operators (Uw f )(t) :=



∞ 

f

.

k=−∞

 k s(wt − k) (w > 0), w

(1)

with the kernel function .s ∈ L1 (R) for the signals .f ∈ C(R) (the space of uniformly continuous and bounded functions on .R endowed with the supremum norm). The term “windowed” comes from the fact that we consider the kernel function 

1

s(t) =

λ(u) cos(π ut)du,

.

(2)

0

which is the cosine Fourier transform of a given continuous even window function λ : R → R+ with .λ(0) = 1 and .λ(u) = 0 for .|u| ≥ 1. The term “cosine-type” comes from the fact that we consider the window function

.

A. Kivinukk () School of Digital Technologies, Tallinn University, Tallinn, Estonia e-mail: [email protected] G. Tamberg Division of Mathematics, Department of Cybernetics, Tallinn University of Technology, Tallinn, Estonia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. D. Casey et al. (eds.), Sampling, Approximation, and Signal Analysis, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-031-41130-4_3

39

40

A. Kivinukk and G. Tamberg

λa (u) =

m 

.

ak cos(kπ u), u ∈ [0, 1],

(3)

k=0

where .a = (a0 , . . . , am ) ∈ Rm+1 , m ≥ 1, or λb (u) =

m 

.

bk cos((k − 1/2)π u), u ∈ [0, 1],

(4)

k=1

where .b = (b1 , . . . , bm ) ∈ Rm , m ≥ 1. As we will show, the operators (1) are well-defined if we assume for .a ∈ Rm+1 that m  .

ak = 1,

k=0

m  (−1)k ak = 0

(5)

k=0

or for .b ∈ Rm that m  .

bk = 1.

(6)

k=1

The approximation properties for (1), using specific kernel functions in (2), were studied in [10–12]. The operators (1), defined by the general kernel function, were introduced by P. L. Butzer and his school since 1977 (see [4] and literature therein). It is shown in [4] that the operator in (1) is well-defined and the equality .

lim (Uw f )(t) = f (t)

w→∞

holds uniformly in .R, which are essentially equivalent to each of the following two assertions: (i)

∞ 

.

s(x − k) = 1, x ∈ [0, 1);

k=−∞

(ii)  s(2kπ ) = 0, k ∈ Z\{0};  s(0) = (2π )−1/2 where the Fourier transform .f of .f ∈ L1 (R) is defined for .v ∈ R by 1 f(v) := √ 2π



.

R

f (t)e−ivt dt.

We assume that the kernel s in (2) is absolutely integrable, i.e. .s ∈ L1 (R). Then by (2), we have

On Generalized Shannon Sampling Operators in the Cosine Operator Function. . .

41

1 v  .s(v) = √ λ( ) π 2π which, by properties of the window function .λ, yields that the condition .(ii) is satisfied. Moreover, by Butzer et al. [4], Theorem 4.1, we have the operator norm of (1) as

.

‖Uw ‖[C(R)] = sup

∞ 

|s(u − k)|.

(7)

u∈R k=−∞

As the working tool to study the cosine-type operators (1), we shall use the framework of the cosine operator function in an abstract Banach space. This approach is introduced in our previous papers [14] and [9]. In this chapter, it is of particular interest to deal with operators defined by window functions λH,n (u) := cosn (

.

πu ) 2

(8)

and λm (u) := 1 − sin2m (

.

πu ). 2

(9)

Here, the subscript H stands for Hann, since in case .n = 2, m = 1, both windows reduce to the Hann window, well-known in several applications in signal analysis (e.g., [5, 6, 16, 17]). Both of these window functions have representations (3) or (4).

2 General Approximation Theorems Using a Framework of Cosine Operator Functions Let X be an arbitrary (real or complex) Banach space and .[X] the Banach algebra of all bounded linear operators . U of X into itself. Then (see, e.g., [2, 15, 19]) Definition 1 An equibounded cosine operator function .Ch ∈ [X] (.h ≥ 0) is defined by the properties: 1. .C0 = I (identity operator). 2. .Ch1 · Ch2 = 12 (Ch1 +h2 + C|h1 −h2 | ). 3. .‖Ch f ‖ ≤ T ‖f ‖, the constant .T > 0 being independent of .h > 0. We shall define the abstract cosine-type approximation operators and the apparatus that is needed for estimating the order of approximation. In this connection, we refer to our previous articles [14] and [9]. The leading idea for definitions below appeared from the trigonometric approximation (see [3, 18], and references cited there).

42

A. Kivinukk and G. Tamberg

An abstract modulus of continuity, defined by the cosine operator function, will play an important role in our work. Definition 2 The modulus of continuity of order .k ∈ N of .f ∈ X is defined for δ ≥ 0 via the cosine operator function by

.

ωk (f, δ) := 2k sup ‖(Ch − I )k f ‖.

.

(10)

0≤h≤δ

For .k = 1, we will denote .ω1 (f, δ) =: ω(f, δ). Remark 1 The coefficient .2k stands for the technical reason: if in a functional space the cosine operator function is defined by Ch (f, x) :=

.

1 (f (x + h) + f (x − h)) , h ≥ 0, 2

(11)

then Definition 2 defines the classical modulus of continuity of order .2k. The next properties are adaptations of the well-known properties of the ordinary modulus of continuity (see, e.g., [3, 21]) Proposition 1 The modulus of continuity .ωk (f, δ) in the previous definition has the following properties: 1. .ωk (f, mδ) ≤ mk (1 + (m − 1)T )k ωk (f, δ), m ∈ N, .(T is the constant in Def 1.). 2. .ωk (f, λδ) ≤ (⎿λ⏌ + 1)k (1 + ⎿λ⏌T )k ωk (f, δ), λ > 0, .(λ ≥ ⎿λ⏌—the integral part of .λ ∈ R). 3. .ωk (f, δ) ≤ (2(1 + T ))k−l ωl (f, δ), k ≥ l and .k, l ∈ N. Structure of the Space X Let .{Aσ }σ >0 be a dense family of subspaces of X union meaning that for the subspaces .Aσ ⊂ X with .Aσ1 ⊂ Aσ2 , 0 < σ1 < σ2 , the  . Aσ is dense in X, i.e., for each .f ∈ X, there exists a family .{gσ }σ >0 ⊂ Aσ σ >0

σ >0

such that .limσ →∞ ‖f − gσ ‖ = 0. We shall approximate .f ∈ X by the elements of the subspaces .Aσ , σ > 0. Moreover, let .Aσ ⊂ X be the subspace consisting of the fixed points of certain operator .Sσ defined over a suitable subspace of X containing .Aσ . Beside the modulus of continuity, we need another quantity, the best approximation. Definition 3 The best approximation of .f ∈ X by elements of .Aσ is defined by Eσ (f ) := inf ‖f − g‖.

.

g∈Aσ

Remark 2 We suppose that there exists an element .g∗ ∈ Aσ of the best approximation, i.e. .Eσ (f ) = ‖f − g∗ ‖.

On Generalized Shannon Sampling Operators in the Cosine Operator Function. . .

43

The following example of the subspaces .Aσ and operators .Sσ is important for the Shannon sampling operators.

Example. Structure of the Space .C(R) Let .X = C(R) be the space of uniformly continuous and bounded functions on .R (for what follows, see, e.g., [7] and [4]) with Bernstein classes .B∞ σ ⊂ C(R), consisting of the bounded functions on .R, which can be extended to the complex plane as an entire function .f (z) (z ∈ C) of exponential type .σ , i.e., .|f (z)| ≤ eσ |y| ‖f ‖C (z = x + iy ∈ C). We specify the class .B∞ σ as ∞ follows: let .B∞ denote the class of functions in .Bσ for which .0 < σ < π w−0 ∞ π w, w > 0. In this case, the linear operator .Sw : B∞ π w−0 → Bπ w−0 is the ∞ classical Whittaker-Kotel’nikov-Shannon operator, for .g ∈ Bπ w−0 defined by sinc .(Sw g)(t)



∞ 

:=

g

k=−∞

 k sinc(wt − k), w

where the kernel function .sinc(t) := sinπ πt t . The fact that for .Swsinc : B∞ π w−0 → ∞ B∞ π w−0 the set of the fixed points is .Bπ w−0 is the statement of the famous Whittaker-Kotel’nikov-Shannon theorem: if .g ∈ B∞ π w−0 , then (Swsinc g)(t) = g(t).

.

We define our approximation operators on the subspaces .Aσ ⊂ X as follows. σ,h,a : Aσ → X (σ > 0, h ≥ 0) are Definition 4 The Blackman-type operators .B defined by σ,h,a g := B

m 

.

ak Ckh (Sσ g) , g ∈ Aσ ,

(12)

k=0

where .a = (a0 , . . . , am ) ∈ Rm+1 , m ≥ 0, and m  .

ak = 1.

(13)

k=0

The Rogosinski-type operators will be defined similarly.  Definition 5 The Rogosinski-type operators .R σ,h,b : Aσ → X (σ > 0, h ≥ 0) are defined by

44

A. Kivinukk and G. Tamberg

 R σ,h,b g :=

m 

.

bk C(k−1/2)h (Sσ g) , g ∈ Aσ ,

k=1

where .b = (b1 , . . . , bm ) ∈ Rm and m  .

bk = 1

(14)

k=1

is supposed to be valid. Here, the case .b = (1) ∈ R leads to the original Rogosinski operator, which in trigonometric approximation was introduced by W. W. Rogosinski [18] and afterward elaborated by S. B. Ste.ckin ˜ in [20]; see also [3]. In our notations, the π (Sn f, x), where .Sn f classical Rogosinski means are in the form .Rn (f, x) = C 2(n+1) denotes the Fourier partial sums of .f ∈ C2π (the space of .2π -periodic continuous functions on .R) and .Ch is the cosine operator function as the central difference in (11). σ,h,a : Aσ → X and .R  So far, our operators .B σ,h,b : Aσ → X are defined only in subspaces .Aσ , but the following bounded linear transformation theorem allows to define our approximation operators in the whole space X as well. Theorem 1 ([8], Sect. 8.2, 8.3) Let .A ⊂ X be a dense subset of a Banach space X  Then .B  : A → X be a bounded linear operator with the operator norm .‖B‖.  and .B  For .f ∈ X, has the unique bounded linear extension .B : X → X with .‖B‖ = ‖B‖.  σ , where .{gσ }σ >0 ⊂ A is an the operator .B ∈ [X] is defined by .Bf = limσ →∞ Bg arbitrary family with .f = limσ →∞ gσ . σ,h,a : Aσ → X are defined, their extensions Thus, if the approximation operators .B will be denoted by .Bσ,h,a : X → X, correspondingly. We use a similar notation for  .R σ,h,b : Aσ → X as well. For the following general approximation theorems, we refer to our previous articles [14] and [9]. Theorem 2 ([14], Theorem 3.7) Assume (13) is valid for .a = (a0 , . . . , am ) ∈ Rm+1 (m ≥ 1). Then for each .f ∈ X for the operators .Bσ,h,a : X → X, we have (.T > 0 is the uniform bound of the cosine operator function .Ch ) . Bσ,h,a f − f ≤

Bσ,h,a

[X]

+T

m 

 |ak | Eσ (f )

k=0

 1 max(T , 1)ω(f, h) l 2 |al | . 2 m

+

l=1

The following theorem is valid for operators with a higher degree of approximation.

On Generalized Shannon Sampling Operators in the Cosine Operator Function. . .

45

Theorem 3 ([14], Theorem 3.13) Let the Blackman-type operators .Bσ,h,a : X → σ,h,a : Aσ → X, X be bounded linear extensions of the operators .B σ,h,a g := B

m 

.

ak Ckh (Sσ g), h > 0, m ∈ N,

k=0

where the cosine operator functions .Ch : X → X are equibounded, i.e., .‖Ch ‖[X] ≤ T . Let us fix the integer q with .m ≥ q ≥ 2, and suppose that the coefficients satisfy the Eqs. (13) and

.

 m   k+p−1 k ak = 0 2p − 1

(15)

k=p

for all .p = 1, 2, .., q − 1. Then for each .f ∈ X, we have . Bσ,h,a f − f ≤



 m  Bσ,h,a + T |ak | Eσ (f ) [X] k=0

CT ,q ωq (f, h) + 2q

m  l=q

  l+q −1 l |al |, 2q − 1

(16)

where .CT ,q = max(T , 1) for .2 ≤ q ≤ m − 1 and .CT ,q = 1 for .q = m. In fact, Rogosinski-type operators are at the same time Blackman-type operators because the equality σ,h,a ' g = σ,2h,b g = B R

2m−1 

.

ak' Ckh (Sσ g) , h ≥ 0

k=0 ' holds, where .b = (b1 , b2 , . . . , bm ) ∈ Rm and .a' = (a0' , a1' , . . . , a2m−1 ) 2m = (0, b1 , 0, b2 , . . . , 0, bm ) ∈ R . However, it makes sense for us to transfer these general Theorems 2 and 3 to Rogosinski-type operators as well.

Theorem 4 ([9], Theorem 4.4) Assume (14) is valid for .b = (b1 , . . . , bm ) ∈ Rm . Then for each .f ∈ X for the Rogosinski-type operators .Rσ,h,a : X → X, we have . Rσ,h,b f − f ≤

Rσ,h,b

[X]

+T

m 

 |bk | Eσ (f )

k=1

h  1 max(T , 1)ω(f, ) (2l − 1)2 |bl | . 2 2 m

+

l=1

46

A. Kivinukk and G. Tamberg

Theorem 5 ([9], Theorem 4.5) Let the Rogosinski-type operators .Rσ,h,b : X → σ,h,b : Aσ → X, X (h > 0) be bounded linear extensions of the operators .R  R σ,h,b g :=

m 

.

bk C(k−1/2)h (Sσ g), m ∈ N,

k=1

where .Ch : X → X are equibounded. Let us fix the integer q with .2 ≤ q ≤ m, and suppose that the coefficients .b = (b1 , . . . , bm ) ∈ Rm satisfy the Eqs. (14) and m  .

(2k − 1)

k=⎿ p2 ⏌+1

  2k + p − 2 bk = 0, 2p − 1

(17)

for all .p = 1, 2, . . . , q − 1. Then for each .f ∈ X, we have f − f . R ≤ σ,h,b +

Rσ,h,b

[X]

+T

m 

 |bk | Eσ (f )

k=1

     m CT ,q h 2l + q − 2 |bl | , ωq f, (2l − 1) 2q 2 2q − 1 q l=⎿ 2 ⏌+1

where .CT ,q = max(T , 1) for .2 ≤ q ≤ m − 1 and .CT ,q = 1 for .q = m.

3 On Approximations by the Powers of Hann Sampling Operators In this section, we apply the results of Sect. 2 in the space .X = C(R) which consists of uniformly continuous and bounded functions on .R. The structure of the space .X = C(R) of Sect. 2 is described in Example 2. In the space .C(R), the natural cosine operator function .Ch ∈ C(R) is defined by the arithmetic mean of shifts of the function .f ∈ C(R), (Ch f )(x) =

.

1 (f (x + h) + f (x − h)) , h ≥ 0, 2

(18)

which is equibounded with the bound .T = 1. In this case, the classical modulus of smoothness of order 2k ([14], cf.[3], p. 76) is defined by ω2k (f, δ) := 2k sup ‖(Ch − I )k f ‖,

.

0≤h≤δ

On Generalized Shannon Sampling Operators in the Cosine Operator Function. . .

47

that is, we have the modulus of continuity of order k in general theorems replaced with the classical modulus of smoothness of order .2k. We need to specify the best approximation of .f ∈ C(R) by .Eπ w−0 (f ) := inf ‖f − g‖ : g ∈ B∞ π w−0 . A recent Jackson-type theorem ([22], Section 4.1, inequality (4.1); compare [21], Subsection 5.1.3) reads Eπ w−0 (f ) ≤ Mk ω2k (f, 1/w) ,

.

(19)

where for the cases .k = 1, 2, 3, 4, the constants .Mk have values .M1 = 5/8, M2 = 65/216, M3 = 0.11613 . . . , M4 = 0.03989 . . . , they could be close to the best possible ([22], Table 1 and Abstract). In order to study the windowed Shannon sampling operators (1), the linear ∞ operator .Sw : B∞ π w−0 → Bπ w−0 is the classical Whittaker-Kotel’nikov-Shannon ∞ operator, for .g ∈ Bπ w−0 defined by ∞ 

(Swsinc g)(t) :=

.

 g

k=−∞

 k sinc(wt − k). w

(20)

Let us denote the operators (1) by .Uw,a : C(R) → C(R), if they are defined via the window function (3). These operators appear to be extensions of the operators w,a : B∞ .U π w−0 → C(R)), defined by (compare with Definition 4) w,a g)(t) := (U

m 

.

  aj Cj/w Swsinc g (t).

(21)

j =0

This extension from (21) to (1) becomes clear when in (21) we use (20) and (18) yielding w,a g)(t) = (U

∞ 

.

 g

k=−∞

 k sa (wt − k), w

where sa (t) = sinc(t)

.

m  (−1)j aj j =0

t2 , t2 − j2

(22)

which is valid due to Eqs. (2) and (3). We will now show why condition (5), second equation, is important. We put for the kernel (22)

48

A. Kivinukk and G. Tamberg



a0 am a1 a2 .sa (t) = t sinc(t) − 2 + 2 − · · · + (−1)m 2 t2 t − 12 t − 22 t − m2  sinc(t) a0 (t 2 − 12 ) . . . (t 2 − m2 ) = 2 (t − 12 ) . . . (t 2 − m2 )



2

− a1 t 2 (t 2 − 22 ) . . . (t 2 − m2 ) + . . . (−1)m am t 2 . . . (t 2 − (m − 1)2 )



 sinc(t) a0 (t 2m + p0 (t)) − a1 (t 2m + p1 (t)) (t 2 − 12 ) . . . (t 2 − m2 )  + . . . (−1)m am (t 2m + pm (t)) , =

where .pj (t), (j = 0, 1, . . . , m) are algebraic polynomials of order .2m − 2. Therefore, by (5), second equation, we obtain sa (t) =

.

sinc(t)P2m−2 (t) , (t 2 − 12 ) . . . (t 2 − m2 )

where .P2m−2 (t) is an algebraic polynomial of order .2m − 2. Thus, .sa (t) = O(t −3 ), t > 0; hence, the kernel function .sa ∈ L1 (R) and the operators (1) are well-defined. We denote the operators (1) by .Uw,b : C(R) → C(R), if they are defined via the window function (4). These operators appear to be extensions of the operators w,b : B∞ .U π w−0 → C(R)), defined by (compare with Definition 5) w,b g)(t) := (U

m 

.

  bj C(j −1/2)/w Swsinc g .

(23)

j =1

This extension from (23) to (1) becomes clear when in (23) we use (20) and (18) yielding w,b g)(t) = .(U

∞  k=−∞

 g

 k sb (wt − k), w

where sb (t) =

.

m j − 1/2 cos π t  (−1)j bj , 2 π t − (j − 1/2)2

(24)

j =1

which is valid due to Eqs. (2) and (4). We see directly from (24) that .sb (t) = O(t −2 ), t > 0; hence, the kernel function 1 .sb ∈ L (R) and the operators (1) are well-defined.

On Generalized Shannon Sampling Operators in the Cosine Operator Function. . .

49

We may have for the kernel function (24) a better rate at infinity if we suppose for that the derivative of .λb satisfies the following: λ'b (1) =

.

m  (−1)k bk (k − 1/2) = 0.

(25)

k=1

Indeed, we have for the sum in (24): m  .

. . . = −b1

j =1

= +

1/2 t2

− (1/2)

2

+ b2

3/2 t2

− (3/2)

2

− · · · + (−1)m bm

m − 1/2 t2

− (m − 1/2)2

 b 1 1 − (t 2m−2 + p1 (t)) 2 (t 2 − (1/2)2 ) . . . (t 2 − (m − 1/2)2 )

 3b2 2m−2 (t + p2 (t)) − . . . (−1)m (m − 1/2)bm (t 2m−2 + pm (t)) , 2

where .pj (t), (j = 1, . . . , m) are algebraic polynomials of order .2m − 4. Therefore, by (24) and (25), we obtain sb (t) =

.

cos π tP2m−4 (t) , π(t 2 − (1/2)2 ) . . . (t 2 − (m − 1/2)2 )

where .P2m−4 (t) is an algebraic polynomial of order .2m − 4. Thus, .sb (t) = O(t −4 ), t > 0. We turn now to the problems of approximations with the powers of Hann sampling operators .Hwn defined via the window function (8). Theorem 6 For each .f ∈ C(R) for the powers of Hann sampling operators .Hwn : C(R) → C(R) (n ∈ N)) defined by the window function .λH,n (u) = cosn ( π2u ), we have .

  n n 1 H f − f ≤ H n ). + 1 Eπ w−0 (f ) + ω2 (f, w w [C(R)] 2 2w

Proof We will deal separately with even and odd powers. We have for the even powers ([23], 1.320, formula 5; here and in the following, .∑ ' means that the first term of that sum is halved) λH,2m (u) := cos

.

2m

 m  2m πu 1 ' cos(π ku); ( ) = 2m−1 m−k 2 2 k=0

hence, the coefficients .ak in the representation (3) are equal to the expressions

(26)

50

A. Kivinukk and G. Tamberg

  2m .a0 = ,. 22m m   2m 1 (k = 1, 2, . . . , m). ak = 2m−1 m−k 2 1

(27) (28)

We see from (26) with .u = 0 and .u = 1 that the conditions (5) are satisfied for the coefficients (27) and (28); hence, the operator .Hw2m : C(R) → C(R) is well-defined. Now we apply Theorem 2 taking .Bw,1/w,a = Uw,a = Hw2m , and we obtain

2m 2m . Hw f − f ≤ Hw

[C(R)]

+

m  k=0



1  2 1 l |al | . |ak | Eπ w−0 (f ) + ω2 (f, ) 2 w m

l=1

Since the coefficients (27) and (28) are positive, we have for (5) m  .

|ak | = 1.

k=0

For the second sum, we differentiate twice the Eq. (26) yielding m  .

k 2 ak cos(π ku) =

k=1

πu m πu cos2m ( ) + sin( )(. . . ). 2 2 2

Taking .u = 0, we get for the second sum m  .

k 2 ak =

k=1

m 2

(29)

and finally  2m 2m . Hw f − f ≤ Hw

 m 1 + 1 Eπ w−0 (f ) + ω2 (f, ). [C(R)] 4 w

(30)

We have for the odd powers ([23], 1.320, formula 7) λH,2m−1 (u) := cos2m−1 (

.

 m  1  2m − 1 πu cos(π(k −1/2)u); ) = 2m−2 m−k 2 2

(31)

k=1

hence, the coefficients .bk in the representation (4) are equal to the expression   2m − 1 (k = 1, 2, . . . , m). .bk = 22m−2 m − k 1

(32)

On Generalized Shannon Sampling Operators in the Cosine Operator Function. . .

51

Again, the operator .Hw2m−1 : C(R) → C(R) is well-defined because (14) is fulfilled (take .u = 0 in (31)) and we may apply Theorem 4. For .Rw,1/w,b = Uw,b = Hw2m−1 , we obtain

 m  2m−1 2m−1 . Hw f − f ≤ Hw + |bk | Eπ w−0 (f ) [C(R)]

+

1 1 ω2 (f, ) 2 2w

m 

k=1

(2l − 1)2 |bl | .

l=1

Here, we differentiate twice the Eq. (31) for the second sum yielding

.

m  πu πu (2k − 1)2 bk cos(π(k − 1/2)u) = (2m − 1) cos2m−1 ( ) + sin( )(. . . ). 2 2 k=1

Taking .u = 0, we get for the second sum m  .

(2k − 1)2 bk = 2m − 1,

k=1

and finally  2m−1 . Hw f − f ≤ Hw2m−1

 1 2m − 1 ω2 (f, ). + 1 Eπ w−0 (f ) + [C(R)] 2 2w (33) Keeping in mind the inequalities (30) and (33) and the property of the modulus of 1 continuity .ω2 (f, w1 ) ≤ 22 ω2 (f, 2w ), we have proved the estimate of our theorem. ⨆ ⨅ n Remark 3 In general, this estimate for . Hw f − f cannot be improved in terms of modulus of continuity, as we will show in Sect. 5. In the case .n = 2, we prove that for a certain constant .c > 0, independent from .f ∈ C(R), the following inequality c ω2 (f,

.

1 ) ≤ Hw2 f − f 2w

is valid. This statement is also confirmed by the fact that neither coefficients (27) and (28) nor coefficients (32) satisfy the conditions of Theorem 3 or Theorem 5, respectively. Operators .Hwn are particularly interesting because it is possible to calculate their operator norms. In one of our earlier articles [10], we have proved the following formula: ‖Hwn ‖[C(R)] =

n 

.

k=0

sH,n−1 (k −

n ), 2

52

A. Kivinukk and G. Tamberg

where the kernel function .sH,n is given by 

1

sH,n (t) =

.

cosn (

0

πu ) cos(π ut)du. 2

4 On Approximations by Certain Linear Combinations of the Powers of Hann Sampling Operators In this section, we apply the results from Sect. 2 for certain linear combinations of the powers of Hann sampling operators. First we consider the operators .Hw,m , defined via the window function λm (u) := 1 − sin2m (

.

πu ), m ∈ N. 2

Let us show that this window is of type, given in (3). This window is introduced in [1], but there may be other sources as well. We recall the trigonometric identity ([23], 1.320, formula 1) (1 − cos x) =

r 1 '

r

.

2r−1

 (−1)

 2r cos kx. r −k

(34)

 2m cos(π ku) m−k

(35)

k

k=0

Using the given identity, we obtain λm (u) = 1 −

.

1 22m−1

m  '

 (−1)k

k=0

with the coefficients .ak in the representation (3) as given via the expressions   2m ,. 2m m 2   (−1)k+1 2m (k = 1, 2, . . . , m). ak = 2m−1 m−k 2 a0 = 1 −

1

.

(36) (37)

The coefficients (36), (37) satisfy the first equation of (5) due to the condition λm (0) = 1. Since .a0 > 0 by (36) and .(−1)k+1 ak > 0 by (37), we obtain for the second equation of (5)

.

.

m m   (−1)k ak = a0 − |ak | = a0 − k=0

k=1

1 22m−1

 m   2m . m−k k=1

On Generalized Shannon Sampling Operators in the Cosine Operator Function. . .

53

Since the sum of the coefficients (27), (28) is equal to one, the equality 1 .

22m−1

 m   2m = a0 m−k

(38)

k=1

holds, and we get

.

m m   (−1)k ak = a0 − |ak | = 0. k=0

(39)

k=1

One reason why the window .λm is interesting and important is that the coefficients (36), (37) also satisfy the condition (15) of Theorem 3 with .q = m. Proof of this is quite technical, which is why we are referring to one of our previous results. Lemma 1 ([9], Lemma 6.3) The coefficients (37) satisfy for all .p = 1, 2, . . . , m − 1 the equation 22m−1

.

    m m    k+p−1 2m k+p−1 k ak ≡ (−1)k+1 k = 0. 2p − 1 m−k 2p − 1

k=p

k=p

Therefore, we can apply Theorem 3 with .q = m to operators .Bw,1/w,a = Uw,a = Hw,m , and we obtain

 m  1 1 Hw,m [C(R)] + ≤ . Hw,m f − f |ak | Eπ w−0 (f ) + ω2m (f, ) |am | . w 2 k=0

We have by (39) m  .

|ak | = 2a0 = 2 −

k=0

  2m . 22m−1 m 1

Therefore, we may state the following result which is covered for .m = 1 by (30). Theorem 7 For each .f ∈ C(R) for the linear combination of Hann sampling operators .Hw,m : C(R) → C(R) (m ∈ N) defined by the window function 2m π u .λm (u) := 1 − sin ( 2 ), we have  ≤ Hw,m [C(R)] + 2 − . Hw,m f − f +

1 1 ω2m (f, ). w 22m

  2m Eπ w−0 (f ) 2m−1 m 2 1

54

A. Kivinukk and G. Tamberg

∗ From our point of view, we could also consider operators .Hw,m whose window function is

λ∗m (u) :=

.

1 − sin2m ( π2u ) , m ∈ N. cos( π2u )

This window function in the given general form is probably new; its special case for .m = 1 defines the classical Rogosinski operator in [18]. Let us show that this window is of type, given in (4). We have by the binomial formula   m 1 − sin2m (x)  k+1 m cos2k−1 (x). = (−1) . k cos x k=1

Now we apply the formula (31)

.

1

cos2m−1 (x) =

22m−2

 m   2m − 1 m−j

j =1

cos((2j − 1)x),

and we obtain ∗ .λm (u)

=

m 

(−1)

k+1

   k  m 1  2k − 1 cos(π(j − 1/2)u). k−j k 22k−2 j =1

k=1

After changing the summation order, we get ∗ .λm (u)

=

m 

bk cos(π(k − 1/2)u),

k=1

where the coefficients .bk here (and in the representation (4)) are equal to the expression     m  1 2j − 1 j +1 m (k = 1, 2, . . . , m). .bk = (−1) j 22j −2 j − k

(40)

j =k

The coefficients (40) satisfy the Eq. (14) due to the condition .λ∗m (0) = 1. We need to simplify the coefficients .bk for future consideration. We go back to the window ∗ .λm in (35), for which we can prove (in similar way as for .λm ) the representation λm (u) = 1 +

m  '

.

k=0

cos(π ku)

m  (−1)j +1 j =k

   m 2j . j −k 22j −1 j 1

On Generalized Shannon Sampling Operators in the Cosine Operator Function. . .

55

Comparing this representation with the formula (35), we get a useful equation (we did not find it in the literature)      m  m 2j (−1)k+1 2m j +1 1 . (−1) = , k = 0, 1, . . . , m. j −k m−k 22j j 22m

(41)

j =k

Now let us subtract from the Eq. (41) the expression of .bk /4 for which we get bk =

.

  (−1)k+1 2m − bk+1 , k = 0, 1, . . . , m − 1. 22m−2 m − k

Moreover, we have from (40) .bm =

(−1)m+1 . 22m−2

Finally, we obtain

 m−k  (−1)k+1  2m k = 1, . . . , m. .bk = j 22m−2

(42)

j =0

Unfortunately, the coefficients (42) do not satisfy the conditions of Theorem 5 in order to obtain higher order of approximations. But we may apply Theorem 4. For ∗ .Rw,1/w,b = Uw,b = Hw , we obtain

∗ H

∗ . Hw,m f − f ≤

w,m [C(R)]

+

m 

 |bk | Eπ w−0 (f )

k=1

1 1  ) ω2 (f, (2k − 1)2 |bk | , 2w 2 m

+

k=1

where the coefficients .bk are given by (42). We have by (42) for the sum m  .

|bk | =

k=1

1 22m−2

m m−k   2m k=1 j =0

j

=

1 22m−2

 m   2m k . m−k k=1

The formula ([23], 0.159, formula 2) m  .

k=1

yields

 k2

     2m 2m 2m − = (m + 1/2) − 22m−1 m−k m−k−1 m

56

A. Kivinukk and G. Tamberg

.

    m  2m 2m (2k − 1) = (m + 1/2) − 22m−1 . m−k m k=1

Since by (38)    m   2m 1 2m 2m−1 =2 , . − m−k 2 m k=1

we get    m   2m m 2m , . k = 2 m m−k k=1

and finally m  .

|bk | =

k=1



m 22m−1

 2m . m

Analogously, 2m−2

2

.

m 

(2k − 1) |bk | = 2

k=1

m 

(2k − 1)

2

m−k  j =0

k=1

    m 2m 2m 2 = . k j m−k k=1

Here, we can use for the last sum equation (29) yielding 1 .

22m−1

m 

 k

k=1

2

 2m m = ; m−k 2

hence, m  .

(2k − 1)2 |bk | = m

k=1

and we get the following statement. Theorem 8 For each .f ∈ C(R) for the linear combination of Hann sampling ∗ operators .Hw,m : C(R) → C(R) (m ∈ N) defined by the window function λ∗m (u) :=

.

1 − sin2m ( π2u ) cos( π2u )

On Generalized Shannon Sampling Operators in the Cosine Operator Function. . .

57

we have  ∗ ∗ ≤ Hw,m . Hw,m f − f + [C(R)] +

  1 m . ω2 f, 2 2w

m 22m−1



2m m

 Eπ w−0 (f )

The estimate obtained with .m = 1 fits well with the estimate in (33) for .m = 1. Remark 4 Let us mention here that in Theorem 8, the sequence of constants   2m .Cm := 22m−1 m m

√ is unbounded. We can show by Stirling’s formula for factorials that .Cm ≈ 2 m/π. This approximation is quite good even for small numbers .m ∈ N. In last section, we ∗ ‖ give an upper bound of the norm .‖Hw,m [C(R)] .

5 Inverse Estimates of the Order of Approximation for Certain Linear Combinations of the Powers of Hann Sampling Operators In this section, we find the approximation estimates below through the modulus of continuity. We start with a general statement. Lemma 2 Suppose a subadditive functional .P : C(R) → (0, ∞) is given such that there exist finite numbers P (f ) , f ∈C(R) ‖f ‖

M0 := sup

.

Nm,σ := sup

g∈B∞ σ

P (g) . ‖g (2m) ‖

(43)

Then for the sampling operators .Uw : C(R) → B∞ π w ⊂ C(R) in (1) for any .f ∈ (0 < ε < π w), the following holds: C(R), gε ∈ B∞ π w−ε   P (f ) ⩽ M0 1 + ‖Uw ‖[C(R)] ‖f − Uw f ‖ + Nm,π w−ε ‖(Uw gε )(2m) ‖.

.

(44)

Proof Take any .g ∈ B∞ π w−ε ; then by (43), we have P (f ) ⩽ P (f − Uw f ) + P (Uw f − Uw g) + P (Uw g) ⩽ M0 ‖f − Uw f ‖

.

+ M0 ‖Uw ‖[C(R)] ‖f − g‖ + Nm,π w−ε ‖(Uw g)(2m) ‖.

58

A. Kivinukk and G. Tamberg

∞ ∗ Now we take .g = gε ∈ B∞ π w−ε ⊂ Bπ w such that . lim ‖gε − g ‖ = 0, where ε→0

g ∗ ∈ B∞ π w is the element of the best approximation of .f ∈ C(R). Then we have

.

‖f − gε ‖ ⩽ ‖f − g ∗ ‖ + ‖g ∗ − gε ‖ ⩽ ‖f − Uw f ‖ + ‖g ∗ − gε ‖.

.

For .ε → 0, we get for any .f ∈ C(R), gε ∈ B∞ π w−ε that ‖f − gε ‖ ⩽ ‖f − Uw f ‖,

.

(45) ⨆ ⨅

and therefore, it follows the inequality (44).

This simple lemma makes possible to study the approximations of the operators in the previous section using a well-known generalization of Bernstein’s inequality [21]

‖g

.

(r)

‖⩽

r

σ

‖δhr (g)‖ (g ∈ B∞ σ ),

σh 2

2 sin

(46)

where .δhr (g) is the central difference m .(δh g)(x)

   m   mh j m g x+ − jh , := (−1) j 2 j =0

and, specifically, using the cosine operator function notation, 2m .(δh g)(x)

=2

m  '

 (−1)

j =0

m−j

 2m (Cj h g)(x). m−j

(47)

Accidently, the operators .Hw,m , defined via the window function λm (u) = 1 − sin2m (

.

πu ), 2

have, by (20), (47) and (36), (37), for .g ∈ B∞ π w−ε the representation Hw,m g = Swsinc g +

.

(−1)m+1 2m sinc δ1/w (Sw g), 22m

∞ and due to the Shannon theorem for .g ∈ B∞ π w−ε (and not in space .Bπ w as it is stated in [4], Th 6.3), we have

Hw,m g = g +

.

(−1)m+1 2m δ1/w g. 22m

(48)

On Generalized Shannon Sampling Operators in the Cosine Operator Function. . .

59

We are now ready to formulate our statement for the approximation estimates below concerning operators .Hw,m . Theorem 9 For each .f ∈ C(R) for the linear combination of Hann sampling operators .Hw,m : C(R) → C(R) (m ∈ N) defined by the window function 2m π u .λm (u) := 1 − sin ( 2 ), we have     2 2 ⩽ 22m Hw,m f − f Hw,m [C(R)] + 3 Hw,m [C(R)] + 1 . .ω2m f, πw ∞ Proof Take .gε ∈ B∞ π w−ε , then also .Hw,m gε ∈ Bπ w−ε , and, on the one hand, Bernstein’s inequality (46) with .h = 1/w yields for the second term in (44) the estimate

‖(Hw,m gε )

(2m)

.

By .sin x ⩾

2x π

(.0 ⩽ x ⩽

π 2 ),

.

‖⩽

πw − ε 2 sin π w−ε 2w

2m 2m ‖δ1/w (Hw,m gε )‖.

we get

πw − ε 2 sin π w−ε 2w

2m ⩽

 π w 2m 2

;

hence, ‖(Hw,m gε )(2m) ‖ ⩽

.

 π w 2m 2

2m ‖δ1/w (Hw,m gε )‖.

On the other hand, by (48), we obtain .

1 ‖δ 2m (Hw,m gε )‖ = Hw,m (Hw,m gε − gε ) . 22m 1/w

Therefore, ‖(Hw,m gε )(2m) ‖ ⩽ (π w)2 Hw,m [C(R)] Hw,m gε − gε .

.

We can estimate by (45) the norm .

  Hw,m f − f , Hw,m gε − gε ⩽ 2 + Hw,m [C(R)]

and finally, we obtain   ‖(Hw,m gε )(2m) ‖ ⩽ (π w)2 Hw,m [C(R)] 2 + Hw,m [C(R)] Hw,m f − f .

.

60

A. Kivinukk and G. Tamberg

Let us take in Lemma 2 .P (f ) = ω2m (f, t). Then by the properties of the modulus of continuity, we have for the constants therein the estimates M0 ⩽ 22m ,

.

Nm,σ ⩽ t 2m . Collecting the previous estimates and taking .t = 2/(π w) from Lemma 2 follow the statement of our theorem. ⨆ ⨅

6 Operator Norm of the Shannon Sampling Operators For the operator norm, we follow an idea from our previous paper [13]. The series of the right-hand side of (7) defines a function with period one; therefore, we may consider (7) in the form  ‖Uw ‖[C(R)] = sup

.

|t|⩽1/2

|s(t)| +

∞  

|s(k + t)| + s(k − t)|

  .

(49)

k=1

For estimation of the second term in (49), we have to estimate how fast the kernel function s in (2) decreases at infinity. Proposition 2 Let for the window function .λ be satisfied following conditions: (i) .λ is continuously differentiable on .[0, 1]. (ii) .λ(0) = 1; .λ(1) = 0. (iii) .λ' ⩽ 0 on .[0, 1], and there exists an .u0 ∈ [0, 1] such that .λ' is decreasing on .[0, u0 ] and it is increasing on .[u0 , 1]. Then we have for the kernel function s in (2) the following estimates: 1 , t > 0, . πt 2λ' (u0 ) |s(t)| ⩽ − , t > 0. (π t)2

|s(t)| ⩽

.

Proof For s in (2), after integrating by parts, we have 1 .s(t) = πt

1  0

 − λ' (u) sin(π tu) du.

(50) (51)

On Generalized Shannon Sampling Operators in the Cosine Operator Function. . .

61

Since .−λ' ⩾ 0 and .λ(0) = 1 and .λ(1) = 0, we obtain the estimate (50). Due to the monotonicity properties of .λ' , we may use the second mean-value theorem for the integrals ⎞ ⎛u  0 1   1 ⎝ + ⎠ − λ' (u) sin(π tu) du. .s(t) = πt u0

0

Therefore, there exist numbers .θ1 and .θ2 such that .0 ⩽ θ1 ⩽ u0 ⩽ θ2 ⩽ 1 and s(t) = −

.

 λ' (u0 )  cos(π tθ ) − cos(π tθ ) , 1 2 (π t)2 ⨆ ⨅

from which (51) follows.

Now we can give upper estimates for the operator norm .‖Uw ‖[C(R)] as follows: Theorem 10 Under assumptions of Proposition 2, we have: 1. ‖Uw ‖[C(R)] ⩽

.

2. For .−λ' (u0 ) ⩾

4 π

 1−

2λ' (u0 ) 3π

 + max |s(t)|. |t|⩽1/2

3π 4 ,

  4λ' (u0 ) 6 2 − 1 + + max |s(t)|. ⩽ ln − π π π |t|⩽1/2

‖Uw ‖[C(R)]

.

Proof Let .m ∈ N, .|t| ⩽ 1/2, and consider Sm :=

m 

.

+

k=1

∞ 

 |s(k + t)|

k=m+1

By (51), we have ∞  .

k=m+1

 ∞ dx 1 2λ' (u0 ) 2λ' (u0 )  ⩽− |s(k + t)| ⩽ − 2 2 2 (k + t) π (x + t)2 π ∞

k=m+1

=

2λ' (u0 ) − π2

m

4λ' (u0 ) 1 ⩽− 2 . m+t π (2m − 1)

(52)

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A. Kivinukk and G. Tamberg

If .m = 1, then by (50) and (51), we have S1 ⩽

.

4λ' (u0 ) 2 1 − ⩽ π π(1 + t) π2

  2λ' (u0 ) 1− . π

(53)

If .m ⩾ 2, .|t| ⩽ 1/2, by (50), we have m  .

|s(k + t)| ⩽

m m 1 1 1 1 1 = + π k+t π(1 + t) π k+t k=1

k=1

k=2

m−1 

1 1 + ⩽ π(1 + t) π

1

  1 1 m−2 dx = + ln 1 + x+t π(1 + t) π 1+t ⩽

2 1 + ln(2m − 3). π π

(54)

Precisely the same estimates we have for the series ∞  .

|s(k − t)|.

k=1

First we show that inequality (54) with .m = 2 improves the estimate (53). Indeed, for .m = 2, we have by (52) and (54) ∞ 

|s(k + t)| ⩽

.

k=1

2 π

  2λ' (u0 ) 1− 3π

(55)

From this follows the statement 1. For the statement 2, by (52) and (54), we obtain .(m ⩾ 2) ∞    2 8λ' (u0 ) 4 |s(k + t)| + |s(k − t)| ⩽ + ln(2m − 3) − 2 . . π π π (2m − 1) k=1

The last expression has the value close to its minimum when 2m − 1 ≈ −

.

4λ' (u0 ) =: V . π

Let us take for the integral part .⎿V ⏌ = 2m − 2 with .m ⩾ 2. Then .2m − 3 < V − 1, and .m ⩾ 2 can be satisfied, if .V ⩾ 3. Moreover, .2m−1 = ⎿V ⏌+1 > V . Therefore, it follows

On Generalized Shannon Sampling Operators in the Cosine Operator Function. . .

.

63

  ∞    4λ' (u0 ) 6 2 −1 + , |s(k + t)| + |s(k − t)| ⩽ ln − π π π k=1

if .−λ' (u0 ) ⩾

⨆ ⨅

3π 4 .

We can apply Theorem 10 for several sampling operators, defined via a window function. In particular:

Example 1. Hann Operator We know by Kivinukk and Tamberg [10] the exact value of the norm of Hann operator, 10 = 1.0610 . . . , 3π   defined by the window function .λ(u) = cos2 π2u = 12 1 + cos π u . We have ‖Hw ‖[C(R)] =

.

2

λ' (u) = − π2 sin π u ⩽ 0 on .[0, 1] and .λ'' (u) = − π2 cos π u with its unique zero .u0 = 1/2. Then .−λ' (u0 ) = π2 , and by Theorem 10, we get

.

‖Hw ‖[C(R)] ⩽

.

8 + max |s(t)|. 3π |t|⩽1/2

Here, 1 s(t) =

cos2

.

0

sin π t πu · cos(π ut) du = 2 2π t (1 − t 2 )

with its maximum value on .[−1/2, 1/2] as .s(0) = 1/2. Therefore, ‖Hw ‖[C(R)] ⩽

.

1 8 + = 1.348 . . . , 2 3π

which is slightly bigger than the true value.

Example 2. Linear Combination of Powers of Hann Operator Here, we study the Shannon sampling operators .Hw,m defined via the window functions (9): (continued)

64

A. Kivinukk and G. Tamberg

λH,m (u) = 1 − sin2m

.

πu . 2

We have (.u ∈ [0, 1]) λ'H,m (u) = −π m sin2m−1

.

πu πu · cos ⩽0 2 2

and   πu π 2 m 2m−2 π u sin 1 − 2m cos2 . 2 2 2

λ''H,m (u) =

.

1 There exists a unique .u0 ∈ [0, 1] such that .cos2 π2u0 = 2m and .λ'H,m is decreasing on .[0, u0 ] and increasing on .[u0 , 1]. Therefore, the assumptions of Proposition 2 are satisfied, and we may use Theorem 10. Here,

.

  1 m mπ 1− − λ'H,m (u0 ) = √ . 2m 2m − 1

The case .m = 1 is covered by Example 1. For .m = 2, we have .

− λ'H,2 (u0 ) =

π33/2 3π < 8 4

and by Theorem 10, ‖Hw,2 ‖[C(R)]

.

4 ⩽ π



√  3 1+ + max |sH,2 (t)| < 2.8245 . . . , 4 |t|⩽1/2

when we have used the very rough estimate .

max |sH,2 (t)| ⩽ 1.

|t|⩽1/2

For .m ⩾ 3 is valid .−λ'H,m (u0 ) > √ increasing to .1/ e, we obtain  ' . − λH,m (u0 )

⩽

3π 4

and, since .ym :=



1−

1 2m

m

is

3 √ π m (m ⩾ 3) 5e

and by Theorem 10, (continued)

On Generalized Shannon Sampling Operators in the Cosine Operator Function. . .

‖Hw,m ‖[C(R)]

.

65

  3 1 6 ⩽ ln m + ln π + + max |sH,m (t)| π 5e π |t|⩽1/2 =

1 ln m + 2.5407 . . . + max |sH,m (t)|. π |t|⩽1/2

To estimate the norm .‖Uw ‖[C(R)] from below, we write −k+1 

1

|s(t)| dt =

.

−k

|s(t − k)| dt

(k ∈ Z)

0

and then ∞ .

|s(t)| dt =

1  ∞

|s(t − k)| dt ⩽ sup

Because the series .

∞ 

|s(t − k)|.

t∈[0,1] k=−∞

0 k=−∞

−∞

∞ 

|s(t − k)| defines a function with period one, we have

k=−∞

∞ ‖Uw ‖[C(R)] ⩾

|s(t)| dt.

.

−∞

But for .Hw,m , we have on hand a result by Babushkin–Zhuk ([1] Th. 5, Remark 1): ∞ .

|sH,m (t)| dt ⩾

−∞

=

2 2 ln m + 2 ln 2 − 0.61 . . . 2 π π 2 ln m − 0.4695 . . . . π2

So we may claim the asymptotic formula ‖Hw,m ‖[C(R)] = c ln m + O(1)

.

with . π22 ⩽ c ⩽

1 π.

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A. Kivinukk and G. Tamberg

∗ Evaluating the norms of operators .Hw,m which are defined in Sect. 4 via the window function

λ∗m (u) :=

.

1 − sin2m ( π2u ) , m∈N cos( π2u )

is a much more complicated story. First, the window .λ∗m does not satisfy the ∗ assumptions of Theorem 10. Second, the kernel function of .Hw,m is too complicated to try to estimate the norm directly from the formula (7). Recall here that the window function .λk,R (u) = cos(π(k − 1/2)u) defines the Rogosinski sampling operator .Rw,k , for which we know from one of our previous articles [10] that ‖Rw,k ‖[C(R)] =

.

2k−2 4  1 . 2l + 1 π l=0

In Sect. 4, we deduced the equality ∗ .λm (u)

=

m 

bk λk,R (u)

k=1

with coefficients (42) as bk =

.

 m−k  (−1)k+1  2m k = 1, . . . , m. j 22m−2 j =0

Thus, we have ∗ Hw,m =

m 

.

bk Rw,k .

k=1

Using the norm .‖Rw,k ‖[C(R)] , we obtain ∗ ‖Hw,m ‖[C(R)] ≤

.

m 2k−2  1 4 . |bk | 2l + 1 π k=1

l=0

If we put .bk into the previous inequality, then we get after complicated (triple sum !) but basic transformations an estimate ∗ ‖Hw,m ‖[C(R)] ≤

.

  m 2m 4 1  (k + 1) . π 22m−1 m−k k=1

(56)

On Generalized Shannon Sampling Operators in the Cosine Operator Function. . .

67

We cannot improve this estimate in general, as the case .m = 1 is equally valid here, ∗ =R since .Hw,1 w,1 . In Sect. 4, we deduced equalities 1 .

22m−1

   m   2m 1 2m = 1 − 2m m m−k 2 k=1

and 1 .

22m−1

   m   2m m 2m . k = 2m m m−k 2 k=1

Putting these equalities into (56), we finally get ∗ .‖Hw,m ‖[C(R)]

4 ≤ π



  m − 1 2m . 1 + 2m m 2

Acknowledgments The first author knew Professor J. Rowland Higgins since SampTA in Riga/Jurmala in 1995. Rowland was one of the keynote speakers and had compiled a written tutorial to describe an important part of mathematics (bases, frames, reproducing kernel spaces, etc.) useful in Shannon’s sampling theory. As a “good student,” I sat in the front row of the lecture room, and so I was pleased to receive his “A Sampling Theorist’s Guide to Bases and Frames: A Tutorial,” Jurmala (Riga) Latvia, September 1995 (36 pages; References include 49 titles), which is still ranked among my books on sampling theory. The work of the second author was partly supported by the Estonian Research Council grant (PRG1483).

References 1. M.V. Babushkin, V.V. Zhuk, On approximation of periodical functions by generalized Rogosinski sums (in Russian). Transactions of Tula State University. Natl. Sci. 2, 5–29 (2014) 2. P.L. Butzer, A. Gessinger, Ergodic theorems for semigroups and cosine operator functions at zero and infinity with rates; applications to partial differential equations. A survey. Contemp. Math. 190, 67–94 (1995). https://doi.org/10.1090/conm/190/02293 3. P.L. Butzer, R.J. Nessel, Fourier Analysis and Approximation (Birkhäuser Verlag, BaselStuttgart, 1971) 4. P.L. Butzer, W. Splettstösser, R.L. Stens, The sampling theorems and linear prediction in signal analysis. Jahresber. Deutsch. Math-Verein 90, 1–70 (1988) 5. A. Ferrero et al. (eds.), Modern Measurements. Fundamentals and Applications (IEEE Press/Wiley, Piscataway/Hoboken, 2015) 6. F.J. Harris, On the use of windows for harmonic analysis with the discrete Fourier transform. Proc. IEEE 66(1), 51–83 (1978) 7. J.R. Higgins, Sampling Theory in Fourier and Signal Analysis (Clarendon Press, Oxford, 1996) 8. L.V. Kantorovich, G.P. Akilov, Functional Analysis, 2nd edn. (Pergamon Press, OxfordElmsford, 1982)

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9. A. Kivinukk, A. Saksa, On Rogosinski-type approximation processes in Banach space using the framework of the cosine operator function. Math. Found. Comput. (2021). https://doi.org/ 10.3934/mfc.2021030 10. A. Kivinukk, G. Tamberg, On sampling operators defined by the Hann window and some of their extensions. Sampl. Theory Signal Image Process. 2, 235–258 (2003) 11. A. Kivinukk, G. Tamberg, Blackman-type windows for sampling series. J. Comput. Analy. Appl. 7(4), 361–372 (2005) 12. A. Kivinukk, G. Tamberg, On Blackman-Harris windows for Shannon sampling series. Sampl. Theory Signal Image Process. 6, 87–108 (2007) 13. A. Kivinukk, G. Tamberg, On window methods in generalized Shannon sampling operators, in ed. by G. Schmeisser, A. Zayed, New Perspectives on Approximation and Sampling TheoryFestschrift in Honor of Paul Butzer’s 85th Birthday (Birkhauser Verlag, Basel, 2014), pp. 63– 86 14. A. Kivinukk, A. Saksa, M. Zeltser, On a cosine operator function framework of approximation processes in Banach space. Filomat 33, 4213–4228 (2019) 15. D. Lutz, Strongly continuous operator cosine functions, in ed. by D. Butkovi´c, H. Kaljevi´c, S. Kurepa, Functional Analysis: Proceedings of a Conference Held at Dubrovnik, Yugoslavia, 1981. Lect Notes in Mathematics, vol. 948 (1982), pp. 73–97 16. R.G. Lyons, Understanding Digital Signal Processing, 3rd edn. (Prentice Hall, Hoboken, 2011) 17. A.H. Nuttall, Some Windows With Very Good Sidelobe Behaviour; Application to Discrete Hilbert Transform. NUSC Technical Report 6239. Surface Ship Sonar Systems Department (1980) 18. W.W. Rogosinski, Reihensummierung durch Abschnittskoppelungen. Math. Z. 25, 132–149 (1926) 19. M. Sova, Cosine operator functions. Rozprawy Mat. 49, 47 (1966) 20. S.B. Stechkin, Summation methods of S. N. Bernstein and W. Rogosinski, in ed. by G.H. Hardy, Divergent Series Moscow (Russian Edition) (1951), pp. 479–492 21. A.F. Timan, Theory of Approximation of Functions of a Real Variable (MacMillan, New York, 1963) 22. O.L. Vinogradov, V.V. Zhuk, Estimates for functionals with a known finite set of moments in term of moduli of continuity, and behavior of constants in the Jackson-type inequalities. St. Petersburg Math. J. 24(5), 691–721 (2012) 23. D. Zwillinger, V. Moll (eds.), Grandshteyn and Ryzhik’s Table of Integrals, Series and Products, Eighth edn. (Academic, Cambridge, 2014)

Bernstein Spaces, Sampling, and Riesz-Boas Interpolation Formulas in Mellin Analysis Isaac Z. Pesenson

1 Introduction In a series of interesting papers by C. Bardaro et al. [2–12], and also by P. Butzer and S. Jansche [16–18], the authors developed in the framework of Mellin analysis such important topics as Sobolev spaces, Bernstein spaces, Bernstein inequality, PaleyWiener theorem, Riesz-Boas interpolation formulas, and different sampling results. Many of their results were obtained by using the notion of polar-analytic functions (see [5, 8–11]). These authors published so many fundamental results devoted to different parts of the Mellin analysis that it makes very difficult to give even a short review of them. In this Introduction, we are going to recall just a few of them. In [3], the authors proved a version of the Paley-Wiener theorem for Mellin transforms by introducing the Mellin-Bernstein spaces, comprising all functions f ∈ Xc2 := {f : R+ I→ C : f (·)(·)c−1/2 ∈ L2 (R+ )},

.

which have an analytic extension to the Riemann surface of the (complex) logarithm and satisfy some suitable exponential type conditions. An equivalent formulation of the Paley-Wiener theorem was given in [5], based on the notion of “polar-analytic” function, which avoids the use of the Riemann surfaces and analytic branches simply by considering functions defined on the half-plane .H := {(r, θ ) : r > 0, θ ∈ R}. This modified notion of analyticity leads naturally to the classical Cauchy-Riemann equations in polar form (see [22, 25]). A number of extensions of the classical sampling theorem to the Mellin analysis with applications to optical physics were given by the same authors in [2, 4]. The so-called Boas interpolation formula

I. Z. Pesenson () Temple University, Philadelphia, PA, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. D. Casey et al. (eds.), Sampling, Approximation, and Signal Analysis, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-031-41130-4_4

69

70

I. Z. Pesenson

[13, 14, 26] which gives expression of the derivative of a classical bounded Bernstein function as a linear combination of its translates was extended to Mellin analysis in [10]. The objective of our chapter is to present a very different approach to the same topics based solely on the fact that the family of Mellin translations defined as Uc (t)f (x) = ect f (et x), Uc (t + τ ) = Uc (t)Uc (τ ), c ∈ R,

.

(1)

forms a one-parameter .C0 -group of isometries in appropriate function spaces (see below). As one can see, its infinitesimal generator is the operator .

d d Uc (t)f (x)|t=0 = x f (x) + cf (x) = Ɵc f (x). dt dx

(2)

In this chapter, we are guided by our abstract theory of sampling and interpolation in Banach spaces which was developed in [27, 28]. At the same time, our approach is very specific and direct, and we are not using the language of one-parameter groups. The fact that we consider a very concrete situation allows us to obtain results which we did not have in our general development. All our results hold true for a general group of translations .Uc with any .c ∈ R. However, for the sake of simplicity, we consider only the case .c = 0 and adapting notations .U0 = U, Ɵ0 = Ɵ. In Sect. 2, we define analog of Bernstein spaces using Bernstein-type inequality for the operator .Ɵ. Our analog of the Paley-Wiener theorem is Theorem 2. In Sect. 3, we prove four sampling theorems. Our formula (26), which is a generalization of the Valiron-Tschakaloff sampling theorem, looks exactly like the one proved in [10]. Three other theorems in this section seem to be new. They all deal with regularly spaced sampling points. In contrast, Sect. 4 contains two sampling theorems which are using irregularly spaced sampling points. One of these theorems is a generalization of a sampling theorem due to J.R. Higgins [23] and another one due to C. Seip [32]. In Sect. 5, we discuss what we call the Riesz-Boas interpolation formulas. The famous Riesz interpolation formula [26, 29, 30] gives expression of the derivative of a trigonometric polynomial as a linear combination of its translates:  .

d dt

 P (t) =

  2n 1 1  2k − 1 (−1)k+1 2 2k−1 P π +t . 4π 2n sin 4n π

(3)

k=1

This formula was extended by Boas [13, 14], (see also [1, 26, 31]) to functions in the Bernstein class .B∞ σ (R) in the following form:  .

d dt

 f (t) =

π  σ  (−1)k−1 f (k − 1/2) + t , σ ∈ R+ . 2 2 σ π (k − 1/2) k∈Z

(4)

Bernstein Spaces, Sampling, and Riesz-Boas Interpolation Formulas in Mellin Analysis

71

In turn, the formula (4) was extended in [21] to higher powers .(d/dt)m . Our objective in Sect. 5 is to obtain similar formulas for .m ∈ N where the operator d .d/dt is replaced by the operator .Ɵ = x dx . When .m = 1, such formula for the operator .Ɵ was established in [10]. Obviously, the goals of the present chapter are quite close to some of the objectives of the papers [2–12]. It would be very interesting and instructive to do a rigorous comparison of our approaches and outcomes. However, the fact that our results are based on rather different ideas makes such a comparison not easy. A serious juxtaposition of our treatments would require substantial increase of the length of the present chapter. We are planning to do such analysis in a separate paper.

2 Bernstein Spaces 2.1 Mellin Translations For .p ∈ [1, ∞[, denote by .‖ · ‖p the norm of the Lebesgue space .Lp (R+ ). In Mellin analysis, the analog of .Lp (R+ ) are the spaces .Xp (R+ ) comprising all functions −1/p ∈ Lp (R ) with the norm .‖f ‖ p .f : R+ I→ C such that .f (·)(·) + X (R+ ) := −1/p ‖f (·)(·) ‖p . Furthermore, for .p = ∞, we define .X∞ as the space of all measurable functions .f : R+ I→ C such that .‖f ‖X∞ := supx>0 |f (x)| < ∞. In spaces .Xp (R+ ), we consider the one-parameter .C0 -group of operators .U (t), t ∈ R, where U (t)f (x) = f (et x), U (t + τ ) = U (t)U (τ ),

.

(5)

whose infinitesimal generator is .

d d U (t)f (x)|t=0 = x f (x) = Ɵf (x). dt dx

(6)

The domain of its power .k ∈ N is denoted as .Dk (Ɵ) and defined as the set of all functions .f ∈ Xp (R+ ), 1 ≤ p ≤ ∞, such that .Ɵk f ∈ Xp (R+ ). The domains k .D (Ɵ), k ∈ N, can be treated as analogs of the Sobolev spaces. The general theory of one-parameter semi-groups of class .C0 (see [15, 24]) implies that the operator .Ɵ is closed in .Xp (R+ ) and the set .D∞ (Ɵ) = ∩k Dk (Ɵ) is dense in .Xp (R+ ). By using the following formula (see [16], p. 355): Ɵk f (x) =

k 

.

r=0

S(k, r)x r f (r) (x),

(7)

72

I. Z. Pesenson

S(k, r) being Stirling numbers of the second kind, one can give more explicit description of the Mellin-Sobolev spaces (see [16], p. 357) in the spirit of Bochner’s definition of the classical one-dimensional Sobolev spaces.

.

Remark 1 The formula (5) simply means that the map R I→ R+ : t I→ et

.

is an isomorphism of the commutative group .(R, +) onto commutative group (R+ , ×). Using this observation, one can easily obtain a few basic formulas in the Mellin analysis. However, more subtle results require a direct approach.

.

2.2 Bernstein Spaces Let’s recall that in the classical analysis, a Bernstein class [1, 26], which is denoted p as .Bσ (R), σ > 0, 1 ≤ p ≤ ∞, is a linear space of all functions .f : R I→ C which belong to .Lp (R) and admit extension to .C as entire functions of exponential type p .σ . A function f belongs to .Bσ (R) if and only if the following Bernstein inequality holds: ‖f (k) ‖Lp (R) ≤ σ k ‖f ‖Lp (R) ,

.

for all natural k. Using the distributional Fourier transform 1 f(ξ ) = √ 2π



.

R

f (x)e−iξ x dx, f ∈ Lp (R), 1 ≤ p ≤ ∞, p

one can show (Paley-Wiener theorem) that .f ∈ Bσ (R), 1 ≤ p ≤ ∞, if and only if p  (in the sense of distributions) is in .f ∈ L (R), 1 ≤ p ≤ ∞, and the support of .f .[−σ, σ ]. p

Definition 1 The Bernstein space .Bσ (Ɵ), σ > 0, 1 ≤ p ≤ ∞, is defined as a set of all functions f in .Xp (R+ ) which belong to .D∞ (Ɵ) and for which ‖Ɵk f ‖Xp (R+ ) ≤ σ k ‖f ‖Xp (R+ ) , k ∈ N.

.

(8)

Theorem 1 A function .f ∈ D∞ (Ɵ) belongs to .Bσ (Ɵ), σ > 0, 1 ≤ p ≤ ∞, if and only if the quantity p

.

sup σ −k ‖Ɵk f ‖Xp (R+ ) = R(f, σ )

k∈N

is finite.

(9)

Bernstein Spaces, Sampling, and Riesz-Boas Interpolation Formulas in Mellin Analysis

73

p

Proof It is evident that if .f ∈ Bσ (Ɵ), then (9) holds. Next, for an .h ∈ Xq (R+ ), 1/p + 1/q = 1, consider a scalar-valued function  Ф(t) =

.

R+

f (et x)h(x)

dx . x

We note that .

 .

d dt

2

d d ' f (et x) = et xf (et x) = x f (et x), dt dx

d f (e x) = dt t

    d d 2 t x f (e x) = x f (et x), dx dx

and in general  .

d dt

k

  d k f (et x). f (et x) = x dx

It implies that  .

d dt

k



 Ф(0) =

R+

d dt

k f (et x)|t=0 h(x)

dx = x

  d k dx x f (x)h(x) . dx x R+



Now for the McLaurin series of .Ф, one has      ∞ ∞   d k 1 k d k 1 k dx x .Ф(t) = Ф(0) = f (x)h(x) . t t k! dt k! dx x R+

(10)

k=0

k=0

This series is absolutely convergent since by the assumption (9),

.

|Ф(t)| ≤

 ∞  1 k dx t Ɵk f (x)h(x) ≤ x k! R+ k=0

R(f, σ )

.

∞  1 k k t σ ‖f ‖Xp (R+ ) ‖h‖Xq (R+ ) = R(f, σ )‖f ‖Xp (R+ ) ‖h‖Xq (R+ ) eσ t . k! k=0

(11) It implies that .Ф can be extended to the complex plane .C by using its McLaurin series (10). Moreover, as the estimate (11) shows, the inequality .

|Ф(z)| ≤ R(f, σ )‖f ‖Xp (R+ ) ‖h‖Xq (R+ ) eσ |z| , z ∈ C,

74

I. Z. Pesenson

will hold. In addition, .Ф is bounded on the real line by the constant p ‖f ‖Xp (R+ ) ‖h‖Xq (R+ ) . In other words, we proved that if .f ∈ Bσ (Ɵ), h ∈ q X (R+ ), 1/p + 1/q = 1, then .Ф belongs to the regular Bernstein space .B∞ σ (R). This fact allows to apply to .Ф the classical Bernstein inequality in the space .C(R) of continuous functions on .R with the uniform norm:    d k d k Ф(0) ≤ sup Ф(t) ≤ σ k sup |Ф(t)| . . dt t dt t

.

Since  .

d dt

k



 Ф(0) =

R+

d dt

k

dx = f (e x)|t=0 h(x) x



t

R+

Ɵk f (x)h(x)

dx x

we obtain .



R+

Ɵk f (x)h(x)

dx ≤ σ k ‖f ‖Xp (R+ ) ‖h‖Xq (R+ ) x

Choosing h such that .‖h‖Xp (R+ ) = 1 and  .

R+

Ɵk f (x)h(x)

dx = ‖Ɵk f ‖Xp (R+ ) x

we obtain the inequality ‖Ɵk f ‖Xp (R+ ) ≤ σ k ‖f ‖Xp (R+ ) , k ∈ N.

.

⨆ ⨅

Theorem is proved.

The following analog of the Paley-Wiener theorem follows from the proof of the previous theorem. Theorem 2 The following conditions are equivalent: p

1. f belongs to .Bσ (Ɵ), 1 ≤ p ≤ ∞. 2. For every .g ∈ Xq (R+ ), 1/p + 1/q = 1, the function  Ф(z) =

.

R+

f (ez x)g(x)

dx , z ∈ C, x

belongs to the regular space .B∞ σ (R), i.e., it is an entire function of exponential type .σ which is bounded on the real line.

Bernstein Spaces, Sampling, and Riesz-Boas Interpolation Formulas in Mellin Analysis

75

3 Sampling Theorems in Mellin Analysis 3.1 A Sampling Theorem in Integral Form Below, we are going to use the following known fact (see [19], p. 46). Theorem 3 If .h ∈ B∞ σ (R), then for any .0 < γ < 1, the following formula holds:     kπ  σ sinc γ −1 z − k , z ∈ C, .h(z) = h γ π σ

(12)

k∈Z

where the series converges uniformly on compact subsets of .C. By using Theorem 3, we obtain a sampling theorem in integral form. p

Theorem 4 If .f ∈ Bσ (Ɵ), 1 ≤ p ≤ ∞, then for all .g ∈ Lq (Ɵ), 1/p + 1/q = 1, and all .0 < γ < 1, the following formula holds:  .

R+

  .

k∈Z

R+

f (e

γ kπ/σ

dx x)g(x) x

f (τ x)g(x) 

dx = x

  σ sinc γ −1 ln τ − k , τ ∈ R+ , π

(13)

where the series converges uniformly on compact subsets of .R+ . p

Proof According to Theorem 2, for any .f ∈ Bσ (Ɵ), 1 ≤ p ≤ ∞, and any q .g ∈ X (R+ ), 1/p + 1/q = 1, the function  Ф(t) =

.

R+

f (et x)g(x)

dx , t ∈ R, x

(14)

belongs to .B∞ σ (R). Applying Theorem 3, we obtain  .

R+

f (et x)g(x)

dx = x

  .

k∈Z

     σ dx sinc γ −1 t − k , t ∈ R, f eγ kπ/σ x g(x) π x R+

(15)

where the series converges uniformly on compact subsets of .R. Setting .τ = et or p q .t = ln τ , we obtain that for any .f ∈ Bσ (Ɵ), g ∈ X (R+ ), 1/p + 1/q = 1,  .

R+

f (τ x)g(x)

dx = x

76

I. Z. Pesenson

 .

k∈Z

    σ dx f eγ kπ/σ x g(x) sinc γ −1 ln τ − k , τ ∈ R+ , x π R+

where the series converges uniformly on compact subsets of .R+ . Theorem is proved.

(16)

⨆ ⨅

3.2 A Sampling Formula for Mellin Convolution According to [16], the Mellin convolution is defined as  F ∗M G(z) =

.

R+

F

z u

G(u)

du . u

p

Theorem 5 For any .f ∈ Bσ (Ɵ) and .h ∈ Xq (R+ ), 1/p + 1/q = 1, 1 < p < ∞, the following formula holds: f ∗M h(τ ) =



.

 f ∗M h

k∈Z

γ kπ σ



  σ sinc γ −1 ln τ − k , τ ∈ R+ , π

(17)

where the series converges uniformly on compact subsets of .R+ . Proof In the formula (16), we replace .g(x), x > 0, by .h(1/x), x > 0, and then perform the substitution .x = 1/y. After all, the formula (16) takes the form   τ dy = .f ∗M h(τ ) = f h(y) y y R+ 

 .

k∈Z R+

 .

k∈Z

 f

eγ kπ/σ y



γ kπ f ∗M h σ





h(y)

  σ dy sinc γ −1 ln τ − k = y π

  σ sinc γ −1 ln τ − k , τ ∈ R+ , π

where the series converges uniformly on compact subsets of .R+ . Theorem is proven.

(18)

⨆ ⨅

Remark 1 Note that (17) is an analog of the formula f ∗ g(t) =



.

  f ∗ g(γ kπ/σ ) sinc γ −1 σ t/π − k ,

(19)

k∈Z p

where .f ∈ Bσ (R), 1 ≤ p < ∞, . g ∈ Lq (R), 1/p + 1/q = 1, . 0 < γ < 1, and

Bernstein Spaces, Sampling, and Riesz-Boas Interpolation Formulas in Mellin Analysis

1 f ∗ g(t) = √ 2π

77



.

R

f (x)g(t − x)dx,

is the classical convolution.

3.3 Valiron-Tschakaloff-Type Sampling Formulas The next theorem contains an analog of the Valiron-Tschakaloff sampling/interpolation formula in Mellin analysis (see [20] for the classical Valiron-Tschakaloff formula). p

Theorem 6 If .f ∈ Bσ (Ɵ), 1 < p < ∞, τ ∈ R+ , then   σ ln τ (∂x f )(1)+ ln τ f (1) + ln τ sinc π π      σ σ ln τ − k . (ln τ )sinc f ekπ/σ π kπ

f (τ ) = sinc

.

.

σ

(20)

k∈Z\{0}

The series converges absolutely and uniformly on compact subsets of .R+ . p

Proof As we know (see Theorem 2), if .f ∈ Bσ (Ɵ), then the function  Ф(t) =

.

R+

f (et x)g(x)

dx , t ∈ R, x

(21)

belongs to .B∞ σ (R). By applying to it the Valiron-Tschakaloff sampling/interpolation formula which holds for functions in .B∞ σ (R) (see [20]), we obtain 

σt .h(t) = t sinc π  sinc

.

σt π

 f (0) +



'

h (0)+

     σt σt kπ sinc −k h , h ∈ B∞ σ (R), kπ π σ k/=0

where convergence is absolute and uniform on compact subsets of .R. Thus,  Ф(t) = t sinc

.

 sinc

.

σt π

 f (0) +

σt π



'

Ф (0)+

     σt σt kπ sinc −k Ф , kπ π σ k/=0

(22)

78

I. Z. Pesenson p

or for .f ∈ Bσ (R), 1 < p < ∞,  .

R+

f (et x)g(x)

     

σt σt dx dx sinc = f (x) + t sinc (x∂x f )(x) g(x) + x π π x R+

 .

k∈Z\{0}

  σt σt dx sinc −k f (ekπ/σ x)g(x) , kπ π x R+

(23)

where convergence is absolute and uniform on compact subsets of .R. Since the following holds:    σ t σ t kπ/σ x) . − k f (e sinc π k∈Z\{0} kπ 

≤ ‖f ‖Xp (R+ )

k/=0, σ t/π

⎛ ‖f ‖Xp (R+ ) ⎝



.

k/=σ t/π

Xp (R+ )

1 1 ≤ |σ t/π − k| |k|

⎞1/p ⎛ ⎞1/q  1 ⎠ 1 ⎠ ⎝ < ∞, |σ t/π − k|p |k|q k/=0

we can rewrite (23) as  .

R+

f (et x)g(x)

     

σt σt dx dx = f (x) + t sinc (x∂x f )(x) g(x) + sinc x π π x R+

⎡





⎣

.

R+

k∈Z\{0}

⎤   σt σt dx − k f (ekπ/σ x)⎦ g(x) . sinc π x kπ

The last equality holds for any .g ∈ Xq (R+ ), 1/p + 1/q = 1, and since for .1 < p < ∞ the space .Xq (R+ ) contains all functionals for .Xp (R+ ), we can conclude that the next equality holds  f (et x) = sinc

.

 .

k∈Z\{0}

σt π



 f (x) + t sinc

σt π

 (x∂x f )(x)+

  σt σt − k f (ekπ/σ x). sinc π kπ

Substituting .x = 1 into (24), we obtain

(24)

Bernstein Spaces, Sampling, and Riesz-Boas Interpolation Formulas in Mellin Analysis



σt .f (e ) = sinc π t

 .

k∈Z\{0}





σt f (1) + t sinc π

79

 (∂x f )(1)+

  σt σt sinc − k f (ekπ/σ ), t ∈ R. kπ π

For .τ = et , .t ∈ R, one has  σ  σ ln τ f (1) + ln τ sinc ln τ (∂x f )(1)+ .f (τ ) = sinc π π      σ σ . f ekπ/σ ln τ − k , τ ∈ R+ . (ln τ )sinc π kπ

(25)

(26)

k∈Z\{0}

Theorem is proven. p

For every .f ∈ Bσ (Ɵ), g ∈ Xq (R+ ), 1/p + 1/q = 1, let’s introduce the function .Ψ defined as follows:  dx f (et x) − f (x) 1 g(x) , (27) .Ψ(t) = (Ф(t) − Ф(0)) = t t x R+ if .t /= 0, and Ψ(0) =

.

d Ф(t)|t=0 = dt

 R+

Ɵf (x)g(x)

dx , x

(28)

if .t = 0. p

Lemma 1 If .f ∈ Bσ (Ɵ), g ∈ Xq (R+ ), 1/p + 1/q = 1, then .Ψ(t) defined in (27) and (28) belongs to .Brσ (R) for any .r > 1.  k Proof We remind [26] that a function .h(t) = ∞ k ak t is an entire function of the exponential type .σ if and only if the following condition holds:  limk→∞ k k!|ak | ≤ σ.

.

Since .Ф belongs to classical Bernstein class .B∞ σ (R) (see Theorem  2),k it is an entire√function of the exponential type .σ .  Thus, .Ф(t) = k ck t with k k−1 , where one clearly has k!|c | ≤ σ . Now we have that .Ψ(t) = c t . limk→∞ k k k  k . limk→∞ k!|ck+1 | ≤ σ , which means that .Ψ is an entire function of the exponential type .σ . Moreover, for any .r > 1, the function .Ψ is in .Lr (R) since  . |Ψ(t)| =

R+

f (et x) − f (x) dx 2‖f ‖Xp (R+ ) ‖g‖Xq (R+ ) . g(x) ≤ x |t| t

80

I. Z. Pesenson

Thus, .Ψ belongs to .Brσ (Ɵ) for any .r > 1. Lemma is proven. p

Theorem 7 If .f ∈ Bσ (Ɵ), 1 < p < ∞, then the following sampling formulas hold for .τ ∈ R+ : f (τ ) = f (1) + ln τ (∂x f )(1) sinc

σ

.

.

 kπ   f e σ − f (1)

ln τ

sinc

kπ σ

k/=0

σ π

π

 ln τ +

 ln τ − k ,

(29)

where the series converges uniformly on compact subsets of .R+ . Proof Consider the same .Ψ as before. Since .Ψ ∈ Brσ (Ɵ), it implies (see [19], p. 46) the following formula: Ψ(t) =



.

 Ψ

k∈Z

  σ k t −k , π sinc π σ

the series being uniformly convergent on each compact subset of .R. Thus,  .

R+

dx f (et x) − f (x) = g(x) x t 



k

R+

f (e σ π x) − f (x)

.

k σπ

R+

k∈Z\{0}

 Ɵf (x)g(x)

dx g(x) x

σ  dx sinc t + x π

 sinc

σ π

 t −k .

The last formula can be rewritten as     σ  dx dx dx t = + sinc t + . f (e x)g(x) f (x)g(x) Ɵf (x)g(x) x x x π R+ R+ R+ t





k

f (e σ π x) − f (x)

.

R+

k∈Z\{0}

k σπ

dx g(x) x

 sinc

σ π

Since the series .

 f (e σk π x) − f (x) k∈Z\{0}

k σπ

sinc

converges in .Xp (R+ ), we can write the equality

σ π

t −k



 t −k .

Bernstein Spaces, Sampling, and Riesz-Boas Interpolation Formulas in Mellin Analysis

 .

R+





.

R+

f (et x)g(x)

f (x) + tƟf (x)sinc

k σπ

k∈Z\{0}

dx = x

σ  t π

 f (e σk π x) − f (x)

+t

81

⎞  dx t − k ⎠ g(x) . sinc π x σ

Because this equality holds true for every .g ∈ Xq (R+ ), 1p+1/q = 1, 1 < p < ∞, we finally come to the formula f (et x) = f (x)+

.

 t (x∂x f )(x) sinc

.

σt π

 +t

 kπ   f e σ x − f (x) kπ σ

k/=0

 sinc

 σt −k . π

(30)

By setting .x = 1, we obtain f (et ) = f (1)+

.

 t (∂x f )(1) sinc

.

σt π

 +t

 kπ   f e σ − f (1) kπ σ

k/=0

 sinc

 σt −k , π

(31)

and for .τ = et , t = ln τ, one has (29). Theorem is proven.

4 Two Theorems Which Involve Irregular Sampling The following fact was proved by J.R. Higgins in [23]. Theorem 8 Let .{tk }k∈Z be a sequence of real numbers such that .

sup |tk − k| < 1/4.

(32)

k∈Z

Define the entire function G(z) = (z − t0 )



.

k∈Z

1−

z tk

  z . 1− t−k

(33)

82

I. Z. Pesenson

Then for all .f ∈ B2π (R), we have f (t) =



f (tk )

.

k∈Z

G(t) , G (tk )(t − tk ) '

uniformly on every compact subset of .R. p

As we know (Lemma 1) for every .f ∈ Bπ (Ɵ), g ∈ Xq (R+ ), 1/p + 1/q = 1, the function .Ψ defined as  dx f (et x) − f (x) g(x) , (34) .Ψ(t) = t x R+ if .t /= 0, and  Ψ(0) =

.

R+

Ɵf (x)g(x)

dx , x

(35)

if .t = 0, belongs to .B2π (R). Applying to it Theorem 8, we obtain the following. Theorem 9 Under assumptions and notations of Theorem 8, for every .f ∈ p Bπ (Ɵ), g ∈ Xq (R+ ), 1/p + 1/q = 1, 1 < p < ∞, the following formula holds: Ψ(t) =



.

Ψ(tk )

k∈Z

G(t) , G (tk )(t − tk )

(36)

'

uniformly on every compact subset of .R. Note that if the sequence .{tk } does not contain zero, then the formula (36) takes the form     f (etk x) − f (x) dx G(t) f (et x) − f (x) dx g(x) = . g(x) . ' t x tk x G (tk )(t − tk ) R+ R+ k∈Z

But in the case .t0 = 0, the formula (36) has the form 

dx f (et x) − f (x) g(x) = t x

.

R+

  .

k∈Z\{0}

R+



dx Ɵf (x)g(x) x R+

f (etk x) − f (x) dx g(x) x tk





G(t) + ' G (0)t

G(t) . ' G (tk )(t − tk )

In the paper by C. Seip [32], the following result can be found.

Bernstein Spaces, Sampling, and Riesz-Boas Interpolation Formulas in Mellin Analysis

83

Theorem 10 Under assumptions and notations of Theorem 8, for any .0 < δ < π and all .f ∈ B∞ π −δ (R), the following holds true: 

f (t) =

.

f (tk )

k∈Z

G(k) , ' G (tk )(t − tk )

uniformly on all compact subsets of .R. This theorem together with Theorem 2 implies the following theorem. p

Theorem 11 If .f ∈ Bπ −δ (Ɵ), 0 < δ < π, 1 ≤ p ≤ ∞, then under assumptions and notations of Theorem 8, the following formula holds uniformly on compact subsets of .C 

Ф(t) =

.

Ф(tk )

k∈Z

G(t) , G (tk )(t − tk ) '

or  .

 dx = f (e x)g(x) x R+



t

k∈Z

dx f (e x)g(x) x R+ tk



G(t) , ' G (tk )(t − tk )

where .g ∈ Xq (R+ ), 1/p + 1/q = 1, 1 ≤ p ≤ ∞.

5 Riesz-Boas Interpolation Formulas We introduce the following bounded operators in the spaces .Xp (R+ ), 1 ≤ p ≤ ∞. R(2m−1) (σ )f (x) =

.

.

 σ 2m−1   π  (−1)k+1 Am,k f e σ (k−1/2) x , π

f ∈ Xp (R+ ), σ > 0, m ∈ N,

k∈Z

(37) and R(2m) (σ )f (x) =

.

 σ 2m  .

π

 πk  (−1)k+1 Bm,k f e σ x ,

k∈Z

where .Am,k and .Bm,k are defined as

f ∈ Xp (R+ ), σ > 0, m ∈ N,

(38)

84

I. Z. Pesenson

 Am,k = (−1)

.

k+1

sinc

(2m−1)

 1 −k = 2

  m−1 1 2j (2m − 1)!  (−1)j π(k − ) . , m ∈ N, 2 (2j )! π(k − 12 )2m

(39)

j =0

for .k ∈ Z, Bm,k = (−1)k+1 sinc(2m) (−k) =

.

m−1 (2m)!  (−1)j (π k)2j +1 , m ∈ N, (2j + 1)! π k 2m+1

(40)

j =0

for .k ∈ Z \ {0}, and Bm,0 = (−1)m+1

.

π 2m , m ∈ N. 2m + 1

(41)

Both series converge in .Xp (R+ ), 1 ≤ p ≤ ∞, and their sums are (see [21]) .

 σ 2m−1  Am,k = σ 2m−1 , π k∈Z

 σ 2m  Bm,k = σ 2m . π

(42)

k∈Z

Since .‖f (et ·)‖Xp (R+ ) = ‖f ‖Xp (R+ ) , it implies that ‖R(2m−1) (σ )f ‖Xp (R+ ) ≤ σ 2m−1 ‖f ‖Xp (R+ ) ,

.

‖R(2m) (σ )f ‖Xp (R+ ) ≤ σ 2m ‖f ‖Xp (R+ ) , f ∈ Xp (R+ ).

.

(43)

Theorem 12 For .f ∈ Xp (R+ ), 1 ≤ p ≤ ∞, the next two conditions are equivalent: p

1. f belongs to .Bσ (Ɵ), σ > 0, 1 ≤ p ≤ ∞. 2. The following Riesz-Boas-type interpolation formulas hold true for .r ∈ N: Ɵr f = R(r) (σ )f,

.

or explicitly

.

   π   σ 2m−1  d 2m−1 x (−1)k+1 Am,k f e σ (k−1/2) x , f (x) = π dx k∈Z

and

(44)

Bernstein Spaces, Sampling, and Riesz-Boas Interpolation Formulas in Mellin Analysis

.

85

   σ 2m   πk  d 2m x f (x) = (−1)k+1 Bm,k f e σ x , dx π k∈Z

where each of the series converges absolutely and uniformly on .R+ . p

Proof We are proving that (1).→ (2). According toTheorem 2, if .f ∈ Bσ (Ɵ), σ > 0, 1 < p < ∞, then the function .Ф(t) = R+ f (et x)g(x) dx x for any .g ∈ (R). Thus, by [21], we have Xq (R+ ), 1/p + 1/q = 1, belongs to .B∞ σ Ф(2m−1) (t) =

.

   σ 2m−1  π (−1)k+1 Am,k Ф t + (k − 1/2) , m ∈ N, σ π k∈Z

Ф

.

(2m)

   σ 2m  πk k+1 , m ∈ N. (t) = (−1) Bm,k Ф t + σ π k∈Z

Together with  .

d dt

k

 Ф(t) =

R+

Ɵf (et x)g(x)

dx , x

it shows  .

R+

.

Ɵ2m−1 f (et x)g(x)

dx = x

  σ 2m−1    π dx (−1)k+1 Am,k f e(t+ σ (k−1/2)) x g(x) , m ∈ N, π x R+ k∈Z

and also  .

R+

Ɵ2m f (et x)g(x)

dx x

  πk     σ 2m  dx t+ σ k+1 x g(x) , m ∈ N. (−1) Bm,k f e = π x R+ k∈Z

Since both series (37) and (38) converge in .Xp (R+ ) and the last two equalities hold for any .g ∈ Xq (R+ ), 1/p + 1/q = 1, we obtain the next two formulas Ɵ2m−1 f (et x) =

.

 σ 2m−1  π

  π (−1)k+1 Am,k f e(t+ σ (k−1/2)) x , m ∈ N,

k∈Z

(45)

86

I. Z. Pesenson

Ɵ

.

2m

  πk    σ 2m  t+ σ k+1 (−1) Bm,k f e f (e x) = x , m ∈ N. π t

(46)

k∈Z

In turn, when .t = 0, these formulas become formulas (44). The fact that (2) .→ (1) easily follows from the formulas (44) and (43). Theorem is proved. ⨆ ⨅ p

Corollary 1 If f belongs to .Bσ (Ɵ), 1 < p < ∞, then for any .σ1 ≥ σ, σ2 ≥ σ , one has R(r) (σ1 )f = R(r) (σ2 )f, r ∈ N.

.

(47)

Let us introduce the notation R(σ ) = R(1) (σ ).

.

One has the following “power” formula which follows from the fact that operators p R(σ ) and .Ɵ commute on any .Bσ (Ɵ).

.

p

Corollary 2 For any .r ∈ N and any .f ∈ Bσ (Ɵ), 1 < p < ∞, Ɵr f = R(r) (σ )f = Rr (σ )f,

.

(48)

where .Rr (σ )f = R(σ ) (. . . (R(σ ))) f. Let us introduce the following notations: R(2m−1) (σ, N )f (x) =

.

 σ 2m−1  π

R(2m) (σ, N )f (x) =

.

 π  (−1)k+1 Am,k f e σ (k−1/2) x ,

|k|≤N

 σ 2m  π

 πk  (−1)k+1 Bm,k f e σ x .

|k|≤N

One obviously has the following set of approximate Riesz-Boas-type formulas. p

Theorem 13 If .f ∈ Bσ (Ɵ), 1 < p < ∞, and .r ∈ N, then Ɵr f ≈ R(r) (σ, N )f + O(N −2 ),

.

(49)

or explicitly

.

   σ 2m−1   π  d 2m−1 x f (x) ≈ (−1)k+1 Am,k f e σ (k−1/2) x + O(N −2 ), dx π |k|≤N

and

Bernstein Spaces, Sampling, and Riesz-Boas Interpolation Formulas in Mellin Analysis

.

87

   σ 2m   πk  d 2m x f (x) ≈ (−1)k+1 Bm,k f e σ x + O(N −2 ). dx π |k|≤N

References 1. J. Akhiezer, Theory of Approximation (Ungar, New York, 1956) 2. C. Bardaro, P.L. Butzer, I. Mantellini, G. Schmeisser, Mellin analysis and its basic associated metric-applications to sampling theory. Anal. Math. 42(4), 297–321 (2016) 3. C. Bardaro, P.L. Butzer, I. Mantellini, G. Schmeisser, On the Paley-Wiener theorem in the Mellin transform setting. J. Approx. Theory 207, 60–75 (2016) 4. C. Bardaro, P.L. Butzer, I. Mantellini, The Mellin-Parseval formula and its interconnections with the exponential sampling theorem of optical physics. Integr. Transforms Spec. Funct. 27(1), 17–29 (2016) 5. C. Bardaro, P.L. Butzer, I. Mantellini, G. Schmeisser, A fresh approach to the Paley-Wiener theorem for Mellin transforms and the Mellin-Hardy spaces. Math. Nachr. 29017–18, 2759– 2774 (2017) 6. C. Bardaro, P.L. Butzer, I. Mantellini, G. Schmeisser, A generalization of the Paley-Wiener theorem for Mellin transforms and metric characterization of function spaces. Fract. Calc. Appl. Anal. 20(5), 1216–1238 (2017) 7. C. Bardaro, P.L. Butzer, I. Mantellini, G. Schmeisser, A generalization of the Paley-Wiener theorem for Mellin transforms and metric characterization of function spaces. Fract. Calc. Appl. Anal. 20(5), 1216–1238 (2017) 8. C. Bardaro, P.L. Butzer, I. Mantellini, G. Schmeisser, Development of a new concept of polar analytic functions useful in Mellin analysis. Complex Var. Elliptic Equ. 64(12), 2040–2062 (2019) 9. C. Bardaro, I. Mantellini, Schmeisser, Gerhard Exponential sampling series: convergence in Mellin-Lebesgue spaces. Results Math. 74(3), 20 (2019). Paper No. 119 10. C. Bardaro, P.L. Butzer, I. Mantellini, G. Schmeisser, Valiron’s interpolation formula and a derivative sampling formula in the Mellin setting acquired via polar-analytic functions. Comput. Methods Funct. Theory 203–4, 629–652 (2020) 11. C. Bardaro, P.L. Butzer, I. Mantellini, G. Schmeisser, Integration of polar-analytic functions and applications to Boas’ differentiation formula and Bernstein’s inequality in Mellin setting. Boll. Unione Mat. Ital. 13(4), 503–514 (2020) 12. C. Bardaro, P.L. Butzer, I. Mantellini, G. Schmeisser, Polar-analytic functions: old and new results, applications. Results Math. 77(2), 26 (2022), Paper No. 64 13. R. Boas, The derivative of a trigonometric integral. J. Lond. Math. Soc. 164, 1–12 (1937) 14. R. Boas, Entire Functions (Academic Press, New York, 1954) 15. P. Butzer, H. Berens, Semi-Groups of Operators and Approximation (Springer, Berlin, 1967) 16. P.L. Butzer, S. Jansche, A direct approach to the Mellin transform. J. Fourier Anal. Appl. 3, 325–375 (1997) 17. P.L. Butzer, S. Jansche, The exponential sampling theorem of signal analysis. Atti Sem. Mat. Fis. Univ. Modena, Suppl. 46, 99–122 (1998) 18. P.L. Butzer, S. Jansche, A self-contained approach to Mellin transform analysis for square integrable functions. Integr. Trans. Spec. Funct. 8, 175198 (1999) 19. P.L. Butzer, W. Splettstösser, R.L. Stens, The sampling theorem and linear prediction in signal analysis. Jahresber. Deutsch. Math. Verein. 90(1), 70 (1988) 20. P.L. Butzer, P.J.S.G. Ferreira, J.R. Higgins, G. Schmeisser, R.L. Stens, The sampling theorem, Poisson summation formula, general Parseval formula, reproducing kernel formula and the Paley-Wiener theorem for bandlimited signals-their interconnections. Appl. Anal. 90(3–4), 431–461 (2011)

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21. P.L. Butzer, G. Schmeisser, R.L. Stens, Shannon sampling theorem for bandlimited signals and their hilbert transform, boas-type formulae for higher order derivatives the aliasing error involved by their extensions from bandlimited to non-bandlimited signals. Entropy 14, 2192– 2226 (2012). https://doi.org/10.3390/e14112192 22. R.V. Churchill, J.W. Brown, R.F. Verhey, Complex Variables and Applications, 3rd edn. (McGraw-Hill, New York, 1974) 23. J.R. Higgins, A sampling theorem for irregular sample points. IEEE Trans. Inform. Theory IT-22, 621–622 (1976) 24. S.G. Krein, Linear Differential Equations in Banach Space, Translated from the Russian by J. M. Danskin. Translations of Mathematical Monographs, vol. 29 (American Mathematical Society, Providence, 1971), v+390 pp. 25. R. Nevanlinna, V. Paatero, Introduction to Complex Analysis (Addison-Wesley Publ. Co., London, 1969) 26. S. Nikol’skii, Approximation of Functions of Several Variables and Imbedding Theorems (Springer, Berlin, 1975) 27. I.Z. Pesenson, Boas-type formulas and sampling in Banach spaces with applications to analysis on manifolds , in New Perspectives on Approximation and Sampling Theory (Springer International Publishing, Switzerland, 2014), pp. 39–61 28. I.Z. Pesenson, Sampling formulas for groups of operators in banach spaces. Sampling Theory Signal Image Proces. 14/1, 1–16 (2015) 29. M. Riesz, Eine trigonometrische Interpolationsformel und einige Ungleichungen für Polynome. Jahresber. Deutsch. Math. Verein. 23, 354–368 (1914) 30. M. Riesz, Les fonctions conjuguees et les series de Fourier. C.R. Acad. Sci. 178, 1464–1467 (1924) 31. G. Schmeisser, Numerical differentiation inspired by a formula of R. P. Boas. J. Approx. Theory 160, 202–222 (2009) 32. K. Seip, An irregular sampling theorem for functions bandlimited in a generalized sense. SIAM J. Appl. Math. 47(5), 1112–1116 (1987)

The Behavior of Frequency Bandlimited Cardinal Interpolants W. R. Madych

1 Introduction 1.1 Prologue If .{cn } = {cn : n = . . . , −2, −1, 0, 1, 2, . . . } is a bi-infinite data sequence of no greater than polynomial growth, there is a frequency bandlimited function that interpolates .{cn } on the integer lattice at the Nyquist rate. In other words, there is an entire function f in .Eπ such that .f (n) = cn , .n = 0, ±1, ±2, . . .. Here and in what follows, for .σ ≥ 0, .Eσ denotes the class of entire functions .F (z) that for some constants C and N satisfy |F (z)| ≤ C(1 + |z|)N eσ |Im z| .

.

Functions F in .Eσ are frequency bandlimited since they are Fourier-Laplace transforms of distributions with support in the interval .[−σ, σ ]. Note that .Eσ1 ⊂ Eσ2 when .σ1 < σ2 . The problem of interpolating the above-described data sequence .{cn } on the integer lattice does not necessarily have a solution F that is in .Eσ if .σ < π . This chapter studies interpolating functions f in .Eπ that also behave roughly like the data, in the sense that .f (x) does not grow significantly faster than the data as .x → ±∞ on the real axis. The basic fact concerning this matter can be formulated rigorously as follows: Theorem 1 Suppose the data sequence .{cn } enjoys the following growth property: cn = O(|n|α ) as .n → ±∞ for some value .α where .α ≥ −1 and .κ is the least integer greater than .α, namely, .κ − 1 ≤ α < κ. Then there is an entire function f in

.

W. R. Madych () Department of Mathematics, University of Connecticut, Storrs, CT, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. D. Casey et al. (eds.), Sampling, Approximation, and Signal Analysis, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-031-41130-4_5

89

90

W. R. Madych

Eπ that satisfies .f (n) = cn , .n = 0, ±1, ±2, . . ., and .f (x) = o(|x|κ ) as .x → ±∞ on the real axis.

.

The functions f that satisfy all the properties mentioned in the conclusion of Theorem 1 enjoy representations in terms of appropriately modified cardinal sine series. Details and results that describe the asymptotic behavior of such functions more precisely are presented in Sect. 2. Here, we only mention that, in general, functions f that satisfy the conclusions of this theorem need not be unique. The restriction .α ≥ −1 is necessary: the statement of Theorem 1 is invalid without this restriction. This can be seen by considering the sequence .{cn } defined by cn =

 1

if n = 0,

0

if n /= 0.

.

Note that .cn = O(|n|α ) for all .α, in particular for all .α < −1. Now, if f is any function in .Eπ that interpolates this sequence on the integer lattice, .f (n) =  namely, k such that cn for .n = 0, ±1, ±2, . . ., then there is a polynomial .P (z) = m a z k k=0 f (z) =

.

sin(π z) + P (z) sin(π z). πz

(1)

Any function f that satisfies identity (1) cannot satisfy .f (x) = o(|x|−1 ) as .x → ±∞ on the real axis. This allows us to conclude that Theorem 1 is not valid without the restriction .α ≥ −1. Since rapid decay of the values .cn as .n → ±∞, in and of itself, does not necessarily allow the data sequence .{cn } to be interpolated on the integer lattice by an entire function in .Eπ that enjoys comparable decay as .x → ±∞ on the real axis, perhaps additional restrictions on the data sequence .{cn } will allow this to happen. The following theorem implies that this indeed is the case. Theorem 2 Suppose .cn = O(|n|α ) as .n → ±∞ for some value of .α where .α < −1 and .κ is the least integer greater than .α, namely, .κ − 1 ≤ α < κ. If, in addition, the sequence .{cn } also satisfies ∞  .

(−1)n nj cn = 0

for j = 0, . . . , −κ − 1,

(2)

n=−∞

then there is an entire function f in .Eπ that satisfies .f (n) = cn , .n = 0, ±1, ±2, . . ., and .f (x) = o(|x|κ ) as .x → ±∞ on the real axis. Note that the integer .κ in the statement of Theorem 2 is negative so that the integer .−κ − 1 can have values .0, 1, 2, . . .. This may seem like a somewhat awkward formulation. However, the negative parameters, .α and .κ, in the statement of Theorem 2 highlight the fact that it can be viewed as a natural extension of Theorem 1.

Frequency Bandlimited Cardinal Interpolants

91

Of course, Theorem 2 can be reformulated with positive parameters. This is done in Sect. 3 where the details and more precise bounds on the rate of decay of f are developed. Here, we only mention that the entire functions f that satisfy the conclusions of Theorem 2 are unique.

1.2 Notation, Conventions, Background, and Viewpoint We use standard mathematical notation and assume the reader is familiar with the rudiments of distribution theory, [5, Sections 1.2–1.7], for example. We also assume that the reader is aware of the basic material surrounding the classical cardinal sine series and its role in the mathematics of signal analysis such as its role in the WKS sampling theorem [3, p. 47, Sampling Theorem (Part B)] and the Plancherel-Polya interpolation theorem [7, p. 152 Theorem 3]. In addition to the items cited above concerning the cardinal sine series and its applications, there are several accessible survey articles and books on various aspects of the subject, for instance, [3, Stories 1-3] and [1, 2, 4, 6, 11, 12]. We say that a function f in .Eσ is in .Lp (R) if its restriction to the real axis, .R, is in .Lp (R). The Fourier transform of a tempered distribution u is denoted by . u: the normalization used here provides   u(ξ ) =

∞

.

−∞

e−iξ x u(x)dx

in the case when u is an integrable function. In the case that the distribution u u(z) is well defined for all complex z and is called the has compact support, . Fourier-Laplace transform of u. The fact that .Eσ consists of the Fourier-Laplace transforms of distributions with support in the interval .[−σ, σ ] is a consequence of the distributional variant of the Paley-Wiener theorem, for example, see [5, Theorem 1.7.7]. In works on mathematical signal analysis, it is customary to regard the data sequences .{cn } to be samples .{F (n)} obtained on the integer lattice of members F of an appropriate subclass .L of entire functions in .Eπ with the objective of obtaining reconstruction formulas for members F in terms of their samples. The classic example is the WKS sampling theorem where .L is .Eπ ∩ L2 (R). In cases where .L contains functions that fail to decay on the real axis appropriately, additional data is frequently used in the reconstruction formulas, such as [3, Formula (9)]; more intricate examples can be found in [8, 10], for instance. Here, the objective is to show that data sequences .{cn } of no greater than polynomial growth can be interpolated on the integer lattice with bandlimited functions in .Eπ that approximately behave like the data asymptotically. The data sequences are not assumed to be samples of any specific functions; see Theorems 1 and 2. Indeed, in the case of Theorem 1 and .α ≥ 0, the solutions are not unique.

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It was pointed out in the Prologue that .Eπ is the smallest class of frequency bandlimited functions that can be used to interpolate arbitrary sequences .{cn } of no greater than polynomial growth on the integer lattice, However, as is known [7, Theorem 3, p. 160] and illustrated by Example 4.5 in Sect. 4, the data can be conformed to more closely by interpolants that are taken from a wider class of entire functions, such as .Eσ where .σ > π . We mention that, as part of standard notation, we use C to denote generic constants whose value depends on the context. Also, it goes without saying that a scaling argument shows that all the results found here are equally valid with the integer lattice replaced with any lattice on the real line, mutatis mutandis.

2 The Case cn = O(|n|α ) for Some Value α Where α ≥ −1 2.1 Introduction The objective of this section is to provide verification for Theorem 1. Thus, in what follows, we assume the following: • The data sequence .{cn } satisfies .cn = O(|n|α ) as .n → ±∞ for some value .α where .α ≥ −1 and .κ is the least integer greater than .α, namely, .κ − 1 ≤ α < κ. If .−1 ≤ α < 0, then the sequence .{cn } is in .𝓁p for .p > −1/α. Hence, in view of the Plancherel-Polya theorem, the entire function f in .Eπ defined by the classical cardinal sine series ∞ 

f (z) =

.

n=−∞

cn

sin π(z − n) π(z − n)

(3)

is in .Lp (R) for values of p that satisfy .−1/α < p ≤ ∞. This f also fulfills .f (x) = o(1) as .x → ±∞, .f (n) = cn , .n = 0, ±1, ±2, . . ., and is the unique entire functions in .Eπ with these properties. If .α ≥ 0, then, defining .c0 /0 to be equal to 0, the sequence .{cn /nκ } is in .𝓁p for .p > 1/(κ − α). It follows that the series .

∞  cn sin π(z − n) nκ π(z − n) n=−∞ n/=0

converges absolutely, defines an entire function in .Eπ that interpolates .{cn /nκ } on the integer lattice, and is in .Lp (R) for values of p that satisfy .p > 1/(κ −α). Hence, f (z) = c0

.

∞  sin π z cn sin π(z − n) + zκ κ π(z − n) πz n n=−∞ n/=0

(4)

Frequency Bandlimited Cardinal Interpolants

93

is an entire function that has the following properties: (i)f is in Eπ . .

(ii)f (n) = cn , n = 0, ±1, ±2, . . . .

(5)

(iii)f (x) = o(|x|κ ) as x → ±∞. The function f defined by (4) is not uniquely characterized by properties (5) since .f (z) + P (z) sin π z, where .P (z) is any polynomial of degree .≤ κ − 1, also enjoys these properties.  If .Pκ (z) is any polynomial with exact degree .κ, namely, .Pκ (z) = κj =0 aj zj with .aκ /= 0, and .N is the set of its distinct integer roots, then f (z) =



.

n∈N

∞  sin π(z − n) cn sin π(z − n) + Pκ (z) cn π(z − n) P (n) π(z − n) n=−∞ κ

(6)

n∈N /

is another expression for a function f that satisfies properties (5). The above observations provide complete verification of Theorem 1 and can be summarized as follows: First, for convenience, we formulate a definition. • For any real number .α, .Gα is the class of all numerical sequences .{cn } with the property that .cn = O(|n|α ) as .n → ±∞. Proposition 1 Suppose for some value of .α, where .α ≥ −1, the sequence .{cn } is in .Gα and .κ is the least integer greater than .α. Then there is an entire function f in .Eπ that interpolates .{cn } on the integer lattice and satisfies .f (x) = o(|x|κ ) as .x → ±∞ on the real axis. If .−1 ≤ α < 0, the entire function with these properties is unique and is represented by an absolutely convergent classical cardinal sine series. If .α ≥ 0, an entire function with these properties is well defined by the absolutely convergent series (4). It is not unique, but the difference of any two functions with these properties is .P (z) sin π z where P is a polynomial of degree no greater than .κ − 1.

2.2 More Precise Bounds on the Asymptotic Behavior of f The example given by the sequence .{cn } defined by .cn = (−1)n sgn(n) shows that, in general, it is not possible to interpolate .{cn } on the integer lattice with a function in .Eπ that has the same order of growth on the real axis as does the data sequence; see Example 4.2 in Sect. 4. However, a better bound, than that provided by Proposition 1, on the rate of growth of f is available.

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Proposition 2 Suppose the sequence .{cn } belongs to .Gα for some value of .α that is ≥ −1. Then any function f that enjoys the properties  guaranteed by conclusions of Proposition 1 also satisfies .f (x) = O |x|α log(|x|) as .x → ±∞ on the real axis.

.

Proof If .α ≥ 0, the difference of any two functions that satisfy the conclusions of Proposition 1 is .P (z) sin(π z) where .P (z) is a polynomial of degree no greater than .κ − 1. Hence, if one such function satisfies the conclusion of Proposition 2 so must all the others. To prove the proposition, it suffices to consider the function f defined by (4). To obtain the desired bound, assume .|z| > 100, and write f (z) = c0

.

sin π z + zκ (A(z) + B(z)) , πz

where A(z) =

.

 cn sin π(z − n) nκ π(z − n)

and

 cn sin π(z − n) . nκ π(z − n)

B(z) =

|n|>2|z|

|n|≤2|z| n/=0

Then  |cn | | sin π(z − n)|  nα−κ−1 eπ |Imz| ≤C κ π |z − n| |n|

|B(z)| ≤

.

|n|>2|z|

|n|>2|z|

which yields |B(z)| ≤ C|z|α−κ eπ |Imz| .

(7)

.

The calculation to estimate .|A(z)| is a bit more involved. First write zA(z) =



|n|≤2|z|

cn z sin π(z − n) π n(z − n) nκ−1

n/=0

.

=



|n|≤2|z|

cn sin π(z − n) π nκ−1



1 1 + z−n n

= A0 (z) +

sin π z A1 (z), π

n/=0

where A0 (z) =



.

|n|≤2|z| n/=0

cn sin π(z − n) nκ−1 π(z − n)

and

A1 (z) =

 (−1)n cn . nκ

|n|≤2|z| n/=0

Frequency Bandlimited Cardinal Interpolants

95

Now, since    sin π(z − n)  eπ |Imz|   . ≤C  π(z − n)  1 + |z − n|

and

|cn | ≤ |n|α−κ+1 , |n|κ−1

we may write 

|A0 (z)| = Ceπ |Imz|

.

|n|≤2|z|

|n|α−κ+1 ≤ C|z|α−κ+1 log(|z|)eπ |Imz| 1 + |z − n|

n/=0

and 



|A1 (z)| ≤ C

|n|

.

α−κ

≤C

|n|≤2|z|

|z|α−κ+1

if α − κ > −1

log(|z|)

if α − κ = −1.

n/=0

The final bounds on .|A0 (z)| and .|A1 (z)| imply that |zκ A(z)| ≤ C|z|α log(|z|)eπ |Imz| ,

(8)

.

and (8) together with (7) yields |f (z)| ≤ C|z|α log(|z|)eπ |Imz|

.

when .|z| > 100. This completes the proof of the proposition. The same example, namely, the sequence .{cn } defined by .cn = (−1)n sgn(n), shows that the bound on the rate of growth of f provided by Proposition 2 cannot be improved. However, if the oscillatory nature of .{cn } is subject to a mild restriction, then the bound on the rate of growth of f can be optimized. This restriction is specified in Proposition 3. Our proof of this proposition makes use of the following two lemmas. n  (−1)j 2z sin π z . Then .|Sn (z)| ≤ Ceπ |Imz| , where Lemma 1 Suppose .Sn (z) = π(z2 − j 2 ) j =1

C is a constant independent of n and z. Proof Assume n is even, say .n = 2m, and write Sn (z) =

m 

.

k=1

2z

sin π z π



1 1 − 2 2 2 z − (2k) z − (2k − 1)2

.

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W. R. Madych

Now

.

z2

1 1 − 2 2 z − (2k − 1)2 − (2k) =

4k − 1 (z − 2k)(z − (2k − 1))(z + 2k)(z + 2k − 1)

and if .Re z ≥ 0

.

    2z(4k − 1)    (z + 2k)(z + 2k − 1)  ≤ C and

    sin π z eπ |Imz|    π(z − 2k)(z − (2k − 1))  ≤ C 1 + |z − 2k|2 .

A combination of the above list of relationships leads to |Sn (z)| ≤ C

m 

.

k=1

eπ |Imz| ≤ Ceπ |Imz| when Re z ≥ 0. 1 + |z − 2k|2

An analogous calculation, mutatis mutandis, shows that |Sn (z)| ≤ C

m 

.

k=1

eπ |Imz| ≤ Ceπ |Imz| when Re z ≤ 0. 1 + |z + 2k|2

Hence, if n is even, |Sn (z)| ≤ Ceπ |Imz|

.

for all z.

(9)

To see that (9) is also valid for odd n, let .n = 2m + 1, and write S2m+1 (z) = S2m (z) −

.

2z sin π z ,  π z2 − (2m + 1)2

which implies that (9) remains valid with .n = 2m + 1. Hence, (9) is valid for all positive integers n. Lemma 2 Suppose .Sn (z) =

n  (−1)j 2j sin π z j =1

π(z2 − j 2 )

. Then .|Sn (z)| ≤ Ceπ |Imz| , where

C is a constant independent of n and z. The proof is completely analogous to that used to prove Lemma 1, mutatis mutandis.

Frequency Bandlimited Cardinal Interpolants

97

For some background on the sums in Lemmas (1) and (2), see the Remark following Example 4.1 in the Appendix. Proposition 3 Suppose the sequence .{cn } is in .Gα for some value of .α that is .≥ −1 and, in addition, satisfies cn+1 − cn = O(|n|α−1 )

.

as n → ±∞ when α is not an integer

(10)

and ∞  .

|(n + 1)−α cn+1 − n−α cn | < ∞

when α is an integer.

(11)

n=−∞

Then any function f that enjoys the properties guaranteed by the conclusions of  Proposition 1 also satisfies .f (x) = O |x|α as .x → ±∞ on the real axis. Proof Without loss of generality, assume that f enjoys representation (4) and that |z| > 100. Using the same notation as in the proof of Proposition 2, write

.



sin π z sin π z κ−1 A0 (z) + +z A1 (z) + zκ B(z). .f (z) = c0 πz πz Then, as before, |B(z)| ≤ C|z|α−κ

(7)

.

and  |A1 (z)| ≤ C

.

|z|α−κ+1

if α − κ > −1

log(|z|)

if α − κ = −1.

(12)

Now suppose .cn+1 − cn = O(|n|α−1 ) as .n → ±∞. To estimate .A0 (z), let .an = n1−κ cn , and note that .an − an+1 = O(nα−κ ). Assume that the sequence .{an } is even, namely, .an = a−n , and let N be the greatest integer .≤ 2|z|. Then A0 (z) =

N 

.

n=1

an

(−1)n 2z sin π z , π(z2 − n2 )

and summation by parts allows us to write A0 (z) = aN SN (z) +

N −1 

.

n=1

(an − an+1 )Sn (z) where Sn (z) =

n  (−1)j 2z sin π z j =1

π(z2 − j 2 )

.

98

W. R. Madych

Now, .|Sn (z)| ≤ Ceπ |Imz| by virtue of Lemma 1. Hence, |A0 (z)| ≤ |aN ||SN (z)| +

N −1 

|an − an+1 ||Sn (z)||

n=1



≤ Ceπ |Imz| N α+1−κ +

.

≤ Ce

nα−κ

n=1



π |Imz|

N −1 

|z|

α+1−κ

+

|z|α+1−κ

if α − κ > −1

log(|z|)

if α − κ = −1

,

or, more succinctly,  |A0 | ≤ Ceπ |Imz|

.

|z|α+1−κ

if α − κ > −1

log(|z|)

if α − κ = −1

(13)

.

If .{an } is an odd sequence, then A0 (z) =

N 

.

n=1

an

(−1)n 2n sin π z , π(z2 − n2 )

and, again, summation by parts allows us to write A0 (z) = aN SN (z) +

N −1 

.

(an − an+1 )Sn (z) where Sn (z) =

n  (−1)j 2j sin π z j =1

n=1

π(z2 − j 2 )

.

Since .|Sn (z)| ≤ Ceπ |Imz| by virtue of Lemma 2, as in the case of even .{an }, inequality (13) follows. If the sequence .{an } satisfies .an − an+1 = O(|n|α−κ ) as .n → ±∞, then its even and odd parts also satisfy this property. Hence, it follows that such a sequence .{an } also gives rise to (13). In view of (7), (12), and (13), we may conclude that if .cn+1 − cn = O(|n|α−1 ) as .n → ±∞, then  |f (z)| ≤ Ce

.

π |Imz|

|z|α

if α > κ − 1

|z|α

log(|z|) if α = κ − 1.

(14)

This implies the conclusion of the proposition in case (10).  −α −α Now suppose that .α = κ − 1 and . ∞ n=−∞ |(n + 1) cn+1 − n cn | < ∞. Then the analogue of the string of inequalities that led to (13), in this case, leads to |A0 (z)| ≤ Ceπ |Imz| .

.

(15)

Frequency Bandlimited Cardinal Interpolants

99

Turning to .A1 (z), if the sequence .{an } is even, then .A1 (z) = 0, and if it’s odd, then

A1 (z) = 2

.

N N −1   (−1)n n−1 an = aN SN + (an − an+1 )Sn n=1

n=1

where Sn = 2

n  (−1)j

j

j =1

and, as earlier, an = n−α cn = n1−κ cn .

 −α −α Since .|an | ≤ C, . ∞ n=−∞ |(n + 1) cn+1 − n cn | ≤ C, and .|Sn (z)| ≤ C, it follows that |A1 (z)| ≤ C.

.

(16)

Since both the even and odd parts of .{an } satisfy property (11) whenever .{an } does, it follows that (16) is valid for all sequences .{cn } that satisfy (11). In view of (7), (15), and (16), we may conclude that if .α = κ −1 and .{cn } satisfies property (11), then |f (z)| ≤ Ceπ |Imz| |z|α .

.

(17)

The statement of (11) part of the proposition now follows from (17).

3 The Case cn = O(|n|α ) for Some Value α Where α < −1 3.1 Introduction The use of the negative integer .κ in the statement of Theorem 2 may be awkward and somewhat confusing, particularly as used in item (2). To avoid this in the development below, we use the convenient change of notation .α = −β. Thus, in what follows, we assume the following: • The data sequence .{cn } satisfies .cn = O(|n|−β ) as .n → ±∞ for some value .β where .β > 1 and .μ is the greatest integer less than .β, namely, .μ < β ≤ μ + 1. We are interested in interpolating the data sequence .{cn } with an entire function in Eπ that, on the real axis, also decays roughly at the same rate as the data. In view of the example mentioned in the introduction, this may not necessarily be possible. To speculate for what additional constraints on sequence .{cn } may make this possible, consider the following heuristic argument. Suppose f is an entire function in .Eπ that satisfies .f (n) = cn , .n = 0, ±1, ±2, . . ., and .f (x) = O(|x|−β ) as .x → ±∞. This implies that (a) f

.

100

W. R. Madych

enjoys the representation (3) as an absolutely convergent cardinal sine series; (b) the Fourier transform of f is f(ξ ) =

∞ 

cn e−inξ χ (ξ ),

.

(18)

n=−∞

where the series converges absolutely and .χ is the indicator function of the interval [−π, π ]; and (c) the functions .x j f (x), .j = 0, . . . , μ − 1 are in .L1 (R). Item (c) implies that .fis .μ − 1 times continuously differentiable that in turn, in view of (b), implies that for .j = 0, . . . , μ − 1,

.

f(j ) (±π ) =

∞ 

(−1)n (−in)j cn = 0.

.

n=−∞

As a consequence, we may conclude that the existence of a function f with the desired properties requires that the sequence .{cn } satisfy the system of equations ∞  .

(−1)n nj cn = 0 for j = 0, . . . , μ − 1.

(19)

n=−∞

To check whether the additional restriction (19) is sufficient to allow for the existence of an entire function in .Eπ that, on the real axis, also decays roughly the case .μ − 1 = 0. In this case, at the same rate as the sequence  .{cn }, consider n c = 0. Since the function f with the (19) reduces to one equation . ∞ (−1) n n=−∞ desired properties enjoys a representation as a classical cardinal series (3), it may also be expressed as f (z) =

.

∞ sin π z  (−1)n cn . π n=−∞ z − n

(20)

 n Now, using . ∞ n=−∞ (−1) cn = 0, write ∞ ∞   (−1)n cn (−1)n cn = − z−n z−n n=−∞ n=−∞ .

=

∞ 

(−1)n cn

n=−∞

∞

n=−∞ (−1)

nc n

z

1 1 − z−n z

=

∞ 1  (−1)n ncn z n=−∞ z − n

or, more succinctly,

.

∞ ∞  1  (−1)n ncn (−1)n cn = . z−n z n=−∞ z − n n=−∞

(21)

Frequency Bandlimited Cardinal Interpolants

101

Substituting (21) into (20) yields f (z) =

.

∞ ∞ sin π(z − n) sin π z  (−1)n ncn 1  ncn = . π z n=−∞ z − n z n=−∞ z−n

(22)

Since .ncn = O(|n|1−β ) and .β − 1 > 0, we may conclude that .xf (x) is in .Lp (R) for p that satisfies .p > 1/(β − 1). Hence, .f (x) = o(|x|−1 ) as .x → ±∞, which is the desired conclusion. In the cases .μ − 1 > 0, an inductive argument allows us to obtain an analogous result. First, we introduce a useful definition. • If .β > 1 and .μ is the greatest integer less than .β, then .Hβ is the class of all numerical data sequences .{cn } that satisfy both .cn = O(|n|−β ) as .n → ±∞ and ∞  .

(−1)n nj cn = 0

for j = 0, . . . , μ − 1.

(19)

n=−∞

The abovementioned analogue of (21) is essentially the content of the following: Lemma 3 Suppose .{cn } is in .Hβ for some .β > 1 and .P (z) is any polynomial of degree .≤ μ. Then ∞ ∞   1 (−1)n P (n)cn (−1)n cn = . z−n P (z) n=−∞ (z − n) n=−∞

whenever P (z) /= 0.

(23)

Proof Assume .μ > 1. In the case when the degree of P is 0, there is nothing to  (−1)n cn (−1)n acn 1 ∞ prove since . ∞ for any constant .a /= 0. n=−∞ n=−∞ z−n = a z−n In the case when the degree of P is 1, say .P (z) = az + b where .a /= 0, argue as in (21). Namely, let .z1 = −b/a, and write ∞ ∞   (−1)n cn (−1)n cn − = z−n z−n n=−∞ n=−∞

.

=

∞  n=−∞

=

(−1)n cn

∞

n n=−∞ (−1) cn

z − z1

1 1 − z − n z − z1

=

∞  (−1)n (n − z1 )cn 1 z − z1 n=−∞ z−n

∞ ∞   (−1)n P (n)cn (−1)n a(n − z1 )cn 1 1 = . z−n P (z) n=−∞ z−n a(z − z1 ) n=−∞

Now assume that (23) is valid for polynomials P of degree .≤ k where .k < μ. If P is a polynomial of degree .k + 1, then .P (z) = (z − z1 )P1 (z) where .z1 is a zero of .P (z) and k j .P1 (z) = j =0 aj z is a polynomial of degree k. Since

102

W. R. Madych ∞  .

∞ 

(−1)n P1 (n)cn =

n=−∞

(−1)n cn

k 

aj nj =

j =0

n=−∞

k 

aj

j =1

∞ 

(−1)n nj cn = 0,

n=−∞

(24) using the induction assumption and (24), we may write ∞ ∞   (−1)n P1 (n)cn (−1)n cn 1 = z−n P1 (z) n=−∞ z−n n=−∞ ∞  1 (−1)n P1 (n)cn = − z−n P1 (z) n=−∞

.

1 P1 (z)

∞

n n=−∞ (−1) P1 (n)cn

z − z1

=



∞  1 1 1 (−1)n P1 (n)cn − P1 (z) n=−∞ z − n z − z1

=

∞  (−1)n (n − z1 )P1 (z)cn 1 (z − z1 )P1 (z) n=−∞ z−n

=

∞  (−1)n P (n)cn 1 , P (z) n=−∞ z−n

which implies the desired result.

These observations provide verification of Theorem 2 and can be summarized as follows: Proposition 4 Suppose .{cn } is in .Hβ for some .β > 1. Then the classical cardinal sine series f (z) =

∞ 

.

n=−∞

cn

sin π(z − n) z−n

(25)

is the unique entire function in .Eπ that satisfies both .f (n) = cn , .n = 0, ±1, ±2, . . ., and .f (x) = o(|x|−μ ) as .x → ±∞. Furthermore, this function f can be reexpressed as f (z) =

.

∞  sin π(z − n) 1 P (n)cn z−n P (z) n=−∞

whenever P (z) /= 0,

(26)

where .P (z) is any polynomial of degree .≤ μ. So .x μ f (x) is in .Lp (R) for p that satisfies .p > 1/(β − μ).

Frequency Bandlimited Cardinal Interpolants

103

3.2 An Additional Observation  Note that if the data sequence .{cn } satisfies . ∞ n=−∞ cn = 0, then it is apparent that the number of non-zero terms must be greater than one. More widely, an analogous phenomenon holds for data sequences in .Hβ . Proposition 5 Suppose the sequence .{cn } is in .Hβ for some .β > 1. Then the number of non-zero terms is either .≥ μ + 1 or all the terms are zero. In other words, if the number of non-zero terms is .≤ μ, then all the terms are zero. Proof Suppose the number of non-zero term is .≤ μ but .> 0. To avoid subscripting the subscripts, let .{x0 , . . . , xμ−1 } be a collection of .μ terms of the sequence n .{(−1) cn } that contains all the non-zero terms of .{cn }, and suppose that, as members of .{(−1)n cn }, the .xj ’s are indexed by the indices .N = {n0 , . . . , nμ−1 }. Re-express (19) as μ−1  .

njν xν

= −Sj

for j = 0, . . . , μ − 1,

where

Sj =

∞ 

(−1)n nj cn .

n=−∞

ν=0

n∈N /

(27) Taking .Sj , .j = 0, . . . , μ − 1, to be known constants, (27) is a system of .μ linear equations in the .μ variables .xν , .ν = 0, . . . , μ − 1, that, in matrix notation, is equivalent to ⎛ ⎜ ⎜ .⎜ ⎜ ⎝

1 n0 .. . μ−1

n0

⎞⎛ ⎛ ⎞ ⎞ 1 ... 1 x0 S0 ⎜ S1 ⎟ ⎟ n1 . . . nμ−1 ⎟ ⎟⎜ ⎜ x1 ⎟ ⎜ ⎟ ⎜ . ⎟ = −⎜ . ⎟. .. . . .. ⎟ ⎟ . . ⎝ ⎝ ⎠ . . ⎠ . . . ⎠ μ−1 μ−1 n1 . . . nμ−1 xμ−1 Sμ−1

(28)

Since the square matrix associated with this system is the transpose of the classical Vandermonde matrix with determinant . 0≤j 1. Then any function f that enjoys the by conclusions  properties guaranteed  of Proposition 4 also satisfies f (x) = O |x|−β log(|x|) as x → ±∞ on the real axis.

104

W. R. Madych

Proof Our argument is essentially the same as that used to prove Proposition 2. Thus, analogously write f (z) =

.

∞ 1  μ sin π(z − n) n cn . zμ n=−∞ z−n

To obtain the desired bound, assume |z| > 100, and write f (z) =

.

1 (A(z) + B(z)) , zμ

where 

A(z) =

.

nμ cn

 sin π(z − n) sin π(z − n) nμ cn and B(z) = z−n z−n |n|≥2|z|

|n|≤2|z|

Then a computation similar to the one used in the proof of Proposition 2 yields |B(z)| ≤ C|z|μ−β eπ |Imz| .

(29)

.

The calculation to estimate |A(z)| is also completely analogous, mutatis mutandis, to the one used earlier. First write A(z) = z−1



nnμ cn sin π(z − n)

|n|≤2|z|

=z

.

−1



|n|≤2|z|

=z

−1

sin π(z − n) nn cn π

z π n(z − n)

μ

1 1 + z−n n





sin π z A0 (z) + A1 (z) , π

where A0 (z) =



nnμ cn

.

|n|≤2|z|

sin π(z − n) π(z − n)

and

A1 (z) =



|n|≤2|z|

Now, computing as earlier yields |A0 (z)| ≤ C|z|μ+1−β log(|z|)e|Imz|

.

and  |A1 (z)| ≤ C

.

|z|μ+1−β

if β < μ + 1

log(|z|)

if β = μ + 1.

nμ (−1)n cn .

Frequency Bandlimited Cardinal Interpolants

105

The final bounds on |A0 (z)| and |A1 (z)| imply that |A(z)| ≤ C|z|μ−β log(|z|)e|Imz| .

.

(30)

As earlier, (29) and (30) imply the desired result. Proposition 7 Suppose the sequence {cn } is in Hβ for some β > 1 and, in addition, satisfies cn+1 − cn = O(|n|−β−1 )

.

as n → ±∞ when β is not an integer

(31)

or ∞  .

|(n + 1)β cn+1 − nβ cn | < ∞ when β is an integer.

(32)

n=−∞

Then any function f that enjoys the guaranteed by conclusions of   properties Proposition 4 also satisfies f (x) = O |x|−β as x → ±∞ on the real axis. Proof Our argument is completely analogous to the one used to establish Proposition 3, mutatis mutandis. Namely, use the representation (26) for f , decompose analogously, etc.

4 Examples 4.1 Example Suppose .{cn } is the sequence defined by cn = (−1)n ,

.

n = 0, ±1, ±2, . . . ,

and f is the entire function defined by (4), namely, f (z) =

.

 cn sin π(z − n) sin π z +z πz n π(z − n) |n|≥1

or, equivalently, f (z) =

z sin π z sin π z  + πz π n(z − n) |n|≥1

.

∞

2 sin π z z sin π z  = . + 2 π πz z − n2 n=1

(33)

106

W. R. Madych

Since .f (n) = (−1)n = cos π n and both .f (z) and .cos π z are in .Eπ and .o(|x|) as .x → ±∞ on the real axis, it follows that .f (z) − cos π z = c sin π z where c is a constant. The fact that .f (z) is even implies that f (z) = cos π z.

.

Note that the use of .

  1 1 1 z = + and = lim n(z − n) z−n n n N →∞ |n|≥1

1≤|n|≤N

1 =0 n

in the first expression for .f (z) in (33) yields f (z) =

.

∞ ∞  sin π(z − n) 1 sin π z  = , (−1)n π(z − n) π n=−∞ z − n n=−∞

which is a classical cardinal sine series with the coefficients .{cn } whose symmetric partial sums converge uniformly on compact subsets of .C. The last equality in the above string, when combined with the  fact that 1.f (z) = cos π z, leads to a peculiar verification of the classical fact that . ∞ n=−∞ π(z−n) = cot π z. Remark Note that the .Sn (z) in Lemma 1 are essentially the partial sums of ∞

1=

.

sin π z sin π z  2(−1)n z sin π z  z sin π(z − n) + = + . πz n π(z − n) πz π z2 − n2 |n|≥1

n=1

Analogously, the .Sn (z) in Lemma 2 are essentially the partial sums of

.

Sgn(z) =

∞  z sgn(n) sin π(z − n) sin π z  2(−1)n n = . π(z − n) π n z2 − n2

|n|≥1

n=1

See [9, Subsection 4.2] for a related discussion of this matter.

4.2 Example Suppose .{cn } is the sequence defined by cn = (−1)n sgn(n),

.

n = 0, ±1, ±2, . . . ,

and f is the entire function defined by (4), namely, f (z) = z

.

 cn sin π(z − n) . n π(z − n)

|n|≥1

Frequency Bandlimited Cardinal Interpolants

107

Then, if z is on the real axis and .|z| = N + 1/2 where N is an integer .≥ 10, |f (z)| ≥

.

2 log(2|z|), 3π

(34)

To see (34), re-express .f (z) as ⎛ ⎞ sin π z ⎝  1 1 ⎠ sin π z .f (z) = z = (zA(z) + zB(z)) π |n| z − n π

(35)

|n|≥1

where A(z) =



.

1≤|n|≤2N

 1 1 = |n| z − n

1≤n≤2N

1 n



1 1 + z−n z+n

=



1≤n≤2N

1 2z n z2 − n2

and B(z) =

.

 1 2z . n z2 − n2

n>2N

Direct estimation yields |zB(z)| ≤ 2/3.

(36)

.

To estimate .zA(z), write

 1 1 1 = A1 (z) + A2 (z), + z−n n z+n

A(z) = 2

.

1≤n≤2N

where A1 (z) = 2



.

1≤n≤2N

1 z2 − n2

and

A2 (z) = 2



1≤n≤2N

1 1 . n z+n

Next, write zA1 (z) =



.

1≤n≤2N

 1 2z 1 . = + z−n z+n z2 − n2 1≤n≤2N

A direct calculation yields  .

1≤n≤2N

1 = 0 when z = N + 1/2, z−n

108

W. R. Madych

while  .

1≤n≤2N

 1 ≥ z+n

1≤n≤2N

1 ≥ 2/3. 3z

Hence, zA1 (z) ≥ 2/3

.

when z = N + 1/2.

Finally, zA2 (z) =



.

1≤n≤2N

2  1 2z ≥ n z+n 3

1≤n≤2N

2 1 ≥ log(2z), n 3

so that zA(z) ≥

.

2 2 + log(2z) when z = N + 1/2. 3 3

(37)

Since .f (z) is an odd function, relations (35), (36), and (37) imply (34).

4.3 Example Suppose the sequence .{cn } is defined by cn = (−1)n nα sgn(n), n = 0, ±1, ±2, . . . , where α is an integer ≥ −1.

.

Then the entire function f defined by (4) that satisfies the conclusions of Theorem 1 can be expressed as f (z) = zα+1



.

|n|≥1

cn sin π(z − n) nα+1 π(z − n)

or, more explicitly, as f (z) =

.

sin π z α+1  1 1 z . π |n| z − n

(38)

|n|≥1

By comparing (38) with (35), it is evident that if z is on the real axis and .|z| = N + 1/2 where N is an integer .≥ 10, then |f (z)| ≥ C|z|α log(2|z|),

.

where C is a positive constant independent of N.

Frequency Bandlimited Cardinal Interpolants

109

4.4 Example Suppose .{cn } is the sequence defined by ⎧ n ⎪ ⎪ ⎨(−1) .cn = 1/2 ⎪ ⎪ ⎩0

when n = 1, 2, 3 . . . , when n = 0, and when n = −1, −2, −3, . . . .

Then the entire function f defined by (4) that satisfies the conclusions of Theorem 1 can be expressed as ∞

f (z) =

.

 (−1)n sin π(z − n) sin π z +z . 2π z n π(z − n) n=1

To investigate the behavior of .f (z), write .cn = an + bn where .{an } and .{bn } are the even and odd parts, respectively, of .{cn }, namely, .2an = (−1)n and .2bn = (−1)n sgn(n) for .n = 0, ±1, ±2, . . ., and use this in the formula for .f (z) to get ⎛ ⎞ 1 1 ⎝  (−1)n sgn(n) sin π(z − n) ⎠ .f (z) = cos π z + z , 2 2 n π(z − n) |n|≥1

where the expression in parenthesis is the entire function considered in Example 4.2. This allows us to conclude that if N is an integer .≥ 10, then |f (x)| ≥ C log(2|x|)

.

when x is real and |x| = N + 1/2,

where C is a positive constant independent of N.

4.5 Example If .{cn } is a bi-infinite data sequence with no faster than polynomial growth, then for every positive .ϵ, there are entire functions in .Eπ +ϵ that interpolate .{cn } on the integer lattice and conform to its behavior better than interpolants from .Eπ . This was alluded to in the introduction, and an example is given below. However, it is beyond the scope of this chapter to provide a comprehensive development of the subject. A readily available class of interpolants of .{cn } from .Eπ +ϵ is described by fϵ (z) =

∞ 

.

n=−∞

cn

 sin π(z − n)  φ ϵ(z − n) π(z − n)

110

W. R. Madych

where .φ is a member of .E1 that satisfies .φ(0) = 1 and other desirable properties, such as sufficiently rapid decay as .z → ±∞ on the real axis to ensure convergence of the series. The function φ(z) =

.

sin(z/k) z/k

k where k is a positive integer,

,

(39)

is a typical example. Note that on the real axis, .|φ(x)| ≤ C(1 + |x|)−k , where C is a constant independent of x. Another example is the case where the Fourier transform , is infinitely differentiable. For instance, of .φ, namely, .φ 

  2 (ξ ) = γ exp 1/(ξ − 1) , when |ξ | < 1 and φ 0, when |ξ | ≥ 1,

.

(40)

where .γ is a constant chosen so that .φ(0) = 1. In this case, on the real axis, .|φ(x)| = C(1 + |x|)−N for any positive N ; here, C is a constant that may depend on N but is independent of x. However, this .φ does not seem to have an expression that allows it to be handily evaluated. To observe how .fϵ (z) can mimic the data better than interpolants from .Eπ , consider the case when .{cn } is the data sequence of Example 4.4 and .φ is the function defined by either item (39) or (40). Then .fϵ (x) is bounded on the real axis. More specifically, |fϵ (x)| ≤ C(1 + ϵ −N ),

.

where N and C are positive constants that depend on .φ. To see this, use the sin π z φ(ϵz), and write substitution .Фϵ (z) = πz |fϵ (x)| ≤

∞ 

|cn Фϵ (x−n)| ≤

.

n=−∞

∞ 

C ≤ C(1 + ϵ −N ). N (1+|x−n|)(1+ϵ|x−n|) n=−∞

Furthermore, not only is .fϵ bounded on the real axis; .limx→−∞ fϵ (x) = 0, which conforms nicely to the fact that .cn = 0 when .n < 0. The rate at which it decays depends on the particular choice of .φ. Namely, if .x < −10, then

|fϵ (x)| ≤

∞ 

.

|Фϵ (x − n)| ≤

n=0

∞  n=0

≤C

⎧ ⎨  ⎩

C 1   |x| + n ϵ(|x| + n) N

0≤n≤|x|

1 1 |x| (ϵ|x|)N

⎫  1 1 ⎬ C + , ≤ n (ϵn)N ⎭ (ϵ|x|)N n>|x|

Frequency Bandlimited Cardinal Interpolants

111

where the positive constant N depends on .φ and the final constant C depends on .φ and the method of estimation but is independent of .ϵ and .x > −10.

References 1. P.L. Butzer, J.R. Higgins, R.L. Stens, Classical and approximate sampling theorems: studies in the LP (R) and the uniform norm. J. Approx. Theory 137(2), 250–263 (2005) 2. P.L. Butzer, P.J.S.G. Ferreira, J.R. Higgins, S. Saitoh, G. Schmeisser, R.L. Stens, Interpolation and sampling: E. T. Whittaker, K. Ogura and their followers. J. Fourier Anal. Appl. 17(2), 320–354 (2011) 3. J.R. Higgins, Five short stories about the cardinal series. Bull. Am. Math. Soc. (N.S.) 12(1), 45–89 (1985) 4. J.R. Higgins, Sampling Theory in Fourier and Signal Analysis: Foundations (Oxford Science Publications, Clarendon Press, Oxford, 1996) 5. L. Hörmander, Linear Partial Differential Operators, Third revised printing (Springer, New York 1969) 6. A.J. Jerri, The Shannon sampling theorem - its various extensions and applications: a tutorial review. Proc. IEEE 65(11), 1565–1596 (1977) 7. B.Ya. Levin, Lectures on Entire Functions, In collaboration with and with a preface by Yu. Lyubarskii, M. Sodin and V. Tkachenko. Translated from the Russian manuscript by Tkachenko. Translations of Mathematical Monographs, vol. 150 (American Mathematical Society, Providence, RI, 1996) 8. W.R. Madych, Band limited functions and extensions of classical interpolation series. Sampl. Theory Signal Image Process. 2(2), 165–189 (2003) 9. W.R. Madych, Convergence of classical cardinal series, in Multiscale Signal Analysis and Modeling, ed. by X. Shen, A.I Zayed. Lecture Notes in Electrical Engineering (Springer, New York, 2012), pp. 3–24 10. G.G. Walter, Sampling bandlimited functions of polynomial growth. SIAM J. Math. Anal. 19(5), 1198–1203 (1988) 11. J.M. Whittaker, Interpolatory Function Theory. Cambridge Tracts in Mathematics and Mathematical Physics, No. 33 (Cambridge University Press, Cambridge, 1935) 12. A.I. Zayed, Advances in Shannon’s Sampling Theory (CRC Press, Boca Raton, 1993)

The Balian-Low Theorem for (Cq )-Systems in Shift-Invariant Spaces

.

Alexander M. Powell

1 Introduction The classical Balian-Low theorem is an uncertainty principle that relates timefrequency localization and spanning structure for systems of time-frequency shifts. Given .g ∈ L2 (R), the Gabor system .{gm,n }m,n∈Z is defined by .gm,n (x) = e2π imx g(x − n). The original version of the Balian-Low theorem [4, 20] states that if .{gm,n }m,n∈Z is an orthonormal basis for .L2 (R), then 

 |x| |g(x)| dx = ∞ 2

.

R

2

or

R

|ξ |2 | g (ξ )|2 dξ = ∞,

 where . g (ξ ) = R g(x)e−2π ixξ dx is the Fourier transform. The Balian-Low theorem has been generalized in many directions, e.g., [5–11, 13, 16, 21]. This chapter will focus on versions of the Balian-Low theorem for systems of translates in shiftinvariant spaces. Given .f ∈ L2 (R), the shift-invariant space generated by f , denoted .V (f ), is the .L2 (R)-closure of the span of integer translates of f , that is, V (f ) = span T(f ), where T(f ) = {f (x − n) : n ∈ Z}.

.

(1)

The shift-invariant space .V (f ) has extra invariance if there exists .γ ∈ R\Z such that for every .h ∈ V (f ), one has .h(x − γ ) ∈ V (f ). See [1, 3] for background on extra invariance in shift-invariant spaces.

A. M. Powell () Department of Mathematics, Vanderbilt University, Nashville, TN, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. D. Casey et al. (eds.), Sampling, Approximation, and Signal Analysis, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-031-41130-4_6

113

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A. M. Powell

Analogues of the Balian-Low theorem for shift-invariant spaces were first studied in [2, 24]. The following sharp result for Riesz bases was proven in [14]. Theorem 1 (Hardin, Northington, Powell) Let .f ∈ L2 (R). If .V (f ) admits extra invariance, and .T(f ) is a Riesz basis for .V (f ), then  |x| |f (x)|2 dx = ∞.

.

R

(2)

The conclusion (2) is sharp in the sense that it fails if .|x| is replaced by .|x|1−ϵ with .ϵ > 0. A version of Theorem 1 holds for multiply generated shift-invariant spaces on general lattices in higher dimensions [14], but we focus our discussion on singly generated shift-invariant spaces in dimension one for the lattice .Z. This chapter focuses on extensions of (2) from the setting of Riesz bases to the settings of exact systems and exact .(Cq )-systems. Background on Riesz bases, exact systems, and exact .(Cq )-systems are recalled in Sect. 2. The next result addresses exact systems; see Theorem 1.11 with .q = ∞ in [22]. Theorem 2 (Nitzan, Northington, Powell) Let .f ∈ L2 (R). If .V (f ) admits extra invariance, and .T(F ) is an exact system for .V (F ), then  .

R

|x|2 |f (x)|2 dx = ∞.

(3)

The conclusion (3) is sharp in the sense that it fails if .|x|2 is replaced by .|x|2−ϵ with .ϵ > 0. The next result addresses exact .(Cq )-systems with general .2 ≤ q ≤ ∞; see Theorem 1.11 in [22]. Theorem 3 (Nitzan, Northington, Powell) Let .2 ≤ q ≤ ∞, and let .f ∈ L2 (R). If .V (f ) admits extra invariance, and .T(f ) is an exact .(Cq )-system for .V (f ), then  .

R

2(1− q1 )

|x|

|f (x)|2 dx = ∞.

(4) 2(1− q1 )

The conclusion (4) is sharp in the sense that it fails if .|x| 2(1− q1 )−ϵ

|x|

.

is replaced by

with .ϵ > 0.

Theorem 3 reduces to Theorem 1 when .q = 2, and it reduces to Theorem 2 when q = ∞, cf. Sect. 2. However, while both Theorems 1 and 2 remain true in higher dimensions [14, 22], the situation for .2 < q < ∞ remains unresolved. A version

.

2d(1− 1 )−d+1

q (see [22]), but this of Theorem 3 holds in .L2 (Rd ) with the weight .|x| higher-dimensional result does not appear to be sharp. The case .2 < q < ∞ merits further investigation, and it is natural to ask if Theorem 3 remains true for all .d ≥ 1.

The Balian-Low Theorem for .(Cq )-Systems in Shift-Invariant Spaces

115

Overview and Main Results The main goal of this semi-expository chapter is to more deeply examine the proof of Theorem 3. Our results are joint work with Shahaf Nitzan and Michael Northington. Our main contributions are: • Section 4 gives an exposition of the proof of Theorem 3 when .2 < q < ∞. For perspective, the proof of Theorem 3 presented in [22] was obtained as part of a more general framework. We extract a short and mostly self-contained proof for singly generated shift-invariant systems in .L2 (R) which will motivate our main result in Sect. 5. • Our main new result, Theorem 6, shows that under the hypotheses of Theorem 3, 1

the function .|x − x0 | q

−1

is in a weighted .L2 space associated with the Gramian 1

−1

∈ L2Pf . In particular, there is a Pf whenever .Pf (x0 ) = 0, i.e., .|x − x0 | q canonical choice of function that violates the Fourier multiplier conditions that arise in the proof of Theorem 3, and this provides an alternate proof of Theorem 3 when .2 < q < ∞. The chapter is organized as follows. Section 2 gives background on .(Cq )-systems and other related spanning systems. Section 3 gives background characterizations of .(Cq )-systems of translates in shift-invariant spaces. Section 4 gives an exposition of the proof of Theorem 3 when .2 < q < ∞. Section 5 proves our main new result, Theorem 6, and uses this to give an alternate proof of Theorem 3 when .2 < q < ∞.

2 Different Types of Spanning Systems This section collects background on different types of spanning systems in a separable Hilbert space H . A system .{fn } ⊂ H is a Riesz basis if it is complete in H and there exist constants .0 < A ≤ B < ∞ such that ∀a ∈ 𝓁2 ,

.

A



|an |2

1 2

    ≤ an fn 

H

≤B



|an |2

1 2

.

(5)

.{fn } ⊂ H is a Bessel system if for every .f ∈ H , one has A system |〈f, fn 〉|2 < ∞ or, equivalently, if the upper inequality in (5) holds; see Theorems 7.2 and 7.4 in [15]. A system .{fn } ⊂ H is minimal if each .fN is not in the closure of .span{fn : n /= N}. A system which is both complete and minimal is called exact. Two systems .{fn }, {gn } ⊂ H are biorthogonal if .〈fj , gk 〉 = δj,k , where .δj,k is the Kronecker delta. Exactness and biorthogonality are related in the following manner: the system .{fn } ⊂ H is exact if and only if there exists a unique sequence .{gn } ⊂ H that is biorthogonal to .{fn }, e.g., see Lemma 5.4 in [15]. .

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Given .2 ≤ q ≤ ∞, a system .{fn } ⊂ H is a .(Cq )-system if there exists .C > 0 such that every .f ∈ H can be approximated arbitrarily well by a finite linear combination . an fn with .‖a‖𝓁q ≤ C‖f ‖H . The following result (see Theorem 3 in [21]) provides equivalent characterizations of exact .(Cq )-systems. Lemma 1 Given .2 ≤ q ≤ ∞, the following are equivalent: (a) .{fn } is an exact .(Cq )-system for H . (b) .{fn } is exact, and there exists .C > 0 such that ∀f ∈ H,



.

|〈f, gn 〉|q

1/q

≤ C‖f ‖H ,

(6)

where .{gn } is the biorthogonal system to .{fn }. (c) .{fn } is complete, and there exists .D > 0 such that D

.



|an |q

1 q

    ≤ an fn 

H

(7)

holds for every finite sum . an fn . If .2 ≤ q ≤ r ≤ ∞, then every .(Cq )-system is also a .(Cr )-system since .‖a‖𝓁r ≤ ‖a‖𝓁q . Also, an exact system is a Bessel (.C2 )-system if and only if it is a Riesz basis. Finally, note that if .{fn } ⊂ H is exact and its biorthogonal system .{gn } ⊂ H is uniformly bounded in norm, i.e., .supn ‖gn ‖H < ∞, then the Cauchy-Schwarz inequality and (6) show that .{fn } ⊂ H is, in fact, an exact .(C∞ )-system. This chapter focuses on systems of translates (1) and exponential systems. Since these systems have the property that biorthogonal systems are uniformly bounded in norm, it follows that for the systems considered in this chapter, exact .(Cq )-systems provide a continuous scale of spanning structures that range from Riesz bases to exact systems.

3 (Cq )-Systems of Exponentials and Translates This section characterizes exact .(Cq )-systems of exponentials and uses this to characterize exact .(Cq )-systems of translates in shift-invariant spaces. Let .T denote the torus, which we identify with .[− 12 , 12 ). Given a nonnegative measurable function .w ≥ 0, let .L2w (T)denote the space of all 1-periodic measurable functions f that satisfy .‖f ‖L2w (T) = ( T |f (x)|2 w(x)dx)1/2 < ∞. For each .n ∈ Z, define .en : T → C by .en (x) = e2π inx , and define the exponential system .E = {en }n∈Z . The following result characterizes when .E is an exact .(Cq )-system in .L2w (T); see Theorem 10.10 in [15].

The Balian-Low Theorem for .(Cq )-Systems in Shift-Invariant Spaces

117

Lemma 2 Suppose .w ∈ L1 (T) and .w ≥ 0. .E is exact in .L2w (T) if and only if −1 ∈ L1 (T). Moreover, the biorthogonal system to .E is given by .h = w −1 e , .w n n with .n ∈ Z. We next require a version of Lemma 2 for exact .(Cq )-systems. Given .f ∈ L1 (T), let the Fourier transform .f = Ff be normalized as .f(n) = 1 Ff (n) = 2 1 f (x)e−2π inx dx for .n ∈ Z. Given .2 ≤ q ≤ ∞, a measurable function −2

v : T → C is an .(𝓁2 , 𝓁q ) Fourier multiplier if the operator .Ψv defined by

.

Ψv g = F(vF −1 g)

(8)

.

is a bounded operator from .𝓁2 (Z) to .𝓁q (Z). The set of all .(2, q)-multipliers is q denoted by .M2 (T). Lemma 3 Let .w : T → C be measurable, and assume that .w > 0 a.e. Given 2 ≤ q ≤ ∞, the following are equivalent:

.

(i) .w −1 ∈ M2 (T). (ii) There exists .C > 0 such that .‖Ff ‖𝓁q (Z) ≤ C‖f ‖L2 q

w2

L2w2 (T).

(T)

holds for all .f ∈

Proof Note that .g ∈ 𝓁2 (Z) if and only if .f = w −1 F−1 g ∈ L2w2 (T). Using Parseval’s equality, one has .‖Ψ 1 g‖𝓁q (Z) ≤ C‖g‖𝓁2 (Z) if and only if .‖Ff ‖𝓁q (Z) ≤ w ⨆ ⨅ C‖wf ‖L2 (T) . Since .‖wf ‖L2 (T) = ‖f ‖L2 (T) , this completes the proof. w2

The following result appears as Proposition 5.2 in [22]; we include a proof for the sake of completeness. Lemma 4 Let .w ∈ L1 (T) satisfy .w > 0 a.e. Given .2 ≤ q ≤ ∞, the following are equivalent: (a) .E is an exact .(Cq )-system for .L2w (T). q (b) .w −1/2 ∈ M2 (T). Proof Note that .w −1 ∈ L1 (T) automatically holds in both directions of the proof. In particular, if (a) holds, then Lemma 2 gives .w −1 ∈ L1 (T). If (b) holds, then .w −1/2 ∈ q 2 −1 ∈ L1 (T). This uses that .M∞ (T) = M2 (T) ⊂ M∞ 2 (T) = L (T), so that .w 2 L2 (T), e.g., see Theorem 1.4 in [19] for a proof with .R instead of .T. Since .w, w −1 ∈ L1 (T), Lemma 2 gives that .E is an exact system in .L2w (T) and the biorthogonal system to .E is given by .hn = en w −1 . Given .f ∈ L2w (T), note that (n). So, by part (b) in Lemma 1, .E is an exact .(Cq )-system for .〈f, hn 〉L2 (T) = f w 2 .Lw (T) if and only if ∀f ∈

.

L2w (T),

 n∈Z

1/q |f(n)|q

≤ C‖f ‖L2w (T) .

(9)

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A. M. Powell

However, since (9) is equivalent to .‖Ff ‖𝓁q (Z) ≤ C‖f ‖L2w (T) , Lemma 3 implies that q E is an exact .(Cq )-system for .L2w (T) if and only if .w −1/2 ∈ M2 (T). ⨆ ⨅

.

.f ∈ L2 (R), we use the Fourier transform normalized as .f(ξ ) =  Given −2π iξ x dx with .ξ ∈ R and let .fq denote the inverse Fourier transform. R f (x)e Define the Gramian Pf of .f ∈ L2 (R) by .Pf (ξ ) = n∈Z |f(ξ + n)|2 .

Theorem 4 Let .f ∈ L2 (R). The system of translates .{f (x + n)}n∈Z is an exact −1/2 ∈ Mq (T). .(Cq )-system for .V (f ) if and only if .(Pf ) 2 | Proof Define the map .Jf : L2Pf (T) → V (f ) by Jf (h) = h f . Theorem 1.2 in [17] shows that .Jf is an isometry and satisfies .Jf (en ) = Tn f , where .Tn f (x) = f (x + n). Thus, given any finite sequence of scalars .{an }, .

    an en  

L2Pf (T)

     = Jf an en 

L2 (R)

    = an Tn f 

L2 (R)

The result now follows from Lemma 4 and Part (c) of Lemma 1.

. ⨆ ⨅

4 Balian-Low Theorem for (Cq )-Systems of Translates This section presents a proof of Theorem 3 in the case .2 < q < ∞. Since the proof given in [22] follows a more general framework, the exposition of this section has been streamlined for the specific setting of Theorem 3.

4.1 Embeddings This section discusses a Sobolev embedding for the Gramian that is needed for the proof of Theorem 3. We use the notation .A ≲ B to indicate that there exists a constant .C > 0 such that .A ≤ CB, and we use the notation .A  B to indicate .A ≲ B ≲ A. Given .0 < s < 1, define Sobolev space .H s (R) to consist of all .f ∈ L2 (R)  the 2s s for which .‖f ‖H (R) = ( R |ξ | |f(ξ ))|2 dξ )1/2 < ∞. An equivalent representation    f(x)|2 of .‖ · ‖H s (R) is given by .‖f ‖2H s (R)  R R |f (x+t)− dxdt, e.g., see Section |t|2s+1

s 2 s 3.5 in [23]. Similarly, .H (T) consists of all .f ∈ L (T) for which .‖f ‖H (T) = ( n∈Z |n|2s |f(n))|2 )1/2 < ∞. An equivalent representation of .‖ · ‖H s (T) is given  1/2   f(x)|2 by .‖f ‖2H s (T)  −1/2 T |f (x+t)− dxdt, e.g., see Lemma 3.2 in [14]. |t|2s+1 √ √ Lemma 5 If .f ∈ H s (R), .0 < s < 1, then . Pf ∈ H s (T) and .‖ Pf ‖2H s (T) ≲ ‖f‖2H s (R) .

The Balian-Low Theorem for .(Cq )-Systems in Shift-Invariant Spaces

119

Proof By Minkowski’s inequality,

1/2    2 Pf (x)| ≤ .| Pf (x + t) − |f(x + t + n) − f(x + n)| . n∈Z

Thus,   ‖ Pf ‖2H s (T) 

1/2



1/2

.

 ≤

−1/2 −1/2 1/2

1/2

−1/2 −1/2

 =



1/2



∞

−1/2 −∞

√ √ | Pf (x + t) − Pf (x)|2 dxdt |t|2s+1 2   n∈Z |f (x + t + n) − f (x + n)| dxdt |t|2s+1

|f(x + t) − f(x)|2 dxdt ≲ ‖f‖2H s (R) . |t|2s+1

We also require the following version of the Sobolev embedding theorem; see Theorem 2.11 in [22]. Theorem 5 If .f ∈ H s (T) with . 12 < s < 1, then for .|x − y| > 0 sufficiently small, 1

|f (x) − f (y)| ≤ C|x − y|s− 2 R(x − y),

.

(10)

where .R : R → [0, ∞) satisfies .limt→0 R(t) = 0 and .|R(t)| ≤ ‖f ‖H s (T) . The constant C only depends on s.

4.2 Proof of Balian-Low Theorem for (Cq )-Systems of Translates We require a few more background lemmas before proving Theorem 3. The following lemma shows that smoothness of .ftogether with extra invariance of .V (f ) implies that the Gramian Pf has a zero; see Equation (2.2) in [2].  Lemma 6 Let .s > 12 . If .f ∈ L2 (R) satisfies . |x|2s |f (x)|2 dx < ∞ and .V (f ) has extra invariance, then there exists .x0 ∈ R such that .Pf (x0 ) = 0.  1, if x ∈ S, Given a set S, define its indicator function .χS by .χS (x) = 0, if x /∈ S. Lemma 7 Let .I ⊂ T be the rectangle .I = [a, b]. If .1 < q ≤ ∞, then the Fourier coefficients of .χI satisfy .‖χI ‖𝓁q (Z) ≳ |I |1−1/q .

120

A. M. Powell

sin(π(b−a)k) Proof Recall that .|χ |, and let .Ja,b = {k ∈ Z : |k(b − a)| ≤ [a,b] (k)| = | πk 1/4}. This gives the required bound q .‖χ  [a,b] ‖𝓁q (Z)

  sin(2π(b − a)k) q  ≳ (b − a)q #(Ja,b ) ≳ (b − a)q−1 .  ≥   πk k∈Ja,b

We are now ready to present a proof of Theorem 3 for the case .2 < q < ∞. Proof of Theorem 3 When .2 < q < ∞ Since .1 − q1 > 12 , Lemma 6 shows there exists .ξ0 ∈ R such that .Pf (ξ0 ) = 0. Without loss of generality, we assume that .ξ0 = 0. Combining Lemma 7, Lemma 4, and Lemma 3 shows that for .ϵ > 0, ϵ

.

1− q1

≲ ‖χ [0,ϵ] ‖𝓁q (Z) ≤ C‖χ[0,ϵ] ‖L2

Pf (T)



ϵ

≲

1/2



|Pf (x)|dx

ϵ

=

0

  | Pf (x) − Pf (0)|2 dx

1/2 .

(11)

0

The assumption (4) gives .f ∈ H s (R) with .s = 1 − q1 . Lemma 5 and Theorem 5 give  .

ϵ

1/2    | Pf (x) − Pf (0)|2 dx ≲

0

ϵ

1/2 |x|2s−1 (R(x))2 dx ≲ ϵ s sup R(x).

0

0≤x≤ϵ

(12) Combining (11) and (12) gives .1 ≲ sup0≤x≤ϵ R(x), which gives a contradiction as ϵ approaches 0.

.

5 Proof Revisited: A Canonical Violator of the Weighted Norm Inequality The proof of Theorem 3 in Sect. 4.2 uses that, as .ϵ → 0, the indicator functions χ[0,ϵ] violate the multiplier condition .‖ g ‖𝓁q (Z) ≤ C‖g‖L2 (T) associated with .T(f ) Pf being an exact .(Cq )-system for .V (f ). The following main new result of this chapter shows there is a canonical choice of function that violates the multiplier condition.

.

Theorem 6 (Nitzan, Northington, Powell) Let .2 < q < ∞, and let .f ∈ L2 (R). Assume that .V (f ) admits extra invariance, and that .T(f ) is an exact .(Cq )-system for .V (f ), and that  .

R

2(1− q1 )

|x|

|f (x)|2 dx < ∞.

(13)

The Balian-Low Theorem for .(Cq )-Systems in Shift-Invariant Spaces

If .− 12 ≤ x0
0 such .|k|s−1 ≲ | 1 .|k| ≥ K. Letting .s = 1 − q gives ∞=

.

 1   q |k|(s−1)q ≲ | ms (k)|q = ‖ ms ‖𝓁q (Z) . = |k|

|k|≥K

|k|≥K

k∈Z

This, together with Lemmas 4 and 3, gives the contradiction

 ∞ = ‖ ms ‖𝓁q (Z) ≲ ‖ms ‖L2

.

Pf (T)

≲

1 2

− 12

|Pf (x)| 2(1− q1 )

|x|

1/2 dx

< ∞.

5.2 A Special Case of Theorem 6 To prove Theorem 6, we first prove the following simpler result which will cover the special case when the Gramian is zero on an interval instead of merely having a zero at a point as guaranteed by Lemma 6. √ Lemma 8 Let . 12 < s < 1. Suppose .G ∈ L2 (T) is nonnegative and satisfies . G ∈ H s (T) and .G(x) = 0 for .− 12 ≤ x ≤ 0. Then

122

A. M. Powell



1 2

.

− 12

|G(x)| dx < ∞. |x|2s

In other words, . |x|1 s ∈ L2√ (T). G

√

√ √ | G(x+t)− G(x)|2 dxdt |t|2s+1 1 that .G(x) = 0 for .− 2 ≤ x ≤ 0

 1/2  1/2 Proof Since . G ∈ H s (T), one has . −1/2 −1/2

< ∞, e.g.,

see Lemma 3.2 in [14]. The assumption

gives that



1/2

.



1 t 2s+1

0

t

1/2  0

 |G(x)|dxdt =

0



0

=

1/2 

−t 0 −t

0

|G(x + t)| dxdt |t|2s+1 √ √ | G(x + t) − G(x)|2 dxdt < ∞. |t|2s+1 (14)

 1/2 Given .ϵ > 0, applying integration by parts to . ϵ 

1/2

.

1 t 2s+1

ϵ

1 = 2s



t

1/2 ϵ

t 0

|G(x)|dxdt gives

|G(x)|dxdt

0



1 t 2s+1

|G(t)| 22s dt − 2s t 2s



1/2

0

1 |G(x)|dx + 2sϵ 2s



ϵ

|G(x)|dx.

(15)

0

Using .G(0) = 0 and Theorem 5 gives 

ϵ

.

0

 |G(x)|dx =

ϵ

   | G(x) − G(0)|2 dx ≲

0

ϵ

|x|2s−1 dx ≲ ϵ 2s .

(16)

0

Combining (15), (16), and .G ∈ L2 (T) ⊂ L1 (T) gives  .

ϵ

1/2

|G(t)| dt ≲ 2s t 2s

 ϵ

1/2

1 t 2s+1

 0

t

|G(x)|dxdt + 22s ‖G‖L1 (T) +

 Since .ϵ > 0 is arbitrary, (17) and (14) give .

1 2

|G(x)| dx − 12 |x|2s

=

 1/2 0

1 . 2s

|G(t)| dt t 2s

< ∞.

(17) ⨆ ⨅

5.3 Proof of Theorem 6: Reduction to Lemma 8 In this section, we prove Theorem 6 by a reduction to Lemma 8. The step (23) in the proof requires some background on weighted norm inequalities for the discrete Hilbert transform; these details are deferred to Sect. 5.4.

The Balian-Low Theorem for .(Cq )-Systems in Shift-Invariant Spaces

123

Proof of Theorem 6 Suppose .f ∈ L2 (R) satisfies the hypotheses of Theorem 6. As in the proof of Theorem 3 in Sect. 4.2, assume without loss of generality that 1 sq  .Pf (0) = 0. Define .sq = 1 − q . By (13), one has .f ∈ H (R), and Lemma 5 √ 2(1− q1 ) | g (n)|2 < ∞. In particular, g is gives .g = Pf ∈ H sq (T). Thus, . n∈Z |n| continuous by Theorem 5. Define the 1-periodic function .h ∈ L2 (T) by h(ξ ) =

 g(2ξ ),

.

if 0 ≤ ξ ≤ 1/2,

g(2ξ − 1),

if 1/2 < ξ ≤ 1.

(18)

A computation shows that the Fourier coefficients of h are  h(n) =

 0,

.

if n ∈ 2Z + 1,

 g (n/2),

if n ∈ 2Z.

In particular,  .

|n|

2(1− q1 )

| h(n)|2 =

n∈Z



|n|

2(1− q1 )

| g (n/2)|2 < ∞,

(19)

n∈2Z

√ so that .h ∈ H sq (T). Since .g(0) = Pf (0) = 0 and g is 1-periodic and continuous, one has .h(0) = h(1/2) = h(1) = 0. Since .h ∈ H sq (T) where .sq = 1 − q1 > 12 , it follows from Theorem 5 that h is .sq -Hölder continuous and that its Fourier series converges absolutely and pointwise. Consequently, 0 = h(0) =



.

 h(n)

0 = h(1/2) =

and

n∈Z



(−1)n  h(n).

(20)

n∈Z

Let .ψ be the 1-periodic function defined by  ψ(ξ ) =

.

1,

if 0 ≤ ξ ≤ 1/2,

−1,

if 1/2 < ξ ≤ 1.

(21)

A computation shows the Fourier coefficients of .ψ satisfy  (n) = .ψ

2 π in ,

if n ∈ 2Z + 1,

0,

if n ∈ 2Z.

(22)

We know that .g, h ∈ H sq (T). The results of Sect. 5.4 will show that ψh ∈ H sq (T).

.

(23)

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A. M. Powell

To see this, recall that (19) gives . h ∈ 𝓁1 (Z). By (19), (20), and . 12 < sq < 1, 2(1− q1 )  applying Lemma 14 to the sequence .α =  h gives . n∈Z |n| |(ψh)(n)|2 = 2(1− q1 )   |ψ ⋆ h(n)|2 < ∞, where .⋆ denotes convolution. n∈Z |n| Define the 1-periodic function G by  Pf (2x),

G(x) =

.

if 0 ≤ x ≤ 1/2, if − 1/2 < x ≤ 0.

0,

Recall .h, ψ are defined by (18) and (21), and note that √ 1 G = (h + ψh) . 2 √ Since .h ∈ H sq (T) and .ψh ∈ H sq (T), it follows that . G ∈ H sq (T). Thus, Lemma 8 gives .



1

.

0

Pf (x) 2(1− q1 )

|x|

 dx ≲

1/2

2(1− q1 )

|x|

0

0 A similar argument gives . −1

Pf (2x)

Pf (x) dx |x|2s

 dx =

1 2

− 12

|G(x)| 2(1− q1 )

|x|

 < ∞, and it follows that .

dx < ∞. 1 2

Pf (x) dx − 12 |x|2s

< ∞.

 ⋆ α and the Discrete 5.4 Weighted Norm Inequalities for ψ Hilbert Transform To complete the proof of Theorem 8 from Sect. 5.3, it remains to prove Lemma 14 which was used for (23). For this, we require some background on the discrete Hilbert transform and .A2 weights. Define the convolution operator T by Tα =

.

πi ), (α ⋆ ψ 2

(24)

 as in (22). The following lemma relates T to the discrete Hilbert transform with .ψ H of a sequence .α, which is defined by (H α)n =



.

k∈Z\{n}

Lemma 9 If .α is any sequence, then

αk . n−k

(25)

The Balian-Low Theorem for .(Cq )-Systems in Shift-Invariant Spaces

(T α)n =

.

 (H α o )n , (H α e )

n,

if n ∈ 2Z, if n ∈ 2Z + 1,

125

(26)

where the sequences .α e , α o are defined by 1 [αn + (−1)n αn ] 2

αne =

.

and

αno =

1 [αn − (−1)n αn ]. 2

(27)

In particular, for any .s > 0,  .



|n|2s |(T α)n |2 ≤

n∈Z

|n|2s |(H α o )n |2 +

n∈Z



|n|2s |(H α e )n |2 .

(28)

n∈Z

Proof If .n ∈ 2Z, then by (22), 

(T α)n =

.

k∈Z with (n−k)∈2Z+1

  αko αk αk = = = (H α o )n . n−k n−k n−k k∈2Z+1

k∈Z\{n}

If .n ∈ 2Z + 1, then by (22), (T α)n =



 αk  αke αk = = = (H α e )n . n−k n−k n−k

.

k∈Z with (n−k)∈2Z+1

k∈2Z

k∈Z\{n}

2s o 2 2s e 2 Now . n∈Z |n|2s |(T α)n |2 = n∈2Z |n| |(H α )n | + n∈2Z+1 |n| |(H α )n | implies (28). ⨆ ⨅ In view of (19), we restrict our attention to sequences .α ∈ 𝓁1 (Z) that satisfy the following for some . 12 < s < 1  .

|n|2s |αn |2 < ∞

(29)

n∈Z

and, in view of (20), which also satisfy  .

n∈Z

αn = 0 =



(−1)n αn .

(30)

n∈Z

Recall that a sequence of nonnegative scalars .{wn }n∈Z is a discrete .A2 weight, e.g., see Section 8 in [18], if there exists .C > 0 such that for every .m, n ∈ Z with .m ≤ n, there holds

126

A. M. Powell



n 

.



wk

k=m

n  1 wk

 ≤ C(n − m + 1)2 .

(31)

k=m

Lemma 10 If .−1 < λ < 1, then the sequence .{wk }k∈Z defined by  wk =

.

|k|λ

if k /= 0,

1,

if k = 0,

(32)

is a discrete .A2 weight. Proof The proof uses that .|x|λ is a standard .A2 (R) weight; see Example 7.1.7 in [12]. We show two representative cases for .m ≤ n; the other cases are handled similarly. If .0 < m ≤ n, then

n  n  n  n    1   1 . wk |k|λ = wk |k|λ k=m

k=m

k=m



n+1

≤

k=m

 

|x| dx λ

m−1

n+1 m−1

1 dx |x|λ



≲ (n − m + 2) ≲ (n − m + 1)2 . 2

If .m < 0 ≤ n and .|m| ≤ |n|, then

n 

.

k=m



wk

n  1 wk



k=m





 n  1 |k| ≤ 1+2 1+2 |k|λ k=1 k=1

n  n  n n     1 1 |k|λ + + |k|λ ≲1+ |k|λ |k|λ n 

λ

k=1

≲1+n

λ+1

k=1

+n

1−λ

k=1



n+1

+

k=1

 

|x| dx λ

0

n+1

0

1 dx |x|λ



≲ n + n + n + (n + 2) ≲ (n − m + 1)2 . 2

2

2

2

Theorem 10 in [18], together with Lemma 10, gives the following weighted norm inequality. Lemma 11 If .− 12 < s <  .

n∈Z\{0}

1 2

and .α is any sequence, then

|n|2s |(H α)n |2 ≲ |α0 |2 +

 n∈Z\{0}

|n|2s |αn |2 .

(33)

The Balian-Low Theorem for .(Cq )-Systems in Shift-Invariant Spaces

127

Proof Since .{wn }n∈Z defined .A2 weight when .λ = 2s, by (32) is a discrete Theorem 10 in [18] gives . n∈Z wn |(H α)n |2 ≲ n∈Z wn |αn |2 . Consequently, 

|n|2s |(H α)n |2 ≤

.

n∈Z\{0}





wn |(H α)n |2 ≲

n∈Z

n∈Z



= |α0 |2 +

wn |αn |2

|n|2s |αn |2 .

n∈Z\{0}

For (23), since .sq = 1 −

∈ ( 12 , 1), we would like a version of (33) when

1 q

1 2

< s < 1. This is not true in general, but does hold when one restricts to sequences {αk }k∈Z that satisfy . k∈Z αk = 0. To see this, first note the following lemma. Lemma 12 If .α ∈ 𝓁1 (Z) satisfies . n∈Z αn = 0, and .βk = kαk , then .(Hβ)n = n(H α)n + αn .

. .

Proof 0=



.



αk = αn +

k∈Z

k∈Z\{n}



= αn + n

k∈Z\{n}





αk = αn +

αk

k∈Z\{n}

n−k n−k



 kαk αk − . n−k n−k k∈Z\{n}

Combining Lemmas 11 and 12 yields the following lemma. Lemma 13 Given . 12 < s < 32 , there exists .C > 0 such that if .α ∈ 𝓁1 (Z) satisfies . k∈Z αk = 0, then  .

|n|2s |(H α)n |2 ≤ C

n∈Z



|n|2s |αn |2 .

(34)

n∈Z

Proof Let .βk = kαk , and note that .β0 = 0. Since .− 12 < s − 1 < Lemmas 11 and 12 gives  .

n∈Z



|n|2s |(H α)n |2 =

n∈Z\{0}

≤2



|n|2s−2 |n(H α)n |2 =



applying

|n|2s−2 |(Hβ)n − αn |2

n∈Z\{0}

|n|2(s−1) |(Hβ)n |2 + 2

n∈Z\{0}

≲0+

1 2,



k∈Z\{0}



|n|2s |αn |2

n∈Z\{0}

|k|2(s−1) |kαk |2 +



n∈Z\{0}

|n|2s |αn |2 ≲

 n∈Z

|n|2s |αn |2 .

128

A. M. Powell

Lemma 14 Given . 12 < s < 32 , there exists .C > 0 such that if .α ∈ 𝓁1 (Z) satisfies k . k∈Z αk = k∈Z (−1) αk = 0, then the convolution operator T defined by (24) satisfies  .

|n|2s |(T α)n |2 ≤ C



n∈Z

|n|2s |αn |2 .

n∈Z

 as in (22), if . n∈Z |n|2s |αn |2 < ∞ and . k∈Z αk = k∈Z (−1)k αk = 0, So, with .ψ then   ⋆ α)n |2 < ∞. . |n|2s |(ψ n∈Z

k e o 1 Proof Since . k∈Z αk = k∈Z (−1) αk = 0, the sequences .α , α ∈ 𝓁 (Z) defined e o by (27) satisfy . k∈Z αk = k∈Z αk = 0. Combining Lemmas 9 and 13 gives  .

|n|2s |(T α)n |2 ≤

n∈Z



|n|2s |(H α e )n |2 +

n∈Z

≲



|n|

2s

|αne |2

+

n∈Z

≲







|n|2s |(H α o )n |2

n∈Z

|n|2s |αno |2

n∈Z

|n|2s |αn |2 .

n∈Z

Acknowledgments The results in this chapter should be considered joint work with Shahaf Nitzan and Michael Northington in connection with the collaborative work published in [22] and during visits at Vanderbilt University, Kent State University, and the Georgia Institute of Technology. The author thanks Chris Heil for valuable discussions on the Balian-Low theorem and Dechao Zheng for sharing his expertise on .Ap weights.

References 1. A. Aldroubi, C. Cabrelli, C. Heil, K. Kornelson, U. Molter, Invariance of a shift-invariant space. J. Fourier Anal. Appl. 16(1), 60–75 (2010) 2. A. Aldroubi, Q. Sun, H. Wang, Uncertainty principles and Balian-Low type theorems in principal shift-invariant spaces. Appl. Comput. Harmon. Anal. 30 (2011), no. 3, 337–347. 3. M. Anastasio, C. Cabrelli, V. Paternostro, Invariance of a shift-invariant space in several variables. Complex Anal. Oper. Theory 5(4), 1031–1050 (2011) 4. R. Balian, Un principe d’incertitude fort en théorie du signal ou en mécanique quantique. C. R. Acad. Sci. 292(20), 1357–1362 (1981) 5. J. Benedetto, C. Heil, D. Walnut, Differentiation and the Balian-Low theorem. J. Fourier Anal. Appl. 1(4), 355–402 (1995)

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6. J. Benedetto, W. Czaja, P. Gadzinski, A.M. Powell, The Balian-Low theorem and regularity of Gabor systems. J. Geom. Anal. 13(2), 239–254 (2003) 7. C. Cabrelli, U. Molter, G. Pfander, Time-frequency shift invariance and the amalgam BalianLow theorem. Appl. Comput. Harmon. Anal. 41(3), 677–691 (2016) 8. A. Caragea, D.G. Lee, F. Philipp, F. Voigtlaender, A quantitative subspace Balian-Low theorem. Appl. Comput. Harmon. Anal. 55, 368–404 (2021) 9. I. Daubechies, A.J.E.M. Janssen, Two theorems on lattice expansions. IEEE Trans. Inf. Theory 39(1), 3–6 (1993) 10. J.-P. Gabardo, D. Han, Balian-Low phenomenon for subspace Gabor frames. J. Math. Phys. 45(8), 3362–3378 (2004) 11. S.Z. Gautam, A critical-exponent Balian-Low theorem. Math. Res. Lett. 15(3), 471–483 (2008) 12. L. Grafakos, Classical Fourier Analysis (Springer, New York, 2014) 13. K. Gröchenig, J.L. Romero, D. Rottensteiner, J.T. van Velthoven, Balian-Low type theorems on homogeneous groups. Anal. Math. 46(3), 483–515 (2020) 14. D. Hardin, M. Northington, A.M. Powell, A sharp Balian-Low uncertainty principle for shiftinvariant spaces. Appl. Comput. Harmon. Anal. 44(2), 294–311 (2018) 15. C. Heil, A Basis Theory Primer. Applied and Numerical Harmonic Analysis (Birkhäuser/Springer, New York, 2011) 16. C. Heil, A.M. Powell, Regularity for complete and minimal Gabor systems on a lattice. Ill. J. Math. 53(4), 1077–1094 (2009) 17. E. Hernandez, H. Sikic, G. Weiss, E. Wilson, On the properties of the integer translates of a square integrable function, Harmonic Analysis and Partial Differential Equations. Contemporary Mathematics, vol. 505 (American Mathematical Society, Providence, RI, 2010), pp. 233–249 18. R. Hunt, B. Muckenhoupt, R. Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform. Trans. Am. Math. Soc. 176, 227–251 (1973) 19. L. Hörmander, Estimates for translation invariant operators in Lp spaces. Acta Math. 104, 93–140 (1960) 20. F. Low, Complete sets of wave packets, in A Passion for Physics—Essays in Honor of Geoffrey Chew, ed. by C. DeTar et al. (World Scientific, Singapore, 1985), pp. 17–22 21. S. Nitzan, J.-F. Olsen, From exact systems to Riesz bases in the Balian-Low theorem. J. Fourier Anal. Appl. 17(4), 567–603 (2011) 22. M. Northington, Uncertainty principles for Fourier multipliers. J. Fourier Anal. Appl. 26(5), Paper No. 76, 38 pp. (2020) 23. E. Stein, Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, No. 30 (Princeton University Press, Princeton, 1970) 24. R. Tessera, H. Wang, Uncertainty principles in finitely generated shift-invariant spaces with additional invariance. J. Math. Anal. Appl. 410(1), 134–143 (2014) 25. A. Zygmund, Trigomometric Series, 3rd edn. (Cambridge University Press, New York, 2002)

Whittaker-Type Derivative Sampling and (p, q)-Order Weighted Differential Operator

.

Tibor K. Pogány

In memory of John Rowland Higgins

MSC 2010: 30B50, 30D15, 41A05, 41A30, 94A12, 94A20

1 Introduction The author’s article [1] gives exhaustive answer to the question about Whittakertype derivative sampling of the stochastic processes which belong to the class α .L (Ω), 0 ≤ α ≤ 2 having spectral representation. The sampling function in this plane sampling procedure is realized with the Weierstraß .σ -function, which certain necessary bounding properties are established in [2–4]. In the case of functions coming from the Leont’ev space .S[2,π q/(2s 2 )) of entire functions of the order 2 which type equals .π q/(2s 2 ); q ∈ N, s > 0, the derivative sampling procedure results are presented in [2]. We refer the reader to Higgins’ work on the Whittaker-type derivative sampling reconstruction [5, 6]. The main difference between Whittaker’s and Higgins’ on the one hand and the recent author’s approach on the other hand is in the shapes of the integration contours in Cauchy residue theorem; Whittaker and Higgins preferred the squared integration path, while the current author used circular contour in his studies. The resulting formulae coincide, but the intermediate steps differ since the obtained truncation errors’ upper bounds. Moreover, the obtained results hold also for harmonizable and wide sense stationary (weakly stationary or stationary in Hinˇcin sense) processes which occur under .α = 2. The related Whittaker-type derivative sampling expansion formulae are interpreted in the .α-mean sense and also in the almost sure .P sense when the

T. K. Pogány () Institute of Applied Mathematics, Óbuda University, Budapest, Hungary Faculty of Maritime Studies, University of Rijeka, Rijeka, Croatia e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. D. Casey et al. (eds.), Sampling, Approximation, and Signal Analysis, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-031-41130-4_7

131

132

T. K. Pogány

initial stochastic processes and their .α-mean derivatives (up to some fixed order) are sampled at the integer lattice .Z2 in the complex plane. The main mathematical tool was the there introduced circular truncation error and its upper bounds, which role was crucial in the truncation error analysis. Finally, derivative sampling sum convergence rate results are provided. As one of the counterpart problems turns out to be the Whittaker-type sampling of the weighted .(p, q)-order derivative,1 the operator’s outcome signal either applied to the Leont’ev class deterministic functions, which consist of the space α .S[2,π q/(2s 2 )) , and the .L (Ω), 0 ≤ α ≤ 2 stochastic processes, and the so-called .α-Piranashvili processes (for the latter consult [7]). The former is treated in the .sup norm, while the latter either in the .α-mean and in almost sure metric. These results are not new; for the first time, some of them were presented on the 2003 Tbilisi conference dedicated to the 100th birth anniversary of Andrey Nikolaevich Kolmogorov [8]; then on the SampTA’03 conference in Strobl, Österreich; and on the 2006 Yadrenko conference held in Kyiv, Ukraine [9]. However, the results concerning the .(p, q)-order derivative operator’s Whittaker sampling reconstruction are not yet presented for wider readership.

2 Overview of Auxiliary Results The Weierstraß .σ -function’s infinite product representation reads [10, p. 868, Eq. 8.172.2] '  1−

σ (z) = z

.

(m,n)∈Z2

   z z2 z exp + , m + in m + in 2(m + in)2

z ∈ C,

where the dashed product means that the zero term is omitted. Assume that f is entire and .

lim sup r→∞

ln Mf (r) qπ < 2, 2 r 2s

q ∈ N, s > 0,

where .Mf (r) := sup|z|=r |f (z)| denotes the maximum modulus, and extending Whittaker’s classical formula, it is already proved [5], [3, Theorem 2] that for all compact z-subsets of .C, we have f (z) = σ (z)

.

q



q q−1 q−1−j   f (q−1−j −k) (m + ni) Rmnj . · j !(q − 1 − j − k)! (z − (m + ni))k+1 2

(m,n)∈Z j =0 k=0

1 In

fact, this is a kind of the Ruscheweyh derivative.

Whittaker Sampling and .(p, q)-Order Differential Operator

133

The convergence of the above series is uniform in z. Here, q

Rmnj =

.

 d j  w − (m + ni q , w→m+in dw σ (w)

q ∈ N.

lim

The case .q = 1 belongs to Whittaker [11]; in turn, for general integer .q ≥ 2, the reconstruction formula was obtained by Higgins [5]. Whittaker’s formula reads [11] 

f (z) = σ (z)

(−1)m+n+mn

.

(m,n)∈Z2

f (m + ni) − π (m2 +n2 ) , e 2 z − (m + ni)

(1)

uniformly in .z ∈ Z. Now, we list bounds for the modulus of the Weierstraß .σ -function. For all .z ∈ C, we have the bilateral bound [3, Theorem 1, Corollary 1.1] π

π

δz K1 (δz ) e 2 |z| ≤ |σ (z)| ≤ δz K2 (δz ) e 2 |z| , 2

2

(2)

.

√ where .δz := inf |z − Z2 | ∈ [0, 1/ 2] and



π 2 δz4  π2 .K1 (δz ) := e 1− G− 6 15   π δz2  π G 2 K2 (δz ) := exp , δz − 1 2 3 − π2 δz2

π 4 δz4 1− 90

2 ,

and .G is Catalan’s constant. Moreover, for .z ∈ C, we have π

δz K1 ≤ |σ (z)|e− 2 |z| ≤ δz K2 , 2

.

where .K2 = K2 (0) = 1 and   2   1 π2 π2 π4 − π4 .K1 = K1 √ . 1− G− 1− =e 360 24 15 2

(3)

Here, .K1 , K2 are the lower and upper (uniform) Hayman’s constants, respectively; see [3, p. 162]. The Leont’ev function space .S[ρ,ψ) consists of all entire functions of the order less than .ρ; in turn, when the order equals .ρ, its type is less than .ψ. That means that .

lim sup r→∞

ln Mf (r) 0 which avoids all points of .Z2 and .[x] stands for the integer part of some .x ≥ 0. Let us recall now the truncation error upper bound result established in [3, p. 162–163, Theorem 2]. Consider function .f ∈ S[2,π qθ/2] , .θ ∈ (0, 1). Then for all .z ∈ C, there holds [3, p. 163, Eq. (17)]    εr z; f ; q  ≤

.

 πq Af r (|z|2 − (1 − θ )r 2 , exp q 2 (r − |z|) (H (r) K1 )

(4)

where .Af characterizes the function .f ∈ S[2,π qθ/2] , that is, |f (z)| ≤ Af e

.

πq 2 2 θ|z|

,

z ∈ C,

and circle .𝚪r0 ; .r0 = √ .K1 is the lower Hayman’s uniform constant. Obviously, the 2 2 (N +1/2) does not contain integer coordinate points since .r0 /∈ N. Accordingly, for all positive integers,   1 |z| +1 + N≥ √ 2(1 − θ ) 2

.

Whittaker Sampling and .(p, q)-Order Differential Operator

135

  and .z ∈ int(𝚪r ) the truncation error’s magnitude becomes .εr0 z; f ; q =   O N q e−2π q(1−θ)N . Finally, the truncation error vanishes with growing N which implies the uniform convergence result .

  lim Ir0 z; f ; σ ; q = f (z) ,

N →∞

being the upper bound in (4) invariant with respect to z.

(p,q)

3 Derivative Sampling of Dσ

[f ](z)

Let f be an entire function coming from the Leont’ev function space .S[2,π ψ/2] ; ψ > 0. For such input function f , define the .(p, q)-order weighted differential operator for .p ∈ N0 , q ∈ N with respect to the Weierstraß .σ -function as

.

D(p,q) [f ](z) = σ

.

(−1)p σ p+q (z)  d p f (z) . 𝚪(p + 1) dz σ q (z)

According to the previous notations, the Whittaker-type q-order derivative sampling (p,q) reconstruction series for .Dσ [f ](z) reads (p,q) .Dσ [f ](z)



=

q−1 q−1−l  

(m,n)∈Z2 l=0 k=0

p+k  p+q (z) Bqkl (zmn )f (l) (zmn ) k σ , l!(q − 1 − l − k)! (z − zmn )p+1+k

(5)

where .zmn = m + ni and  d q−1−l−k  ζ − z q mn .Bqkl (zmn ) = lim . ζ →zmn dζ σ (ζ ) The related truncated sampling series with respect to the already introduced index set .Z(r) becomes 

(p,q) .Ir z; Dσ [f ]



=



q−1 q−1−l  

(m,n)∈Z(r) l=0 k=0

p+k 

σ p+q (z) Bqkl (zmn )f (l) (zmn ) , l!(q − 1 − l − k)! (z − zmn )p+1+k k

which defines the associated sampling truncation error as the difference     εr z; D(p,q) [f ] = D(p,q) [f ](z) − Ir z; D(p,q) [f ] . σ σ σ

.

Now we formulate and prove our main result.

136

T. K. Pogány

Theorem 1 Let .f ∈ S[2,π ψ/2] and .0 < ψ < θ q − (1 − θ )p, θ ∈ (0, 1] for all p, q ∈ N. Then we have

.

      A exp − π2 θ q − (1 − θ )p − ψ r 2 . εr (D(p,q) [f ]; z) ≤ √ f . √ σ p+q p  2 r H (r) K1 )q (1 − 1 − θ )p+1 Moreover, the series expansion 

D(p,q) [f ](z) = σ p+q (z) σ

q−1 q−1−l  

.

(m,n)∈Z2 l=0 k=0

  p+k Bqkl (zmn )f (l) (zmn ) k l!(q − 1 − l − k)! (z − zmn )p+1+k

holds uniformly for all compact z-sets from .C. Proof Consider the integration path .𝚪r = {ζ : |ζ | = r} which encircles the index set .Z(r) = {(m, n) ∈ Z2 : m2 + n2 < r 2 } in positive direction, and let .z ∈ int(𝚪r ) fixed. Now, having in mind that .|f (z)| ≤ Af exp{ π2ψ |z|2 } and we have the bounding inequality (2) upon the modulus of the Weierstraß sigma function, and constant (3), according to the Cauchy integral formula for derivatives, we conclude p+q p!    |f (ζ )| |dζ | . εr (D(p,q) [f ]; z) ≤ |σ (z)| σ 2π 𝚪(p + 1) 𝚪r |σ (ζ )|q |ζ − z)|p+1  max |f (ζ )|  δz K2 (δz ) exp{ π2 |z|2 }]p+q ζ ∈𝚪r  |dζ | ≤ 2π min δζ K1 (δζ ) exp{ π2 |ζ |2 }]q (|ζ | − |z|)p+1 𝚪r ζ ∈𝚪r

√  Af r K2 / 2]p+q

 π 2 2 [(p + q)|z| − (q − ψ)r ] exp ≤ 2 H (r) K1 )q (r − |z|)p+1   π Af r exp − 2 (q − ψ)r 2 − (p + q)|z|2 ≤ . √ p+q  (r − |z|)p+1 H (r)K1 )q 2 √ Fixing .|z| = r 1 − θ , we infer the upper bound       A exp − π2 θ q − (1 − θ )p − ψ r 2 εr (D(p,q) [f ]; z) ≤ √ f , √ σ p+q p  2 r H (r) K1 )q (1 − 1 − θ )p+1

.

which is our stated inequality. It remains to prove the convergence modality of the Whittaker-type sampling differential operator, specifying reconstruction √ series of the .(p, q)-order weighted √ .r = r0 = 2(N + 1/2); N ∈ N in .|z| = r0 1 − θ where .θ ∈ (0, 1], we have Af (4N + 3)q e−π [θq−(1−θ)p−ψ](N +1/2) ≤ . √ √ (1 − 1 − θ )p+1 ( 2K1 )q (2N + 1)p 2

(p,q) .|εr0 (Dσ [f ]; z)|

Whittaker Sampling and .(p, q)-Order Differential Operator

137

Since this upper bound is not containing z, and varying the scaling parameter .θ we exhaust the whole .int(𝚪r ), letting .N → ∞, the truncation error vanishes which proves the uniform convergence of the series (5). ⨆ ⨅

4 Discussion A. During a month-long visit in 1993 to the Anglia Polytechnic University, Cambridge, my host Rowland Higgins introduced me to the Whittaker formula (1) and his own extensions of this result to the derivative plane sampling series, published as [5]. I remain extremely grateful to him for the warm hospitality and also for (indeed not to long) reference list concerning this topic; among others, he drew my attention to the monograph Interpolatory Function Theory by J.M. Whittaker, encouraging me for continuous study sampling theoretical questions, specially the ones which occur for stochastic signals, which leads to the book chapter [13] in the 1999 monograph [14]. Few years later, he sent me the short note [15] written by E.T. Whittaker, the father of J.M. Whittaker, about the existence of an automorphic function analogous to Weierstraß σ function in the sense, when it becomes double-periodic, it becomes exactly the Weierstraß σ -function. Let me recall that in fact σ is not doubleperiodic, but quasi-periodic (see, e.g., [16, p. 447]). The function discussed in [15] could be used as the building block of the novel approach to the Whittaker-type plane sampling reconstruction procedure. B. The next goal in the plane sampling research would be to “translate” the deterministic results of Theorem 1 to the quasi-norm space Lα (Ω); 0 ≤ α ≤ 2 of stochastic processes, and the possible extensions to α-Piranashvili processes, for which we have already “classical” Kotel’nikov–Shannon derivative sampling theorems; see, for instance, [1, 7]. C. The asymptotic behavior of the σ -function was studied by several authors. The first result known to me is given in the monograph by Hurwitz and Courant [17], where it is proved that ln |Mσ (r)| ∼ π2 r 2 , as r → ∞; also consult the PólyaSzeg˝o problem book [18, Chapter 4, §1, Problem 49]. Hayman [19, p. 436–437] proved the existence of absolute constants A, A1 , A2 for which A

A1 δz ≤ |σ (z)| e− 2

.

|z|2

≤ A2 δz .

In turn, the value A = π2 was erroneously given there; the exact value is A = π, which means that the type of σ (z) is π2 ; see [2] and [20]. Finally, exact numerical values of Hayman’s constants are derived by the current author in [3, p. 161, Corollary 1.1.] where the constants A1 = K1 ≈ 0.26574548;

.

A2 = K2 = 1 .

Further detailed treatment of this problem is elaborated in [3].

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References 1. T.K. Pogány, Whittaker-type derivative sampling reconstruction of stochastic Lα (Ω)processes. Appl. Math. Comput. 187(1), 384–394 (2007) 2. T. Pogány, Derivative uniform sampling via Weierstraß σ (z). Truncation error analysis in [2, π q/(2s 2 )). Georgian Math. J. 8, 129–134 (2001) 3. T. Pogány, Local growth of the Weierstraß σ –function and Whittaker–type derivative sampling. Georgian Math. J. 10, 157–164 (2003) 4. T. Pogány, Derivative Whittaker sampling in Leont’ev spaces, in International Workshop on Sampling Theory and Applications—SampTA’03 (Extended Abstracts book, Department of Mathematics, University of Vienna and Numerical Harmonic Analysis Group, Vienna, 2003), pp. 73–74 5. J.R. Higgins, Sampling theorems and the contour integral method. Appl. Anal. 41, 155–171 (1991) 6. J.R. Higgins, Sampling Theory in Fourier and Signal Analysis—Foundations (Oxford Science Publications, Clarendon Press, Oxford, 1996) 7. Z.A. Piranashvili, T.K. Pogány, On generalized derivative sampling series expansion, in Current Trends in Mathematical Analysis and its Interdisciplinary Applications, ed. by H. Dutta, L.j. Koˇcinac, H.M. Srivastava (Birkhäuser Verlag, Springer Basel AC, 2019), pp. 491– 519 8. Z.A. Piranashvili, T.K. Pogány, Some new generalizations of the Kotel’nikov–Shannon formula for stochastic signals, in Probability Theory and Mathematical Statistics Conference Dedicated to 100th Annyversary of A.N.Kolmogorov, held at Tbilisi, Georgia, September 21–27 (2003) 9. T.K. Pogány, Whittaker type sampling reconstruction of Ruscheweyh derivatives for stochastic processes, in International Conference Modern Stochastics: Theory and Applications (Dedicated to the 60th annyversary of the Department of Probability Theory and Mathematical Statistics, Taras Ševˇcenko University, Kyiv and to the memory of Yadrenko M.Y. (1932–2004)), Kyiv, Ukraine (2006) 10. I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products. (Corrected and enlarged edition prepared by A. Jeffrey and D. Zwillinger), 6th edn. (Academic Press, New York, 2000) 11. J.M. Whittaker: Interpolatory Function Theory (Cambridge University, Cambridge, 1935) 12. A.F. Leont’ev, Generalization of Series of Exponentials (Nauka, Moscow, 1981) (in Russian) 13. T. Pogány, Almost sure sampling reconstruction of band–limited stochastic signals, in Sampling Theory in Fourier and Signal Analysis, ed. by J.R. Higgins, R.L. Stens, vol. 2 (Oxford University Press, Oxford, 1999), pp. 209–232, 284–286 14. J.R. Higgins, R.L. Stens (eds.) Sampling Theory in Fourier and Signal Analysis, vol. 2. Advanced Topics (Oxford University Press, Oxford, 1999) 15. E.T. Whittaker, Note on a function analogous to Weierstrass’ sigma–function. The Messenger Math. 31, 145–148 (1902) 16. E.T. Whittaker, G.N. Watson, A Course of Modern Analysis (Cambridge University Press, Cambridge, 1962) 17. A. Hurwitz, R. Courant, Allgemeine Funktionentheorie und elliptische Funktionen. Geometrische Funktionentheorie (J. Springer, Berlin, 1922) 18. Gy. Pólya, G. Szeg˝o, Aufgaben und Lehrsätze aus der Analysis II: Funktionentheorie. Nullstellen. Polynome. Determinanten. Zahlentheorie. Dritte berichtigte Auflage. Die Grundlehren der Mathematischen Wissenschaften, Band, vol. 20 (Springer, Berlin, 1964) (in Russian: Nauka, Moscow (1978)) 19. W.K. Hayman, On the local growth of power series: a survay of the Wiman–Valiron method. Canad. Math. Bull. 17(3), 317–358 (1974) 20. K. Seip, R. Wallstén, Density theorems for sampling and interpolation in the Bargmann–Fock space II. J. Reine Angew. Math. 429, 107–113 (1992)

Shannon Sampling Theorem via Poisson, Cauchy, Jacobi, and Levin Stephen D. Casey

1 Prologue “The Poisson summation formula and Cauchy’s integral and residue formulas are two different aspects of a broad gauged duality formula which lies athwart most of analysis.” S. Bochner

Claude Shannon was born over a century ago. His genius gave us many things, e.g., contributions to switching theory and cryptography and the creation of information theory. He placed what we commonly refer to as the Classical (a.k.a., Whittaker-Kotel’nikov-Shannon (WKS)) sampling theorem as a cornerstone of information theory [68]. The harmonic analysis and signal processing communities have adopted Shannon’s point of view, as is evidenced in Benedetto [6], Gröchenig [38], Higgins [41], Jerri [47], Körner [51], Levin [52], and Papoulis [60, 61] et al. Our “Age of Information” presents us with new challenges, in particular, with respect to sampling. We are in the time of ultra-wideband systems, such as miniature and hand-held devices for communications, robotics, and microaerial vehicles, cognitive radio, radar, etc. We are presented with the challenge of powering these and other systems. We need to develop systems that can communicate information more efficiently in terms of energy. Digital circuitry has provided dramatically enhanced digital signal processing operation speeds, but there has not been a corresponding dramatic energy capacity increase in batteries to operate these circuits; there is no Moore’s law for batteries. Sampling theory provides insight on how to best process signals for these new systems and remains as timely as it ever was [22, 23, 28, 29].

S. D. Casey () Department of Mathematics and Statistics, American University, Washington, DC, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. D. Casey et al. (eds.), Sampling, Approximation, and Signal Analysis, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-031-41130-4_8

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Shannon was careful to note that the sampling theorem did not originate with him. In fact, as Higgins carefully outlines in his paper “Five short stories about the cardinal series” [39] and in Chapter 1 of his book Sampling Theory in Fourier and Signal Analysis: Foundations [41], the ideas of sampling have been derived by many, going back to Poisson and Cauchy. This chapter explores some of these connections and provides machinery (via the Cauchy residue calculus and Jacobi interpolation) to develop not only the sampling theorem but generalizations and extensions of this theory. In particular, we look at the work of six mathematicians • • • • • •

Siméon Denis Poisson (1781–1840) Augustin-Louis Cauchy (1789–1857) Carl Gustav Jacob Jacobi (1804–1851) Boris Yakovlevich (B. Ya.) Levin (1906–1993) Claude Elwood Shannon (1916–2001) Athanasios Papoulis (1921–2002)

and show how their work contributes to sampling theory. We develop connections between some of the most powerful theories in analysis—the Poisson summation formula, Cauchy’s integral and residue formulae, the Jacobi interpolation, and Levin’s sine-type functions—to the Shannon sampling theorem formula and its generalizations. The main techniques in this chapter use tools from complex analysis, and, in particular, the Cauchy theory and the theory of entire functions, to realize sampling sets Λ as zero sets of well-chosen entire functions (sampling set generating functions). We then reconstruct the signal from the set of samples using the Cauchy-Jacobi machinery. These add to a remarkable set of connections of formulae to sampling, as given in a series of papers by Butzer, Higgins, Ferreira et al. Higgin’s paper [39] “Five short stories about the cardinal series,” Bull. AMS, Volume 12, Number 1, pp. 45–89 (1985), is one of the most referenced articles about sampling theory. It establishes the truly international connections of sampling theory. Other papers give connections to summation formulae of Euler-Maclaurin, Abel-Plana, Poisson, Hermite polynomials, Plancherel’s theorem, the maximum modulus principle, and the Phragmen-Lindelöf principle. We cite four of them—[5, 21, 40, 42]. All of these connections give us powerful tools for creating a variety of general sampling formulae, e.g., allowing us to derive Shannon sampling theorem and Papoulis generalized sampling via Jacobi interpolation. Additionally, the techniques developed by these connections are also manifest in solutions to the analytic Bezout equation associated with certain multi-channel deconvolution problems. These in turn lead to another connection to general sampling, leading to multi-rate sampling. We give specific examples of unions of non-commensurate lattices associated with multi-channel deconvolution and use a generalization of B. Ya. Levin’s sine-type functions to develop interpolating formulae on these sets. Classical sampling theory is the relation of the signal, a function f in PaleyWiener space, its values on a sampling set Λ = {λn }, and the recovery of the signal f from the values {f (λn )}. In the cases where multiple components of information

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are required at given sample points (as in, e.g., Papoulis generalized sampling), we also require corresponding derivative sample values {f (j ) (λn )}. We work with two different types of sampling sets throughout the chapter, those that are separated and those that are not. Definition 1 A sampling set Λ is separated if .

inf{|λ − η| : λ, η ∈ Λ, λ /= η} > 0 .

(1)

For Shannon sampling theorem and Papoulis generalized sampling, the set Λ is separated. However, as the work on deconvolution shows, one can relax the condition that Λ is separated and still recover f from {f (j ) (λ)}—conditionally. When the sampling sets are not separated, we have to “piece together” the reconstruction basis. This block summation technique generates the function from a generalized basis or a basis with braces. These ideas were developed by Levin in his seminal paper [54] “On bases of exponential functions in L2 ,” Zap. Mekh.-Mat. Fak. i Khar’kov. Mat, Obshch., 27, pp. 39–48 (1961). We discuss conditional convergence in Sect. 7.

1.1 Preliminary Definitions In this chapter, all functions considered are absolute and square integrable functions on the real line .(f ∈ L1 ∩ L2 (R)), unless noted otherwise. Likewise, all integrals are assumed to be over the whole domain (either .R or .C depending on the context) unless noted otherwise. References for the material on harmonic analysis and sampling include Benedetto [6], Dym and McKean [33], Grafakos [37], Gröchenig [38], Higgins [41], Hörmander [45], Jerri [47], Körner [51], Levin [52], Meyer [57], Papoulis [60, 61], and Young [75]. References of the theory of distributions include Barros-Neto [4], Benedetto [6], Hörmander [45], and Horváth [46].

Fourier Series: Hilbert Spaces We start by defining Fourier series as in [6] and [33]. Let .exp(·) = e(·) . Definition 2 (Fourier Series) Let f be a periodic, absolute, and square integrable function on .R, with period .2Ф, i.e., .f ∈ L1 ∩ L2 (T2Ф ). The Fourier coefficients of f , .f[n], are defined by 1 f[n] = 2Ф



Ф

.

−Ф

f (t) exp(−iπ nt/Ф) dt .

(2)

If .{f[n]} is absolute and square summable (.{f[n]} ∈ l 1 ∩ l 2 ), then the Fourier series of f is

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f (t) =



.

f[n] exp(iπ nt/Ф) .

(3)

n∈Z

The space .L2 (T2Ф ) is the canonical example of a separable Hilbert Space, and the set .{exp(−iπ nt/Ф) : n ∈ Z} is the canonical example of an orthonormal basis for .L2 (T2Ф ). Let .H be a Hilbert space with inner product .〈·, ·〉 and norm .‖ · ‖. We need some of the basics of Hilbert space theory.1 Refer to the elements of .x ∈ H as vectors. We say that a sequence of vectors .{xn } in .H is a Schauder  basis for .H if for every .x ∈ H, there is a unique set of scalars .{cn } such that .x = cn (x)xn , with the series converging in .‖ · ‖. Hereafter, we will refer to this as a basis. We say that the sequence .{xn } in .H is orthonormal if .〈xi , xj 〉 = δi,j for all .i, j in the sequence, where .δi,j is the Kronecker delta. We say that the sequence .{xn } in .H is complete if given .x ∈ H such that .〈x, xn 〉 = 0, then .x = 0. A sequence .{xn } is minimal if every element in the sequence lies outside of the closed linear span of the other elements. A sequence that is both minimal and complete is called exact. If .{xn } is a basis for .H, the mapping .xn I−→ cn (x) is a continuous linear functional. The examples of Hilbert spaces we discuss in this chapter are function spaces. The fact that these functions are also analytic adds a tremendous amount of information and allows us access to the tools of both harmonic and complex analysis. We say that .H is a functional Hilbert space if for every .z ∈ C, the point I → f (z) on .H is bounded, i.e., for every .f ∈ H and evaluation functional .f − .z ∈ C, there exists .Mz such that .|f (z)| ≤ Mz ‖f ‖. By the Riesz representation theorem, every bounded linear functional arises from an inner product. Thus, for .z ∈ C, there exists .Kz ∈ H such that .f (z) = 〈f, Kz 〉 for every .f ∈ H. The function K on .C × C defined by .K(z, w) = 〈Kz , Kw 〉 = Kw (z) is called the reproducing kernel of .H. ∞ A sequence .{ym }∞ m=1 in a Hilbert space .H is biorthogonal to a sequence .{xn }n=1 if .〈xn , ym 〉 = δn,m . By the Hahn-Banach theorem, a given .{xn } will have a biorthogonal sequence .{ym } if and only if .{xn } is minimal. Thus, a basis .B = ∗ ∞ {xn }∞ n=1 for .H possesses a biorthogonal basis .B = {ym }m=1 . A fact that we shall find useful is that .1 ≤ ‖xn ‖‖yn ‖ ≤ M (see [75],  there exists M such that for all n, pp. 19–20). If . n cn xn is convergent if and only if . n cn yn is convergent, then the ∞ bases .A = {xn }∞ n=1 and .B = {ym }m=1 are said to be equivalent. Equivalent bases have equivalent biorthogonal bases. Bessel sequences and Riesz bases are key to our work in this chapter. Basic results are discussed in Chapter 1, Section 8, of Young [75]. A sequence .{xn } ∈ H is sequence if there is a constant B such that for all .x ∈ H,  called a Bessel 2 2 . n |〈x, xn 〉| ≤ B‖x‖ . The sequence .{xn } ∈ H is a Bessel sequence if and only if given scalars .{cn }, there exists a constant .A > 0 such a finite set of arbitrary  ∞ for .H is a bounded that .‖ n cn xn ‖2 ≤ A n |cn |2 . A Riesz basis .B = {xn } n=1 basis. It is also unconditional in that for all .x ∈ H, .x = n 〈x, xn 〉xn converges unconditionally, i.e., regardless of the order in which the terms are summed. There 1 Young

provides an excellent discussion of this material in [75].

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are many characterizations of Riesz bases. A set .B is a Riesz basis if and only if it is equivalent to .A, an orthonormal basis for .H. Also, basis if and  .B is a Riesz 2 ≤ B‖x‖2 . If only if there exists .A, B > 0 such that .A‖x‖2 ≤ |〈x, x 〉| n n ∗ .B ={ym }∞ to .B, then .B∗ is a basis, and, for every .x ∈ H, m=1 is biorthogonal  .x = n 〈x, yn 〉xn = n 〈x, xn 〉yn , where both sums converge unconditionally.  2 −1 2 Moreover, .B −1 ‖x‖2 ≤ n |〈x, yn 〉| ≤ A ‖x‖ . Finally, we note that a set .{xn } = B is a Riesz basis if and only if both .{xn } and its biorthogonal sequence .{yn } are Bessel sequences. We will find this fact useful in our work on multi-rate sampling. For Shannon sampling theorem, sampling sets .Λ = {λn } are separated. The corresponding sets .{e2π iλn t } are Riesz bases. In fact, we have the following. Lemma 1 Suppose that .{xn } is a basis for a Hilbert space .H with the property that infn {xn } > 0. Then there is a .σ > 0 such that .‖xn − xm ‖ ≥ σ for all .n, m.

.

Proof Let .{xn } be a basis, and let .{yn } be the basis that is biorthogonal to .{xn }. Then there exists a constant .A > 0 such that for all n, 1 ≤ ‖xn ‖‖yn ‖ ≤ A .

.

Therefore, there exists a constant .B > 0 such that .supn ‖yn ‖ = B < ∞. Thus, 1 = 〈(xn − xm ), yn 〉 ≤ ‖yn ‖‖xn − xm ‖ ≤ B‖xn − xm ‖ .

.

Let .σ = 1/B. Then .‖xn − xm ‖ ≥ σ for all .n, m.

⨆ ⨅

If the set .{xn } is not separated, the angle between subsets of vectors that get arbitrarily close can get arbitrarily small as .n → ∞. This then causes the norms .‖yn ‖ to become arbitrarily large.

Fourier Transforms The Fourier transform .f(ω) can give us information about continuous spectra. The definition, again from [6] and [33], follows. Definition 3 (Fourier Transform and Inversion Formulae) Let f be a function in .L1 ∩ L2 (R). The Fourier transform of f is defined as (ω) = .f

 R

f (t)e−2π itω dt

(4)

for .t ∈ R (time), .ω ∈  R (frequency). The inversion formula, for .f ∈ L1 ∩ L2 ( R), is )∨ (t) = .f (t) = (f

  R

f(ω)e2π iωt dω.

(5)

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Formally, we can think of the Fourier series coefficient integral (2) and the Fourier transform (4) as analysis and the Fourier series (2) and the inverse transform (5) as synthesis. The choice to have .2π in the exponent simplifies certain expressions, e.g., for .f, g ∈ L1 ∩ L2 (R), .f,  R), we have the Parseval-Plancherel g ∈ L1 ∩ L2 ( equations  g〉 . ‖f ‖L2 (R) = ‖f‖L2 ( R) and 〈f, g〉 = 〈f , 

.

We extend the transform from .L1 ∩ L2 to .L2 via a density argument.

Generalized Functions We can also define the transform for certain classes of generalized functions or distributions (for additional information, see [4, 26, 45]). We define three sets of test functions—.D, S, E. Let .N denote the natural numbers, and let .k ∈ N. We define k .C (R) = {complex valued φ(t) such that φ is k- times continuously differentiable}  and .C ∞ (R) = k C k (R). Then, D(R) = Cc∞ (R) = {φ ∈ C ∞ (R) with compact support },

.

   dk  S(R) = {φ ∈ C ∞ (R) with lim t n k φ(t) = 0} , for all n, k ∈ N, |t|→∞ dt

.

and E(R) = C ∞ (R).

.

Note that the spaces are nested .D ⊂ S ⊂ E. Each of these spaces of test functions has an associated space of distributions. A distribution is a continuous linear functional on a space of test functions whose action on a test function is given formally by .〈λ; φ〉 = R φ(t) · λ(t) dt. This is a linear mapping from the space of test functions into .C which is continuous in terms of the topology on the space of test functions, i.e., if .{φj } is a sequence of test functions converging to zero, then .limj →∞ 〈λ; φj 〉 = 0. Such maps are also called continuous linear functionals. The spaces of distributions associated with the spaces of test functions above are denoted .D' , .S' , and .E' . The more restrictive the notion of convergence on the space of test functions, the broader the class of distributions. We have that .E' ⊂ S' ⊂ D' . The set .S' is called the space of tempered distributions. Let U be an open subset of .R. We say that a distribution .λ ∈ E' is zero on U if 〈λ; φ〉 = 0 for all φ ∈ C∞ (R) such that supp(φ) ⊂ U.

.

The support of a distribution .λ is then defined to be the complement of the largest open subset U where .λ = 0. For example, .supp(δ) = {0}. It can be shown that .E' is the set of compactly supported distributions.

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The operation of convolution extends to distributions, and for two distributions .λ and .η, .λ ∗ η is defined as 〈λ ∗ η; φ〉 = 〈λy ; 〈ηx ; φ(x + y)〉〉 = 〈ηx ; 〈λy ; φ(x + y)〉〉.

.

Convolution exists only if one of the distributions is compactly supported. This definition reduces to the standard integral representation if the two distributions are functions. The operation is linear and commutative and is associative if two of the three distributions are compactly supported. For distributions in .E' , convolution is associative, and so .E' is a commutative algebra with the identity .δ. Convolution commutes with translations and differentiations. To define the Fourier transform for distributions, we proceed as follows. For a general distribution .λ, we can attempt to define it as 〉. 〈 λ; φ〉 = 〈λ; φ

.

 is not compactly supported unless .φ = 0. This is circumvented by However, .φ  is in .S. Thus, for .λ ∈ S' , the definition above makes noting that for .φ ∈ S, .φ sense. In fact, the Fourier transform from .S' to .S' is a topological isomorphism. The transform  .λ and inverse transform ˇ 〈λˇ ; φ〉 = 〈λ; φ〉

.

are continuous linear maps with respect to the strong topology on .S' . For .λ ∈ E' , the Fourier transform continues as an analytic function to the entire complex plane. This continuation defines the Fourier-Laplace transform on ' .E , which is given by −2π itζ  .λ = 〈λ; e 〉.

Properties of the function  .λ are given in the Paley-Wiener theorem.

2 Shannon sampling theorem via Poisson Classical sampling theory applies to functions that are both square integrable and bandlimited. We need the concepts of the support and the periodization of a function. Definition 4 (Support) Let .f : R −→ C. Then the support of f , denoted .supp f , is the smallest closed subset outside of which f vanishes. Definition 5 (Periodization) Let . T > 0 and let .g(t) ∈ L1 (R). Then, .[g]◦ (t), the ◦ T -periodization of g, is .[g] (t) = ∞ n=−∞ g(t − nT ) a.e. .

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A function that is both .Ω bandlimited and .L2 has several smoothness and growth properties given in the Paley-Wiener theorem (see, e.g., [4, 33]). We denote this class of functions by .PWΩ . The sampling theorem [50, 67, 71, 72] applies to functions in 2 .PWΩ . Definition 6 (Paley-Wiener Space .PWΩ ) PWΩ = {f : f, f ∈ L2 , supp(f) ⊂ [−Ω, Ω]} .

.

Theorem 1 (WKS Sampling Theorem) Let .f ∈ PWΩ and let .T > 0 be a fixed sin( π t) sampling rate. Let .sincT (t) = π Tt and .δnT (t) = δ(t − nT ), a shifted Dirac delta. (a) If .T ≤ 1/2Ω, then for all .t ∈ R, 

f (t) = T

.

n∈Z

sin( Tπ (t − nT )) =T f (nT ) π(t − nT )





 δnT · f

∗ sinc(t) .

n∈Z

T

(6)

(b) If .T ≤ 1/2Ω and .f (nT ) = 0 for all .n ∈ Z, then .f ≡ 0. A beautiful way to prove the WKS sampling theorem is to use the Poisson summation formula. We sketch these ideas, using them to provide a baseline for further discussion. Let .T > 0, and suppose that for an .η > 0 and .C > 0, we have 1 1 , |f(ω)| ≤ C . 1+η (1 + |t|) (1 + |ω|)1+η

|f (t)| ≤ C

.

 Let .[f ]◦ (t) = n∈Z f (t − nT ) be the T -periodization of f . We can then expand ◦ .[f ] (t) in a Fourier series. The sequence of Fourier coefficients of this T -periodic  

function is given by .[f ]◦ [n] = T1 f − Tn . We have  .

f (t + nT ) =

n∈Z

1  f (n/T )e2π int/T . T

(PSF)

n∈Z

Therefore, when .t = 0,  .

n∈Z

f (nT ) =

1  f (n/T ) . T

(PSF1)

n∈Z

Thus, the Poisson summation formula allows us to compute the Fourier series of [f ]◦ in terms of the Fourier transform of f at equally spaced points. This extends to the Schwartz class of distributions .S' as follows. Let .f ∈ S and let .〈T ; f 〉 be

.

2 Please

also see [31, 34, 36, 55, 56, 58, 59, 66, 69, 73, 74, 76].

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the action of a distribution .T ∈ S' on f . Then, applying Parseval-Plancherel and (PSF1),         1  f (nT ) = . δnT ; f = δnT ; f = f (n/T ) T n∈Z n∈Z n∈Z n∈Z    1 1  δn/T (f) = δn/T ; f . = T T n∈Z

n∈Z

Thus,  1   δn/T . δnT = T

.

(PSF2)

n∈Z

n∈Z

We recall the Fourier transform pairings .f ∗ g ←→ f ·  g , .f · g ←→ f ∗  g, χ and . [−1/2T ,1/2T ) ←→ sincT (see Benedetto [6]). If .f ∈ PWΩ , .f is compactly supported, and we can periodically extend the function. If .T ≤ 1/2Ω,  

n χ (ω) = δn/T ∗ f · χ [−1/2T ,1/2T ) . .f f(ω − ) · [−1/2T ,1/2T ) (ω) = T n∈Z n∈Z (7) 

By computing the inverse transform (using the pairings listed above) and applying (PSF2), we get that (7) holds if and only if f (t) = T





.





δnT · f

∗ sincT (t) .

(8)

n∈Z

3 Cauchy and Jacobi: Preliminaries 3.1 Cauchy Theory We need some basics from complex analysis and, in particular, Cauchy’s theorem. References for this subsection include Ahlfors [1, 2], Apostol [3], Conway [30], and Levin [52, 53]. Recall that a domain .Ψ is an open, simply connected subset of the complex plane .C and that a holomorphic function on .Ψ is a complex-valued function on .Ψ that has a continuous complex derivative. Theorem 2 (Cauchy Integral Formula) Let f be holomorphic in a domain .Ψ ⊂ C. Let .𝚪 be a simple closed positively oriented rectifiable curve in .Ψ with .z ∈ interior .𝚪. Then,

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1 .f (z) = 2π i

 𝚪

f (ζ ) dζ (ζ − z)

(9)

By Leibnitz’s theorem on differentiating under the integral sign, we get f (n) (z) =

.

n! 2π i

 𝚪

f (ζ ) dζ (ζ − z)n+1

(10)

Analytic functions have Taylor expansions in neighborhoods of any points in the domains in which they are analytic. By expanding the kernel of Cauchy integral formula, we get that holomorphic functions are analytic. (See [30], pp. 83–86.) If f is analytic in a domain .Ψ, and .ζ ∈ Ψ, if .|f (ζ )| ≥ |f (z)| for all .z ∈ Ψ, then f is a constant [52, 53]. The is the maximum principle ([30], pp. 128–129). We also have that a function g is a meromorphic function in a region if it is analytic in region except for poles. We can expand meromorphic functions in neighborhoods of poles in a Laurent series. This expansion, in an annular neighborhood .{z : 0 < r1 ≤ |z − w0 | ≤ r2 }, is g(z) =

∞ 

.

am (z − w0 ) + m

m=0

𝓁 

bn (z − w0 )(−n) ,

n=1

with coefficients given by the integrals am =

.

1 2π i

 𝚪

g(z) 1 dz , bn = m+1 2π i (z − w0 )

 g(z)(z − w0 )n−1 dz , 𝚪

for .𝚪 = {z : |z − w0 | = r}, where .r1 < r < r2 . The poles of g can be classified based on the number of non-zero .bn coefficients (see [30], Chapter V). The largest index of n of non-zero .bn is the order of the pole. If this index n is unbounded, the pole is essential, and the upper index of summation .𝓁 in the Laurent expansion of g is .∞. Residues are given in terms of the Laurent coefficient .b1 . For a simple Jordan curve .𝚪 about an isolated singularity .z0 of a meromorphic function f , b1 = Res(f, z0 ) =

.

1 2π i

 f (ζ ) dζ . 𝚪

Given n isolated singularities .z1 , . . . , zn , contained in the interior of a Jordan curve 𝚪, the residue theorem (see [30], Chapter V) then evaluates

.

1 . 2π i

 f (ζ ) dζ = 𝚪



Res(f, zk ) .

k

Entire functions are analytic everywhere and thus have convergent Taylor expansions throughout .C. Questions concerning the growth of a function and the

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149

distribution of its roots are essential to the theory of entire functions. Liouville’s theorem tells us that a bounded entire function is a constant ([30], page 77). For a general characterization of growth, let .Mf (r) = max|z|=r |f (z)| . By the maximum principle, this function increases monotonically with r. By the fundamental theorem of algebra, there is a direct link between the growth of a polynomial and the number of its roots, which are both given by its degree. The situation for general entire functions is considerably more complicated. The extensive theory studying this relationship between growth and roots will provide some powerful tools to interpolate discrete data and reconstruct functions. For a general entire function, its growth is described by two parameters order ρ = lim sup

.

r→∞

log log Mf (r) log Mf (r) ; type σ = lim sup . log r rρ r→∞

as exp(σ r ρ ) If we expand f in a Asymptotically,3 we have that f satisfies .Mf (r) <  n Taylor series about .z = 0 (.f (z) = cn z ), the coefficients asymptotically satisfy

|cn | 0, there exists .βη > 0 such that for all z such that .|z−n| > η, βη e2π |z| ≤ | sin π z| .

.

(32)

Proof First note that for .z = x + iy, | sin(π z)|2 = 1/2 cosh(2πy) − cos(2π x)) , e−2π |y| cosh(2πy) = 1/2(1 + e−4π |y| ) .

.

Therefore, e−2π |y| | sin(π z)|2 = 1/2(e−2π |y| (cosh(2πy) − cos(2π x)))

.

= 1/4(1 + e−4π |y| + 2e−2π |y| (cosh(2πy)) = 1/4(1 + e−2π |y| )2 + 1/2e−2π |y| (1 − cos(2π x)) . (33) Now, suppose that .δ < 1/4 and .|z − n| > δ. If .|y| ≥ δ/2, then .(1 − e2π |y| )2 ≥ (1−eπ δ )2 > 0. If .|y| < δ/2, then .(e2π |y| (1−cos(2π x)) ≥ (eπ δ )(1−cos(2π δ)) > 0. Let .βη be the minimum of these two estimates. ⨆ ⨅ Lemma 6 The set .Z is a set of uniqueness for .PW(1/2) . Proof Suppose that there exists .g ∈ PW(1/2) such that g(n) = 0 for all n ∈ Z .

.

Then the function ϕ(z) =

.

f (z) g(z)

is entire and is of exponential type .π . Note that .g(z) satisfies the Paley-Wiener growth bound, i.e., there exists .C = C(n) such that for all .n ∈ N,

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|g(z)| ≤ C(n)(1 + |z|)−n eπ |z| .

.

Given .η, .0 < η < 1/4, define the covering .Cη by Cη =



.

n∈Z

{z : |z − n| < η} .

Thus, .Cη is a countable collection of disjoint open sets in .C. If .z ∈ / Cη , then dist(z, Z) > η. Moreover, on the boundaries of .Cη , namely,

.

∂Cη =



.

n∈Z

{z : |z − n| = η} ,

by Lemma 5, |ϕ(z)| ≤

.

C . βη

By the maximum modulus principle, this estimate extends to .Cη . Therefore, .ϕ is a bounded entire function and so, by Liouville’s theorem, is a constant. Since .ϕ(n) = 0 for all .n ∈ Z, .ϕ(z) ≡ 0. Thus, .Z is a set of uniqueness for .PW(1/2) . ⨆ ⨅ This completes the proof of the theorem.

⨆ ⨅

Remark Given .G(z) = sin(π z) and .λ ∈ Z, the Lagrange interpolants are Hk (z) =

.

sin(π z) sin(π(z − k)) G(z) = = = sinc(z − k) , G' (λk )(z − λk ) π cos(π k)(z − k) π(z − k)

the canonical basis for .PW(1/2) , and are the biorthogonal basis to the canonical basis {exp(−iπ nt) : n ∈ Z} for .L2 [−1/2, 1/2].

.

Remark There are several ways to connect Poisson summation to the Cauchy integral. The general method is to connect Poisson summation to WKS sampling and then connect sampling to a restricted Cauchy integral (fixed contour, either a circle or rectangle). We again note that in a series of papers by Butzer, Higgins et al., sampling is connected to summation formulae of Euler-Maclaurin, AbelPlana, Poisson, Hermite polynomials, Plancherel’s theorem, the maximum modulus principle, and the Phragmen-Lindelöf principle. Four such papers are [5, 21, 40, 42].

5 Papoulis Generalized Sampling via Cauchy and Jacobi Papoulis gave a generalization of WKS sampling in the paper “Generalized sampling expansion,” IEEE Trans. Circuits and Systems, 24 (11), 652–654 (1977). His technique was to write sampling down in terms of linear systems and then solve

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the resulting system of equations (see [61, 62]). This gave us formulae for derivative and bunched samples. We will use Jacobi interpolation to derive the Papoulis theorem for derivative sampling. In particular, we derive the “double point formula,” where the sampling rate is half the rate of WKS sampling, but for which twice the information, namely, the values of f and .f ' , is required at each of the sample points. Theorem 6 Let .f ∈ PWΩ and let .T > 0 be a fixed sampling rate. Let .sinc(t) = sin(π t) πt . (a) If .T ≤ 1/2Ω, then for all .t ∈ R,  2 

 

(t − 2nT ) ' f (2nT ) + (t − 2nT )f (2nT ) sinc .f (t) = . 2T

(34)

n∈Z

(b) If .T ≤ 1/2Ω and .f (2nT ) = 0, .f ' (2nT ) = 0 for all .n ∈ Z, then .f ≡ 0. To simplify notation, we again assume .Ω ≤

1 2

.

Proof of Papoulis via Cauchy and Jacobi Given f which satisfies the hypothesis of the theorem, we have that .f (t) is real analytic and has an analytic continuation .f (z) to .C. Moreover, .f (z) satisfies the Paley-Wiener growth bound for .Ω, i.e., there exists .C = C(n) such that for all .n ∈ N, |f (z)| ≤ C(n)(1 + |z|)−n e2π Ω|z| .

.

Let .T = 1. Then .T ≤

1 2Ω .

Also, let G(z) = sin2

.

πz 2

(35)

.

The function .G(z) will be the Jacobi interpolating function and has zeros .Z = 2k, k ∈ Z. To avoid these zeros, let .𝚪m be a circle centered at the origin with radius .(2m + 1), for .m ∈ N. We again apply the Jacobi interpolation formula. .

f (z) =

.

=

1 2π i 1 2π i

 

𝚪m

f (ζ ) dζ (ζ − z)

𝚪m

1 f (ζ ) G(ζ )−G(z) · dζ + (ζ −z) G(ζ ) 2π i

 𝚪m

f (ζ ) G(z) · dζ . (36) (ζ −z) G(ζ )

Here, we see the structure similar to (11), with the first integral providing the residues and the second integral playing the role of the remainder. Let .Rm equal the second integral. We will show below that .|Rm | −→ 0 as .m −→ ∞. The estimation provides a bound for the truncation error in our sampling formula. In Eq. (36), we again see the structure similar to (11) and (23), with the first integral playing the role

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of (27) and the second integral playing the role similar to that of (29). Let .Rm equal the second integral. We will show below that .|Rm | −→ 0 as .m −→ ∞. Again, the estimation of this second integral provides a bound for the truncation error in our sampling formula. Computing, 

f (ζ ) G(ζ ) − G(z) · dζ + Rm G(ζ ) 𝚪m (ζ − z)

  1 1 f (ζ ) G(z) · − dζ + Rm . = 2π i 𝚪m (ζ − z) G(z) G(ζ )

f (z) =

.

1 2π i

(37)

We again use the Weierstrass product representation of a sine function, getting  G(ζ ) = sin

2

.

πζ 2



2 N   ζ2 1− 2 = lim gN (ζ ) = lim (π ζ ) . N →∞ N →∞ 2j 2

j =1

The terms of the Mittag-Leffler partial fraction are repeating and therefore telescope. The formula generalizes to the expansion 1 1 = (ζ − z) · gN (ζ ) (ζ − z)gN (z)  . 

(−1) 1 + '' (n) + (2n − z)(ζ − 2n)2 g '' (n) . (2n − z)2 (ζ − 2n)gN N

(38)

|n|≤N

Finally, for f analytic in a neighborhood of .z0 , the residues needed are Res

.

f (z) f (z) = f (z0 ) , Res = f ' (z0 ) . (z − z0 ) (z − z0 )2

Then, if .m < N, G(z) .f (z) = 2π i (1.)

 𝚪m

 1 1 f (ζ ) · − dζ + Rm (ζ − z) G(z) G(ζ )

gN (z) = lim N →∞ 2π i

(2.)

gN (z) = lim N →∞ 2π i

(3.)

 𝚪m



 1 1 f (ζ ) · − dζ + Rm (ζ − z) gN (z) gN (ζ ) 

𝚪m |n|≤N

1 '' (n) (2n − z)2 (ζ − 2n)gN  (−1) + '' (n) dζ + Rm (2n − z)(ζ − 2n)2 gN

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(4.)

= lim gN (z) N →∞

|n|≤N

= lim gN (z)

 𝚪m



f (ζ ) '' (n) (2n − z)2 (ζ − 2n)gN  (−1)f (ζ ) + '' (n) dζ + Rm (2n − z)(ζ − 2n)2 gN



(5.)

N →∞

1 2π i

|n|≤N

f (2n) f ' (2n) + '' (n) '' (n) (z − 2n)gN (z − 2n)2 gN

 + Rm

    f (2n) 4 f ' (2n) 2 πz + Rm = 2 sin + 2 (z − 2n) π (z − 2n)2

(6.)

|n|≤m

 2 

 

(z − 2n) ' f (2n) + (z − 2n)f (2n) sinc + Rm = 2

(7.)

(39)

|n|≤m

where (1.) is the Jacobi interpolation, (2.) is the Weierstrass product, (3.) is the Mittag-Leffler decomposition, (4.) is the switch between integration and a finite sum, (5.) is the Cauchy residue calculus, (6.) is the Weierstrass product (and trigonometric evaluation), and (7.) is the definition of the sinc function. By showing that .|Rm | −→ 0 as m −→ ∞, we get the sampling formula. We now have to estimate .|Rm |, i.e., the second integral above. Lemma 7    1   2π i

.

𝚪m

 G(z)  f (ζ ) · dζ  −→ 0 as m −→ ∞ . (ζ − z) G(ζ )

(40)

Proof We have that .𝚪m is a circle centered at the origin with radius .(2m + 1) for m ∈ N. Thus,     1 G(z)  f (ζ )  · dζ  .  2π i 𝚪m (ζ − z) G(ζ ) 

   2π  1  sin(π z) f ((2m + 1)eiθ ) iθ =  i((2m + 1)e ) dθ  . iθ 2π i 0 (ζ − z) sin(π(2m + 1)e )

.

Now let z be contained in the closed ball of R centered at the origin, with .R < 2m + 1. Since .ζ ∈ 𝚪m , |ζ − z| ≥ (2m + 1) − R .

.

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Since .f (z) satisfies the Paley-Wiener growth bound for .Ω ≤ 1/2, there exists .C = C(n) such that for all .n ∈ N, |f (z)| ≤ C(n)(1 + |z|)−n eπ |z| .

.

For .z = x + iy, | sinh(y)| ≤ | sin(z)| ≤ | cosh(y)| .

.

Choose .δ sufficiently small so that for .0 < |θ | < δ, | sin(π(2m + 1) cos(θ ))| ≥

.

1 . 2

Then, for .0 < |θ | < δ, | sin(π(2m + 1) cos(θ ))| ≥

.

1 exp(π(2m + 1)| sin(θ )|) . 2

Thus,  G(z)  f (ζ ) · dζ  𝚪m (ζ − z) G(ζ ) 

   2π  1  sin(π z) f ((m + 1/2)eiθ ) iθ  = i((2m + 1)e ) dθ  iθ 2π i 0 (ζ − z) sin(π(2m + 1)e )

   1  .|(Rm )| =  2π i

≤ | cosh(πy)|

C(n) exp(π(2m + 1)(Ω − 1)) . [2m + 3/2]n

(41)

   G(z)  f (ζ ) · dζ  = O em(Ω−1) (ζ − z) G(ζ )

(42)

For .0 < Ω < 1,    1 .  2π i

𝚪m

which .−→ 0 exponentially as .m −→ ∞ and .δ −→ 0+ . For .Ω = 1,    1 .  2π i

𝚪m

   1 G(z)  f (ζ ) · dζ  = O (ζ − z) G(ζ ) [3/2 + 2m]n

(43)

which .−→ 0 with polynomial decay as .m −→ ∞ and .δ −→ 0+ . This completes the proof of the lemma. ⨆ ⨅ We can show uniqueness by an argument similar to Lemma 6. This completes the proof of the theorem.

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Remark A straightforward way to write down the Lagrange-Hermite interpolating formula for double points is as follows. Given an analytic function f and a set of interpolating points .{λ0 , . . . , λn }, let .Lk (z) be the Lagrange interpolant at .λk . Let Hk (z) = [Lk (z)]2 (1 − 2L'k (λk )(z − λk )) , Kk (z) = [Lk (z)]2 (z − λk ) .

.

Then we have Hj (λk ) = δj k , Hj' (λk ) = 0 , Kj (λk ) = 0 , Kj' (λk ) = δj k ,

.

and the Lagrange-Hermite interpolating formula has the form .

 [f (λk )Hk (z) + f ' (λk )Kk (z)] .

Given that .Λ = 2Z is a set consisting of double points and .G(z) = sin2 .f ∈ PW(1/2) , f (z) =



.

f (2k) + (z − 2k)f ' (2k)



 sinc

k∈Z

Remark Given .G(z) = sin2  Hk (z) = sinc

.

 πz  2

(z − 2k) 2

(z − 2k) 2

 πz  2 , for

 2 .

and .λ ∈ 2Z, the Lagrange interpolants are 2

 , Kk = (z − 2k) sinc

(z − 2k) 2

2

which form a basis for .PW(1/2) . They are the biorthogonal basis to the basis {t (j ) exp(−iπ nt) : j = 0, 1, n ∈ Z}

.

for .L2 [−1/2, 1/2]. Remark The result generalizes. Using the Jacobi interpolating function

 π z K G(z) = sin , K

.

we can sample at .1/K the rate, namely, sample points at .KZ. However, at each point, we now require a “K-tuple” of information, namely, the values of f , .f ' , .f '' , (K−1) at the sample points. The sampling formulae are as follows. .. . . f K = 1 : f (t) =



.

n∈Z

K = 2 : f (t) =



 (t − nT ) f (nT ) sinc T



n∈Z

f (2nT ) + (t − 2nT )f ' (2nT )



 sinc

(t − 2nT ) 2T

 2 ,

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165

and, for general .K ∈ N, 

(t−KnT )2 '' f (KnT )+(t−KnT )f ' (KnT )+ .f (t) = f (KnT ) 2! n∈Z

(t−KnT )(K−1) (K−1) f +...+ (KnT ) (K−1)!





(t−KnT ) sinc KT

 K . (44)

This last formula naturally leads to a discussion of the coding of information for functions .f ∈ PWΩ . We are exchanging a slower rate of gathering information with a requirement of an exactly corresponding increase in the amount of information gathered at each sample point. As .K −→ ∞, we are approaching encoding all of the information of the function at a single point. However, we are requiring an infinite amount of information about the function f at that point, namely, the values of .f, f ' f '' . . .—the information in the Taylor series. Let .Δ be an open, simply connected domain in .C, and let h be an analytic function .h : Δ −→ C. Then the following are equivalent. • .h ≡ 0. • There exists .z0 ∈ Δ such that .h(n) (z0 ) = 0 for all .n ∈ N ∪ {0}. • The zeros of h, .Zh , have a limit point in .Δ. (See [30], pp. 78–79.) The complete unique encoding of h at a single point. Eldar [35], pp. 228–233, has a discussion of similar ideas in her writing on Papoulis generalized sampling.

6 Multi-Channel Deconvolution via Cauchy and Jacobi We consider, in this section, an overview of the problem of recovering information from linear translation-invariant systems (deconvolution). The details of this work are presented in two papers of Casey and Walnut [26, 27]5 . A key step in our solutions of deconvolution problems is the interpolation from discrete data, using the Cauchy residue calculus and Jacobi interpolation. This key step essentially boiled down to a sampling problem. Multi-channel deconvolution utilizes information recovery from a given signal by taking several “looks” at the signal, each of which recovers information possibly missed by one of the other “looks.” The “looks” are sensors and can be modeled as ' d a collection of compactly supported distributions .{μi }m i=1 ⊆ E (R ). We discuss the first two steps in this recovery process. The first is how one chooses .{μi }m i=1 . The framework of how this is done is given in a theorem of Hörmander [44]. This first

5 Please

also see [7, 8, 10–12, 15, 19, 20, 24, 25, 70].

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step gives conditions on the sensors .{μi }m i=1 which allow for this reconstruction. This step gives us the discrete sets which will act as our sampling sets. The second step in multi-channel deconvolution is to recover an arbitrary signal, m a function .f ∈ C ∞ (Rd ) from the data .{si }m i=1 = {f ∗ μi }i=1 This second step is a sampling problem, an interpolation from discrete data. This step involves the construction of deconvolvers, which come in a variety of types but which are essentially a collection of distributions which (1) depend only on the convolvers m and (2) allow for the solution with only simple linear operations on the data .{μi } i=1 m .{si } i=1 . The deconvolvers are constructed via interpolation from the discrete sets, using the Cauchy residue calculus and Jacobi interpolation. ' d We construct a set of distributions .{νi }m i=1 ⊆ D (R ) which satisfy m 

μi ∗ νi = δ

(45)

μ i (γ ) νi (γ ) = 1 .

(46)

.

i=1

or, equivalently, m  .

i=1

The collection .{νi }m i=1 is a set of deconvolvers. In this case, f may be recovered by m  .

i=1

si ∗ νi =

m m   (f ∗ μi ) ∗ νi = f ∗ (μi ∗ νi ) i=1

=f ∗

i=1 m 

μi ∗ νi = f ∗ δ = f ,

(47)

i=1

provided that the associative law holds. Equation (46) is a type of Bezout equation. Many theorems in elementary number theory, such as Euclid’s lemma or Chinese remainder theorem, are derived from the basic Bezout equation, which holds in principle ideal domains. Equation (46) is an analytic Bezout equation, in which we are dealing with transcendental entire functions, rather than finite number theoretic problems. Bezout problems involving transcendental entire functions have been extensively studied in a variety of contexts, including the study of division problems, interpolation, analytic continuation, complexity theory, number theory, and solution to systems of PDEs [9, 16–18, 44, 49]. For the purposes of this chapter, we require the following result of Hörmander which gives necessary and sufficient conditions under which compactly supported solutions of (45) exist. Hörmander’s result gives a framework for solving the first step of the problem. Theorem 7 ([44]) There exist compactly supported distributions

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167

' d {νi }m i=1 ⊆ E (R )

.

solving (45) if and only if there exist constants .A, B, N > 0 such that m  .

|μ i (z)| ≥ A(1 + |z|)−N e−B|z| for all z ∈ Cd .

(48)

i=1 ' d 6 A collection .{μi }m i=1 ⊆ E (R ) which satisfies (48) is said to be strongly coprime. Several different classes of strongly coprime convolving systems are discussed in [26, 27]. The most relevant systems for our discussion on sampling are due to Petersen and Meisters.

Definition 7 A real number .α is poorly approximated by rationals provided that there exist constants .C, N > 0 such that for all integers .p, q, |α − p/q| ≥ C|q|−N .

.

(49)

√ a perfect For example, quadratic irrationals of the form . n, where .n ∈ N is not √ square, are poorly approximated by rationals. The golden mean .ϕ = (1 + 5)/2 is the most poorly approximated, as discussed in Hardy and Wright (see [26]). Theorem 8 ([63]) Let .0 < r1 < · · · < rm , .m ≥ d + 1 satisfy the condition that ri /rj is poorly approximated by rationals whenever .i /= j . Then the collection m is a strongly coprime set. .{χ [−r ,r ]d } i=1 i i .

The next step of the problem involves solving an interpolation problem, recon' ) from structing functions (the deconvolvers) in a space of restricted growth (.E discrete data (their values on the zero sets of the convolvers). This gives solutions to the Bezout equation [26, 27]. Note that this step is essentially a sampling problem. First, we “smooth out” our deconvolvers using an approximate identity family. Let k d .ϕ ∈ Cc (R ), for a sufficient degree of smoothness k. We solve a modified version of (45), namely, m  .

μi ∗ νi,ϕ = ϕ

(50)

i=1

or, equivalently, the modified Bezout equation m  .

μ

νi (ω) =  ϕ (ω) i,ϕ (ω)

(51)

i=1

6 It is of interest to compare the envelope condition above with the Paley-Wiener-Schwartz growth condition. Note that Hörmander’s envelope condition is essentially the inversion of the PaleyWiener-Schwartz growth condition.

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where .ϕ ∈ Cck (Rd ). If there exist compactly supported solutions to (45), then certainly there exist compactly supported solutions to (50), namely, .νi,ϕ = νi ∗ ϕ. In this case, the deconvolvers are themselves in .Cck (Rd ). If (50) can be solved, then ∞ d the problem can  be solved locally as follows. Choose .ψ ∈ Cc (R ) sodthat .ψ(t) ≥ 0 for all t and . Rd ψ(t) dt = 1. For .λ ≥ 1, define .ϕ(t) = ψλ (t) = λ ψ(λt), i.e., m to be a solution of (50) .{ψλ } is an approximate identity family. Define .{νi,ψλ } i=1 with .ϕ(t) = ψλ (t). Then, f = lim (f ∗ ψλ ) =

m 

.

λ→∞

i=1

lim (f ∗ μi ∗ νi,ψλ ) =

λ→∞

m  i=1

lim (si ∗ νi,ψλ )

λ→∞

with convergence in .E' . The solution is local in light of the fact that λ 〉 = f (t0 ) = lim 〈f, τt0 ψ

.

λ→∞

m  〈τt0  si , νi,ψλ 〉. i=1

Thus, recovery of f at .t0 requires knowledge of .si on the compact set .{t0 } + (supp νi + supp ψ). Let .{χ [−ri ,ri ] } , i = 1, . . . , m , where .{ri }m i=1 is such that .(ri /rj ) is poorly approximated by rationals for .i /= j . We first note that the Fourier-Laplace transri ζ ) . As noted in the previous section, i (ζ ) = sin(2π form of .μi (t) = χ [−ri ,ri ] (t) is .μ πζ the strongly coprime condition reduces to the ratio .(ri /rj ) being poorly approximated by rationals. The optimal case when .N = 2 is given by quadratic irrationals. √ √ √ m−1 is a set of primes, .{1, p1 , p1 p2 , . . . , p1 p2 · · · pm−1 } For example, if .{pi }i=1  will work. Let .R = ri . We will assume, for technical reasons, that .f ∈ C ∞ ∩ L2 (R). A density argument in [26, 27] extends the result to all of .L2 . We choose .ϕ to be a .C ∞ function with support in .[−A, A] with .0 < A < R such that .ϕ ≥ 0 and . R ϕ(t) dt = 1 (.ϕ is an approximate identity). We refer to .ϕ as the auxiliary function of the construction. Let  〈f, ϕ〉 =

.

R

f (t) ϕ(t) dt

be the .L2 inner product of f and .ϕ, and let  〈ν; ϕ〉 =

.

R

ϕ(t) ν(t) dt

formally denote the dual product between .ϕ ∈ E and .ν ∈ E' . Let  Λk =

.

be the zero set of .μ k (ζ ).

±n 2rk

 , n∈N

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169

 Let’s first solve the problem on .Λ = m i=1 Λi . Let .σ be any permutation of .{1, 2, . . . , m} such that .σ (i) /= i. We want solutions to m  .

 μi (z) νi,ϕ (z) =  ϕ (z)

(52)

i=1

for .z ∈ Λ of the form  .νi,ϕ (z) =  ασ (i),ϕ (z)



 μj (z) .

(53)

j /=σ (i)

Substituting back and rearranging, we get m  .

 αi,ϕ (z)

i=1



 μj (z) =  ϕ (z) .

(54)

j /=i

 Now, fix k, and let .z ∈ Λk . Then, if .i /= k, . j /=i  μj (z) = 0 for all .z ∈ Λk . Therefore, the only non-zero term in (54) is the .i = k term. Thus, the solution to (54) is  ϕ (z) . μj (z) j /=k 

 αk,ϕ (z) =  .

(55)

Substituting back into (53) gives  .νk,ϕ =  ασ (k),ϕ



 μj .

(56)

j /=σ (k)

Thus, we can completely solve the Bezout equation on .Λ. The question now becomes one of interpolation. In this section, we will use the Jacobi interpolation formula and the Cauchy residue calculus to do this interpolation. Remark 1 For each  .νk,ϕ , the zero set .Λk is a sampling lattice. Because the sampling rates are non-commensurate, .Λ is then a union of these non-commensurate sampling grids. This idea is what led to the exploration of these grids for general sampling problems. Theorem 9 ([26, 27]) Let .{ri }m i=1 be a set such that .(ri /rj ) is poorly approximated by rationals for .i /= j . Then {χ [−ri ,ri ] } , i = 1, . . . , m

.

 is a set of strongly coprime convolvers. Let .R = ri , and let .ϕ be a .C ∞ function with support in .[−A, A] with .0 < A < R such that .ϕ(t) ≥ 0 for all t and

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S. D. Casey

 .

/ i, R ϕ(t) dt = 1. Let .σ be any permutation of .{1, 2, . . . , m} such that .σ (i) = and let    1 μ k (ζ )  ϕ (z)  . . αk,ϕ (ζ ) = (57) d j (z) dζ ζ −z μ k (z) j /=k μ z∈Λk

Given any .f ∈ L2 , the deconvolvers .νi,ϕ such that f ∗ϕ =

m 

.

f ∗ μi ∗ νi,ϕ

i=1

are given in the transform domain by the formulae  .νk,ϕ (ζ ) =  ασ (k),ϕ (ζ )



μ j (ζ ) .

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j /=σ (k)

Remark 2 (a) We have that for ϕ(t) = ψλ (t) = λ ψ(λt) , ϕ(t) −→ δ as supp(ϕ) −→ {0}

.

and so  ϕ (ζ ) −→ 1 as supp(ϕ) −→ {0}

.

in the sense of distributions. The function .ϕ can be chosen arbitrarily close to the .δ. The advantage of constructing .ϕ instead of the .δ is that we can express our formulae for the deconvolvers as functions, and not as distributions. (b) The formulae for the .νi are a type of Lagrange interpolation, where the analytic functions .νi (ζ ) were constructed from known values on a discrete set of data. The Cauchy residue theory is used to convert these discrete values to analytic functions. The formulae are also related to Shannon sampling theorem, with the functions   1 μ k (ζ ) ∨ (t) . d k (z) ζ − z dζ μ acting as interpolators. Sketch of Proof The fact that {χ [−ri ,ri ] } , i = 1, . . . , m

.

Shannon Sampling Theorem via Poisson, Cauchy, Jacobi, and Levin

171

is strongly coprime is given by Theorem 8. Now, we have that .μ k (ζ ) =

sin(2π rk ζ ) πζ

.

d Note that . dζ μ k (ζ ) /= 0 for .ζ ∈ Λk and that

  d  2rk  μ   (ζ )  dζ k  = |ζ | for ζ ∈ Λk .

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.

The strongly coprime condition allows us to construct a sequence of concentric circles .{𝚪n } with centers at the origin and radii .ρn such that .limn→∞ ρn = ∞ and such that there exist constants .ck > 0 with | sin(2π rk ζ )| ≥ ck e2π rk |ζ |

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.

for .ζ ∈ 𝚪n (see [14]). As . ϕ (ζ ) is entire, we may represent . ϕ by the Cauchy integral formula, i.e.,   ϕ (z) 1 dz. . ϕ (ζ ) = 2π i 𝚪n z − ζ We need the following. Lemma 8 Let .σ be any permutation of .{1, 2, . . . , m} such that .σ (i) /= i, and let  αk,ϕ (ζ ) =



.

z∈Λk

 ϕ (z)  j (z) j /=k μ





1 d k (z) dζ μ

μ k (ζ ) ζ −z

 .

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For .ζ ∈ {ζ : |ζ | < ρn } and .z ∈ Λk such that .|z| < ρn ,  ϕ (ζ ) =

m 

.

μ k (ζ ) ασ (k),ϕ (ζ )



μ j (ζ ) + Rn (ζ ),

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j /=σ (k)

k=1

where .Rn (ζ ) −→ 0 as .n −→ ∞. Proof For .|ζ | < ρn and .z ∈ Λk such that .|z| < ρn , the Jacobi interpolation formula gives 1 . ϕ (ζ ) = 2π i =

1 2π i +

 𝚪n



𝚪n

 ϕ (z) dz z−ζ    ϕ (z) m j (z) −  ϕ (z) m j (ζ ) j =1 μ j =1 μ   dz m (z − ζ ) μ  (z) j j =1

 m  ϕ (z) 1    dz μ j (ζ ) m 2π i 𝚪 n (z − ζ ) μ  (z) j j =1 j =1

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S. D. Casey

=

 ϕ (z)



1 2π i

m

j (z) − j =1 μ

(z − ζ )

𝚪n





m

j (ζ ) j =1 μ

m j (z) j =1 μ



dz

 m  ϕ (z) 1    dz . μ j (ζ ) m 2π i 𝚪n (z − ζ ) μ  (z) j j =1 j =1

+

(63)

  Now, for .z ∈ Λk for any k, . m j (z) = 0, but j =1 μ ⎞ ⎛ m m    d ⎝ d d  μi (z)  μk (z) . μ j (z)⎠ = μ j (z) = μ j (z) /= 0 . dζ dζ dζ j =1

j /=i

i=1

j /=k

Thus, the function .

has simple poles at .z ∈ Λ =



 ϕ (z) .

has simple poles in .Λ =



1 m j (z) j =1 μ



Λk . Therefore,

m

 j (z) − m j (ζ ) j =1 μ j =1 μ   m (z − ζ ) j (z) j =1 μ



Λk , and so by the Cauchy residue theorem,

 m  μ (z) −  μ (ζ ) k=1 k k=1 k m  . dz (z − ζ ) μk (z) 𝚪n k=1    m    1  ϕ (z) μ k (ζ )  . μ j (ζ ) = d j (z) dζ ζ −z μ k (z) j /=k μ 

 ϕ (z)

m

k=1 j /=k

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z∈Λk

This is also valid if indices of the transforms of the convolvers are permuted, i.e., m   .

μ j (ζ )

k=1 j /=k

=

m  k=1

 z∈Λk

μ k (ζ )



 ϕ (z)  j (z) j /=k μ

μ j (ζ )

j /=σ (k)



z∈Λσ (k)





1 d k (z) dζ μ

 ϕ (z)  j (z) j /=σ (k) μ



μ k (ζ ) ζ −z



1 d  σ (k) (z) dζ μ



μ  σ (k) (ζ ) ζ −z

 . (65)

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173

Let Rn (ζ ) =

.

 m  ϕ (z) 1    dz . μ j (ζ ) m 2π i 𝚪n (z − ζ ) μ  (z) j j =1 j =1

Next, we need to show that .Rn (ζ ) −→ 0 as .n −→ ∞. We were able to choose .𝚪n such that for .z ∈ 𝚪n , | sin(2π rk z)| ≥ ck e2π rk |z| .

.

By the Paley-Wiener-Schwartz theorem, there exists a positive constant .C = C(N) such that for any positive integer N and any .z ∈ C, | ϕ (z)| ≤ C(1 + |z|)−N e2π A|z| .

.

Therefore, for .ζ fixed and any fixed .N > 0,     m       1     ϕ (z)      dz   .|Rn (ζ )| ≤  μ  (ζ ) j m    2π i    𝚪n  (z − ζ ) j =1   μ  (z) j j =1     m         ϕ (z) 1       2πρn ≤ μ j (ζ ) sup  m j =1  z∈𝚪n 2π  (z − ζ ) j (z)  j =1 μ   ⎤ ⎡   −N e2π A|z| ρ m  C(1 + |z|) n  |z|2 ⎦  ≤  μ j (ζ ) sup ⎣ m 2π rj |z| j =1  z∈𝚪n (||z| − |ζ ||) c e j =1 j  ( ) m  −N ρ 3   C(1 + ρ ) n n 2π(A−R)ρ n e =  μ j (ζ ) m . j =1  ( j =1 cj ) (ρn − |ζ |)

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Since .A < R, this last quantity .−→ 0 as .n −→ ∞. This completes the proof of the lemma. ⨆ ⨅ We again see the techniques of Jacobi interpolation and the Cauchy residue calculus and have shown how a key step in solving the problem—computing the deconvolving functions—essentially boiled down to a sampling problem. The remaining details in the proof can be found in two papers of Casey and Walnut [26, 27].

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7 Multi-rate Sampling via Cauchy and Jacobi The heart of the multi-channel theory involves solving an interpolation problem, ' ) from reconstructing functions (the deconvolvers) in a space of restricted growth (.E discrete data (their values on the zero sets of the convolvers). This gives solutions to the Bezout equation [26, 27], utilizing the zero sets of the .μ i as different sampling rates. This connection between sampling and multi-channel deconvolution naturally leads to sampling schemes on properly chosen non-commensurate lattices. We m consider the following: Let .{ri } i=1 be positive real numbers such that .ri /rj is irrational for .i /= j , and let .R = m i=1 ri . Let Λ=

.

 m   ±k  i=1

2ri

{0}

for .k ∈ N be a sampling grid, made up of a union of sampling grids with noncommensurate generators .{ri }. Given a R bandlimited function f , we reconstruct f —conditionally—from samples of f taken on .Λ. We give a specific example of a non-commensurate sampling lattice and use a generalization of Levin’s sine-type functions to develop interpolating formulae on this lattice. Our interpolators are products of sine-type functions. We show that our reconstruction formulae converge conditionally. Recall that, given a Hilbert space .H, a sequence .{xn } ∈ H is a Bessel sequence if and only if  given a finite set  of arbitrary scalars .{cn }, there exists a constant .A > 0 such that .‖ n cn xn ‖2 ≤ A n |cn |2 . Also, a Riesz basis .B = {xn }∞ n=1 for .H is a bounded basis. The set .B is a Riesz basis if and only if both .{xn } and its biorthogonal sequence .{yn } are Bessel sequences. We have been careful to distinguish between sampling sets .Λ that are separated and those that are not. Again, as shown in Lemmas (1) and (3), when the sampling set .Λ is not separated, it does not generate a Riesz basis. We then have to “piece together” the reconstruction basis. This block summation technique generates the function from a generalized basis or a basis with braces. These ideas were developed by Levin [52, 54]. Definition 8 Let .H be a Hilbert space and .H = {Hi } be a collection of subspaces of .H. Then .H is a generalized basis if there exist projection  operators .Pn such that .Pn restricted to .Hk equals .δn,k , and given .x ∈ H, .x = Pn x unconditionally. The projection operator can be defined as follows. Suppose that .{xn } is a Riesz basis with biorthogonal system .{yn }. Define .Hn to be the one-dimensional subspace of .H spanned by .xn . Then the span of .{Hn } is dense in .H, and the projection operators .Pn are given by .Pn x = 〈x, yn 〉xn . This converges unconditionally. In our reconstruction, we have both isolated sampling elements and clusters of elements arbitrarily close. Each cluster has basis elements .xk,j with corresponding

Shannon Sampling Theorem via Poisson, Cauchy, Jacobi, and Levin

175

biorthogonal elements .yk,j . The projection operator for these cluster is .Pk x =  〈x, y 〉x . We have to be careful to keep the projection operator for the blocks k,j k,j j separate from the isolated points (and from separate blocks). We use techniques from both real and complex analysis, several of which have been built up in the context of the Hardy spaces.

7.1 Hardy Spaces The Hardy, or .H p , spaces are spaces of analytic functions in a domain .Δ with p .L boundary values. The most common domains are the unit disc and .D = {z : |z| < 1} and the upper half plane .U = {z : (z) > 0}. For .1 ≤ p < ∞, .H p is a Banach space. The space .H ∞ is a Banach algebra. The .H p spaces blend very nicely with a sub-theme of this chapter, namely, the interplay between the “realvariable feel” of harmonic analysis and the power of complex analysis. Excellent references for this material are Duren [32], Hoffman [43], and Young [75]. The majority of the books focus on .H p spaces on .D. The Hardy spaces for the upper half plane are discussed in Duren [32], Chapter 11, and Hoffman [43], Chapter 8, while Hardy spaces for general domains are discussed in Duren  [32], nChapter 10. on .D with The space .H 2 (D) consists of all analytic functions .f (z) = ∞ n=0 cn z  n square summable Taylor coefficients. The mapping .(c0 , c1 , c2 , . . .) −→ ∞ n=0 cn z 2 2 is a Hilbert space isomorphism from .𝓁 and .H . The space .H 2 (U) is defined as follows. Definition 9 The Hardy space .H 2 (U) is the set of all functions analytic in .U such that  ‖f ‖H 2 = sup

∞

.

−∞

y>0

1/2 |f (x + iy)| dx 2

0, and .Λ = {λn }n∈Z be a separated sequence. Then there exists a constant .C = C(h, k) such that  .

1/2 |f (k)(λn + ih)|

2

≤ C‖f ‖H 2 .

n

7.2 Multi-rate Sampling: An Example The following example captures √ the spirit and technique of multi-rate sampling. Let t ∈ R, and let .ϕ = (1 + 5)/2. This choice of .ϕ will allow us to demonstrate results √using results which relate the Fibonacci numbers with the golden mean .ϕ = (1 + 5)/2.

.

Remark It is also important to note that, despite the strongly coprime condition needed for multi-channel deconvolution, the constructions in this section work for any irrational .α > 1. Let .μ1 (t) = χ [−1,1] (t) , μ2 (t) = χ [−ϕ,ϕ] (t) model the impulse response of the channels of a two-channel system. Then μ 1 (ζ ) =

.

sin(2π ζ ) sin(2π ϕζ ) , μ 2 (ζ ) = . πζ πζ

(Note, since .ϕ is poorly approximated by rationals, .{μi } is strongly coprime.) Now, for .m, n ∈ N, let     ±m ±n , Λ2 = and Λ = Λ1 ∪ Λ2 ∪ {0} = {λm,n } ∪ {0} . .Λ1 = 2 2ϕ The information contained in the original signal is reconstructed by creating deconvolvers defined initially on .Λ. Note, for a .(1 + ϕ)-bandlimited function, the Nyquist rate is .1/(2(1 + ϕ)). However, our individual sampling rates are .1/2 and .1/(2ϕ). Both these rates are below Nyquist. Let .G1 (z) = sin(2π z), let .G2 (z) = sin(2π ϕz), and let our sampling generating function be G(z) = G1 (z) · G2 (z) = sin(2π z) · sin(2π ϕz) .

.

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177

Then .G(z) is an entire function, which is almost periodic on .R, and has simple zeros on .Λ \ {0} and a double zero at .{0}. The sampling set .Λ has infinitely many pairs of “clustering points” of the form j k .{ , 2 2ϕ }—.Λ is not separated. This interpolation problem requires tools beyond the “standard toolbox.” These new tools can be found in Levin [54] and are also discussed in [65]. By the now familiar Cauchy-Jacobi machinery, f (z) =

.

1 2π i

 𝚪m

f (ζ ) G(ζ ) − G(z) 1 · dζ + (ζ − z) G(ζ ) 2π i

 𝚪m

f (ζ ) G(z) · dζ , (ζ − z) G(ζ )

where .𝚪m is a sequence of circles with increasing radii chosen to avoid zeros of G. Let .ρm be the radius of .𝚪m . We choose the circles so that .ρm −→ ∞ as .m −→ ∞. We also let the second integral be denoted by .Rm (the remainder), which will .−→ 0 as .m −→ ∞.   1 f (z)[G1 (ζ )G2 (ζ ) − G1 (z)G2 (z)] f (ζ ) 1 dζ = dζ +Rm . .f (z) = 2π i 𝚪m (ζ − z) (G1 (ζ )G2 (ζ )) 2π i 𝚪n (ζ − z) d (G1 (z)G2 (z)) /= 0. Thus, Now, for .z ∈ Λ1 or .z ∈ Λ2 , .(G1 (z)G2 (z)) = 0, but . dζ

.

f (z)[G1 (ζ )G2 (ζ ) − G1 (z)G2 (z)] (ζ − z)(G1 (ζ )G2 (ζ ))

has simple poles in .Λ1 ∪ Λ2 , and so by the Cauchy residue theorem, 

.

f (ζ )[G1 (ζ )G2 (ζ ) − G1 (z)G2 (z)] dζ (ζ − z) (G1 (ζ )G2 (ζ )) 𝚪m    G1 (ζ )G2 (ζ ) f (z) = d (ζ − z) G2 (z) G1 (z) dζ z∈Λ 2 |z| 0 such that .G(t + ih) = sin(2π(t + ih)) · sin(2π ϕ(t + ih)) never vanishes on .R. Also, .G(t + ih) is bounded. Computing for .λk , .

  G(t + ih) cλk ,n 1 = , cλ,n G(t + ih) (z − (λk − ih)) (z − (λk − ih)) λk ,n

λk ,n

and so             G(t + ih)  cλk ,n  . ≤ B . c λk ,n 1    (t − (λk − ih))  λk ,n λk ,n (t − (λk − ih))  Similarly, for .λk ' ,             cλk' ,n G(t + ih)    . ≤ B2  . cλk' ,n    (t − (λk ' − ih))  λk' ,n (t − (λk ' − ih))  λk' ,n Again, choose .ψ ∈ H 2 (U) with norm 1. Then              ψ(t) c λk ,n = . , ψ c . dt λk ,n     R (t − (λk − ih))  λk ,n   λk ,n (z − (λk − ih)) = ψ ' (ξ1 − (λk + ih). By the mean value theorem, there exists .ξ1 such that . ψ(t) (t−(λk −ih)) Substituting this in gives

184

.

S. D. Casey

            c λk ,n ' (ξ − (λ + ih) =  , ψ c ψ 1 k λk ,n      λk ,n   λk ,n (z − (λk − ih)) ⎛ ≤⎝



⎞1/2 ⎛ |cλk ,n |2 ⎠

λk ,n

⎝



⎞1/2 |ψ ' (ξ1 − (λk + ih)|2 ⎠

λk ,n

⎛ ≤ A1 ⎝



⎞1/2 |cλk ,n |2 ⎠

.

λk ,n

Similarly, for .λk '  ⎛ ⎞1/2       cλk' ,n , ψ  ≤ A2 ⎝ |cλk' ,n |2 ⎠ . .    λk' ,n (z − (λk ' − ih)) λk ' ,n The result follows by the triangle inequality. ⨆ ⨅ Since, for elements .λ ∈ Λλk ,λk' , .{e2π iλm,n t } and its dual basis are both Bessel sequences, they form a generalized basis for the subspace generated by .Λλk ,λk' . ⨅ ⨆ Theorem 12 Let f be a .(1+ϕ)-bandlimited function. Then f can be reconstructed conditionally (in the sense of a generalized basis) by .{f (λm,n )} ∪ {f (0), f ' (0)} . In other words, .Λ is a set of conditional reconstruction for .PW(1+ϕ) , and so 2π iλm,n t } ∪ {t, t 2 } is minimal. .{e Let .G(z) = sin(2π z) · sin(2π ϕz). The reconstruction formula is  G(z) G(z) G(z) + f (λ) ' + f ' (0) 2 4π ϕz G (λ)(z − λ) 4π ϕz λ∈Λσ  

G(z) G(z) f (λk ) ' . (74) + f (λk ' ) ' G (λk )(z − λk ) G (λk ' )(z − λk ' )

f (z) = f (0)

.

+

λk ,λk ' ∈Λη

Proof Let  G(z) G(z) G(z) + f ' (0) + f (λ) ' 2 G (λ)(z − λ) 4π ϕz 4π ϕz λ∈Λσ  

G(z) G(z) f (λk ) ' + f (λk ' ) ' . G (λk )(z − λk ) G (λk ' )(z − λk ' )

h(z) = f (0)

.

+

λk ,λk ' ∈Λη

Then .h ∈ PW(1+ϕ) , converging in the sense of a generalized basis. Moreover, h(z) = f (z) for all z ∈ Λ = Λ1 ∪ Λ2 ∪ {0} = {λm,n } ∪ {0} .

.

Therefore, by Theorem 11, .h(z) = f (z).

⨆ ⨅

Shannon Sampling Theorem via Poisson, Cauchy, Jacobi, and Levin

185

Remark The points .λk , λk ' ∈ Λη can be treated as double points. Note that Pλk ,λk' f =

.

 G(z) f (λk ) + (z − λk )f ' (λk ) . ((z − λk )(z − λk ' ))

For .{λk , λk ' } ⊂ Λη , .

f (λk )Hλk (z) + f (λk ' )Hλk' (z)

⎤

⎡

= f (λk )Hλk (z) + f ' (ξλk ) ⎣ 

G' (λk )−G' (λk ' ) 2(λk −λk ' )

G(z) ⎦ + R2 ,  (z − λk )(z − λk ' )

where .ξλk −→ λk and .R(λk − λk ' )2 −→ 0 as .η −→ 0.  .

G(z) G(z) , G' (λ)(z − λ) (G'' (λk )/2!)(z − λk )2

 λk ∈Λη

is a Bessel sequence. For .λ ∈ Λσ , the projection operator for the generalized basis is given by .Pλ f = f (λ)Hλ (z). This converges unconditionally. For .λ ∈ Λη are pairs .{λk , λk ' } with projection operator .Pλk ,λk' f = f (λk )Hλk (z) + f (λk ' )Hλk' (z). The result generalizes. We can create sampling sets on .𝓁 lattices  using a set of 𝓁 numbers .{ri } such that .(ri /rj ) is irrational for .i /= j . Let .R = k rk , and let , - i=1 ±n .Λk = 2rk for .n ∈ N. Let Λ=

𝓁 

.

Λk = Λ = {λk } .

(75)

k=1

We reconstruct on .Λ ∪ {0}, letting G(z) =

𝓁 

.

sin(2π rk z)

(76)

G(z) . m )(z − λm )

(77)

k=1

and letting Hm (z) =

.

G' (λ

We let .z = 0 be a sample point of multiplicity .𝓁. By a generalization of Theorem 11, Λ will be a set of uniqueness for .PWR . Clusters of sample point will occur in different combinations, from clusters of k points requiring the data .f j (λ) , 0 ≤ j ≤

.

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S. D. Casey

k − 1, down to pairs of points. For example, if .k = 3, we will have clusters of three points containing . 2rn1 , 2rm2 , 2rp3 , for some .n, m, p ∈ Z, and three different set of clusters of two points generated by different pairs of rates .ri , rj , .i /= j . For   general .𝓁, we will have . 2𝓁 different types of clustering pairs of sample points, . 3𝓁 different types of triples, etc. Convergence again is in the sense of a generalized basis. Once again, develop .Λ = Λσ ∪ Λη , and construct .f ∈ PWR with .z = 0 being a sample point of multiplicity .𝓁, single element generalized basis elements on .Λσ , and appropriate clusterings on .Λη .

8 Epilogue Young [75] defines the concept of sine-type function on page 143. Definition 10 An entire function .G(z) of exponential type .π is said to be of sine type if: (a) The zeros of G are separated (b) There exist positive constants A, B, and H such that Aeπ |y| ≤ |f (x + iy)| ≤ Beπ|y|

.

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whenever .x, y ∈ R and .|y| > H Levin [52] has a similar definition given at the beginning of Lecture 22, pp. 163– 168. Both require that the zeros of G be separated. For Shannon sampling theorem and Papoulis generalized sampling,  our sampling generating functions, namely, 2 πz .G(z) = sin(π z) and .G(z) = sin 2 , have separated zero sets. Our work on multichannel deconvolution pointed to the possibility of signal recovery from samples on sets which were not separated. For .ϕ equal to the golden mean, G(z) = sin(2π z) · sin(2π ϕz)

.

is a “sine-type” function for these types of sampling sets. Levin also relaxed the separated condition in his work. These are presented in his seminal paper [54] “On Bases of Exponential Functions in .L2 ,” Zap. Mekh.-Mat. Fak. i Khar’kov. Mat, Obshch., 27, pp. 39–48 (1961). A translation of Levin’s paper will appear separately. Acknowledgments The author thanks Professors J. Rowland Higgins, Carlos Berenstein, John Benedetto, Radu Balan, and David Walnut for several conversations related to the contents of the chapter. Professor Higgins’ passing was a sad event for our community. The author wishes to acknowledge his deep respect for J. R. Higgins. He was a truly excellent mathematician, an extremely knowledgeable math historian, and an even better person. He was a true blessing to our community. The author’s research was partially supported by US Air Force Office of Scientific Research Grant Number FA9550-20-1-0030.

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56. F. Marvasti, Nonuniform sampling, in Advanced Topics in Shannon sampling theorem and Interpolation Theory, ed. by R. Marks (Springer, New York, 1993) 57. Y. Meyer, Wavelets: Algorithms and Applications. Ryan, R.D, translator (SIAM, Philadelphia, PA, 1993) 58. H. Nyquist, Certain topics in telegraph transmission theory. AIEE Trans. 47, 617–644 (1928) 59. R. Paley, N. Wiener, Fourier Transform in the Complex Domain (American Math Society Colloquim Publications, New York, 2000), 19 60. A. Papoulis, The Fourier Integral and its Applications (McGraw-Hill, New York, 1962) 61. A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977) 62. A. Papoulis, Generalized sampling expansion. IEEE Trans. Circuits and Systems 24(11), 652– 654 (1977) 63. E. Petersen, G. Meisters, Non-Liouville numbers and a theorem of Hörmander. J. Funct. Anal. 29, 142–150 (1978) 64. P.M. Prenter, Splines and Variational Methods (Wiley, New York, 1975) 65. B. Rom, D. Walnut, Sampling on unions of shifted lattices in one dimension, in Harmonic Analysis and Applications, ed. by C. Heil (2006), pp. 289–323 66. K. Seip, Interpolation and Sampling in Spaces of Analytic Functions. Iniversity Lecture Series, vol. 33 (American Mathematical Society, Providence, RI, 2004) 67. C.E. Shannon, A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423 (1948) 68. C.E. Shannon, Communications in the presence of noise. Proc. IRE. 37, 10–21 (1949) 69. M. Unser, Sampling–50 years after Shannon. Proc. IEEE 88(4), 569–587 (2000) 70. D.F. Walnut, Nonperiodic sampling of bandlimited functions on unions of rectangular lattices. J. Fourier Anal. Appl. 2(5), 435–452 (1996) 71. E.T. Whittaker, On the functions which are represented by the expansions of the interpolation theory. Proc. Roy. Soc. Edinb. 35, 181–194 (1915) 72. J.M. Whittaker, Interpolatory Function Theory (Cambridge University Press, Cambridge, 1935) 73. N. Wiener, The Fourier Integral and Certain of its Applications (MIT Press, Cambridge, MA, 1933) 74. N. Wiener, Extrapolation, Interpolation, and Smoothing of Stationary Time Series (MIT Technology Press, Cambridge, MA, 1949) 75. R. Young, An Introduction to Nonharmonic Fourier Series (Academic Press, New York, 1980) 76. A.I. Zayed, Advances in Shannon’s Sampling Theory (CRC Press, Boca Raton, FL, 1993)

Part II

Theoretical Extensions

Schoenberg’s Theory of Totally Positive Functions and the Riemann Zeta Function Karlheinz Gröchenig

1 Introduction John Rowland Higgins had a lifelong fascination with THE sampling theorem—the Shannon-Whittaker-Kotelnikov sampling theorem—and the cardinal series [25]. He was intrigued by the manifold connections of the sampling series to other fields [10, 11]. With a celebrated team of specialists on sampling and the cardinal series, he showed that many fundamental theorems of Fourier analysis, numerical analysis, and complex analysis are equivalent. A particular gem is the equivalence of the sampling theorem to the functional equation for the Riemann zeta function [26]. What better way to commemorate J. R. Higgins than by following his spirit with a contribution to sampling theory and its interaction with number theory. For many years the author’s interest has been the concept of totally positive function and its use in sampling theory. After digging into Schoenberg’s work to find properties of totally positive functions that might help solve some open problems about sampling and interpolation in shift-invariant spaces, the author came on a surprising, in retrospect perhaps not so surprising, connection between total positivity and number theory. One of the numerous characterizations of total positivity contains a hidden connection to the Riemann zeta function. In fact, Riemann’s hypothesis is equivalent to a statement that a function related to the zeta function is totally positive. While the Riemann hypothesis is equivalent to several positivity conditions, the connection to total positivity seems to be new.

K. G. was supported in part by the project P31887-N32 of the Austrian Science Fund (FWF). K. Gröchenig () Faculty of Mathematics, University of Vienna, Vienna, Austria e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. D. Casey et al. (eds.), Sampling, Approximation, and Signal Analysis, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-031-41130-4_9

193

194

K. Gröchenig

This chapter mirrors the interests of J. R. Higgins in that it passes from sampling theory to number theory. The focus is on the notion of totally positive functions. After describing Schoenberg’s characterizations of totally positive functions, we will discuss why totally positive functions are of interest in sampling theory. In the next step we will explain the connection between total positivity and number theory, more precisely the Riemann zeta function, which is again in line with Higgins’s own interests. The goals are mathematical connections and mathematical entertainment. Therefore, we do not strive for completeness nor provide proofs of existing results.

2 Totally Positive Functions In a series of papers in the 1950s, Schoenberg investigated the properties of totally positive functions [17, 44–46, 49]. He found several characterizations and used total positivity to prove fundamental properties of splines [48, 49]. A measurable function .Λ on .R is totally positive, if for every .n ∈ N and every two sets of increasing numbers .x1 < x2 < · · · < xn and .y1 < y2 < · · · < yn , the matrix .(Λ(xj − yk ))j,k=1,...,n has nonnegative determinant: .

det(Λ(xj − yk ))j,k=1,...,n ≥ 0 .

(1)

If in addition .Λ is integrable, then .Λ is called a Polya frequency function. If .Λ is totally positive and not equal to .eax+b , there exists an exponential .ecx , such that .Λ1 (x) = ecx Λ(x) is a Polya frequency function, i.e., .Λ1 is totally positive and integrable [46, Lemma 4]. It is usually no loss of generality to restrict to Polya frequency functions. The class of totally positive functions played and plays an important role in approximation theory, in particular in spline theory [49], and in statistics [19, 30]. In a different and rather surprising direction, totally positive functions appear in the representation theory of infinite dimensional motion groups [38]. Recently, totally positive functions appeared in sampling theory and in time-frequency analysis [5, 21–23], where they were instrumental in the derivation of optimal results. Elementary Examples The one-sided exponential function .Λ(x) = δ −1 e−x/δ 2 χ[0,∞) (x) for .δ > 0 and the Gaussian .Λ(x) = γ −1/2 e−π x /γ for .γ > 0 are the most basic examples of totally positive functions. It is easy to check directly that these functions are totally positive.

2.1 The Laguerre-Polya Class An entire function .Ψ of order at most 2 belongs to the Laguerre-Polya class, if its Hadamard factorization is of the form

Totally Positive Functions and the Zeta Function

Ψ (s) = Cs m e−γ s

.

2 +δs

∞ 

195

(1 + δj s)e−δj s

s ∈ C,

(2)

j =1

where .−δj−1 ∈ R are the zeros of .Ψ , m is the order of the zero at 0, .γ ≥ 0, .δ ∈ R, and 0 0. (ii) Conversely, if .Ψ is in the Laguerre-Polya class with .Ψ (0) > 0, then its reciprocal .1/Ψ is the Laplace transform of a Polya frequency function .Λ. This is a fascinating theorem, because it relates two function classes that seem to bear absolutely no resemblance to each other. Schoenberg’s theorem establishes a bijection between the class of Polya frequency functions, the Laguerre-Polya class, and yields a parametrization .(C, γ , δ1 , . . . , δn , . . . ) in the set .(0, ∞) × [0, ∞) × 𝓁2 (N , R).

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K. Gröchenig

By using the Fourier transform instead of the Laplace transform, Schoenberg’s theorem can be recast as follows: A function .Λ is totally positive and integrable, if and only if its Fourier transform possesses the factorization ˆ ) = Ce−γ τ 2 +2π iδτ Λ(τ

∞ 

.

(1 + 2π iδj τ )−1 e2π iδj τ

(5)

j =1

where .C > 0, .γ ≥ 0, .δ, δj ∈ R, and

∞

< ∞ (and the product in (5) ˆ )= may also be finite). From (5) we recover the elementary examples, because .Λ(τ −1 −1 −x/δ (1 + 2π iδτ ) yields the one-sided exponential .Λ(x) = δ e χ[0,∞) (x), and the Gaussian factor alone in (5) yields the Gaussian as a totally positive function. If we drop the condition of integrability and exclude exponential functions, then the representation (5) still holds for every totally positive function, but the Laplace transform of .Λ converges in some vertical strip .{z ∈ C : α < Re z < β} that does not contain 0. A similar result holds for one-sided totally positive functions [46, Thm. 2]. .

2 j =1 δj

Theorem 2 (i) If .Λ is a Polya frequency function with support in .[0, ∞), then its Laplace transform converges in a half-plane .{z ∈ C : −α < Re z}, α > 0, and 

∞

.

Λ(x)e−sx dx =

0

1 Ψ (s)

(6)

is the reciprocal of an entire function .Ψ with Hadamard factorization Ψ (s) = C

.

∞ 

(1 + δj s) ,

(7)

j =1

 with .δj ≥ 0, 0 < δj < ∞. (ii) Conversely, if .Ψ possesses the factorization (7), then its reciprocal .1/Ψ is the Laplace transform of a Polya frequency function .Λ with support in .[0, ∞). The proof of the implication (ii) . ⇒ (i) of Theorem 1 is based on the (nontrivial) fact that the convolution .Λ = Λ1 ∗ Λ2 of two Polya frequency functions .Λ1 , Λ2 is again a Polya frequency function. The converse in Theorem 1 lies much deeper, and Schoenberg used heavily several results of Polya about functions in the LaguerrePolya class [39, 42]. The essential step of this argument is explained at the end of this chapter. Schoenberg’s motivation was the characterization and deeper understanding of totally positive functions, and in this regard, the implication (i) . ⇒ (ii) and the factorization (5) can be considered his main insight about totally positive functions. However, instead of reading Schoenberg’s theorem as a characterization of totally positive functions, one may read it as a characterization of the Laguerre-Polya class.

Totally Positive Functions and the Zeta Function

197

Corollary 3 A function .Ψ with .Ψ (0) > 0 and .Ψ = / eas+b is in the Laguerre-Polya class, if and only if the Fourier transform of .1/Ψ is a Polya frequency function.

3 Totally Positive Functions in Sampling Theory 3.1 Sampling in Shift-Invariant Spaces With the rise of wavelet theory, signal processing has begun to study alternative signal spaces in addition to bandlimited functions. The underlying idea is to imitate the cardinal series and use it for the definition of a signal space. Precisely, fix a generating function .g ∈ L2 (R) with some smoothness and decay and define a space V (g) = {f ∈ L2 (R) : f (x) =



.

ck g(x − k), (ck )k∈Z ∈ 𝓁2 (Z)} .

(8)

k∈Z

 Under the mild condition .0 < A ≤ k∈Z |g(ξ ˆ + k)|2 ≤ B < ∞, the .L2 -norm of 2 .f ∈ V (g) is equivalent to the .𝓁 -norm of the coefficient sequence. In the theorems below we will assume that this condition of “stable translates” is fulfilled. The spaces .V (g) form the starting point for the construction of a multiresolution analysis and wavelet bases [18, 35]; therefore, they are often called wavelet subspaces. In approximation theory and sampling theory, one also uses the terminology shift-invariant space, or spline-type space, or space with finite rate of innovation. If .g(x) = sinπ πx x , then one obtains the cardinal series in (8), and thus, V

.

 sin π x  πx

= {f ∈ L2 (R) : suppfˆ ⊆ [−1/2, 1/2]}

is the space of bandlimited functions. Bandlimited functions and their sampling theory are thus a special case of the sampling theory in shift-invariant spaces. If the generator g is smooth, or equivalently, if its Fourier transform .gˆ decays rapidly, then .V (g) consists of functions that are “almost bandlimited.” In this sense the sampling theory in shift-invariant spaces can be understood as an adaption of the classical sampling theory to neatly defined spaces of almost bandlimited functions. For a survey of the general theory of sampling in shift-invariant spaces we refer to [3]. The variations and generalizations are almost as extensive as for the theory of bandlimited functions. The analysis of uniform sampling on .Z, i.e., at the critical rate, is easy. Let  2π ikξ be the Fourier series of the samples of the generator .γ (ξ ) = g(k)e k∈Z g on .Z. If g is a smooth function with decay .O(|x|−1−ϵ ), then the Fourier series converges absolutely, and .γ is a continuous periodic function on .[0, 1].

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K. Gröchenig

Theorem 4 Assume that there exist constants .A, B > 0, such that .A ≤ |γ (ξ )| ≤ B for almost all .ξ ∈ [0, 1]. Then the sampling map .f I→ f (k) k∈Z is an isomorphism from .V (g) onto .𝓁2 (Z). Thus, every .f ∈ V (g) is uniquely determined by its samples on .Z and can be reconstructed with a bounded operator.  Proof Let .Sf = f (k) k∈Z be the sampling operator. Since we assume .‖f ‖2 2 ‖c‖2 , it suffices to show  that the map .c I→ Sf is bounded and invertible on .𝓁 (Z). We note that .f (k) = l∈Z cl g(k − l), and therefore, the sampling operator can be written as a convolution: Sf = c ∗ Sg .

.

= cˆ γ . By assumption on .γ we obtain Taking Fourier series, we obtain .Sf

‖2 ≤ B‖c‖ A‖c‖ ˆ 2 ≤ ‖cˆ γ ‖2 = ‖Sf ˆ 2.

.

With Plancherel’s theorem, this yields A‖c‖2 ≤ ‖Sf ‖2 =

 

.

1/2 |f (k)|

2

≤ B‖c‖2 .

k∈Z

⨆ ⨅

Thus, the sampling map is an isomorphim.

/γ more explicitly, one obtains a reconstrucBy writing the reconstruction .cˆ = Sf tion formula similar to the cardinal series. See, e.g., [29] for more details. We next formulate a very general sampling theorem with hardly any assumptions on the generator. Its obvious advantage is the generality and the flexibility of the sampling set. It is not necessary to sample on a lattice (as in the classical sampling theorem favored by J. R. Higgins), but arbitrary nonuniform sampling sets may be used. The proxy for the sampling set is the maximum gap between samples defined by δ(Λ) = sup

.

λ∈Λ

inf

μ∈Λ:μ/=λ

|λ − μ| .

(9)

For uniform sampling on .Λ = αZ, the maximum gap is .δ(Λ) = α. The following result is due to A. Aldroubi and H. Feichtinger [2].  Theorem 5 Assume that the generator .g satisfies the condition . k∈Z supx∈[0,1] |g(x + k)| < ∞. Then there exists maximum gap .δ0 = δ0 (g) ≤ 1 depending on the generator g with the following property: If .δ(Λ) < δ0 and .Λ is separated, i.e., .infλ,μ∈Λ,λ/=μ |λ − μ| > 0, then .Λ is a set of stable sampling for .V (g) and there exist positive constants .A, B > 0, such that A‖f ‖22 ≤



.

λ∈Λ

|f (λ)|2 ≤ B‖f ‖22

for all f ∈ V (g) .

(10)

Totally Positive Functions and the Zeta Function

199

Not only is .f ∈ V (g) uniquely determined by its samples on .Λ, but standard algorithms from frame theory yield a stable approximation or reconstruction of f from the samples. Similar to bandlimited functions, a set of stable sampling must satisfy some necessary conditions that express that there are enough samples for a reconstruction. The following result in the style of Landau [33] holds for all shift-invariant spaces with a mild condition on the generator.  Theorem 6 Assume that the generator .g satisfies the condition . k∈Z supx∈[0,1] |g(x + k)| < ∞. If .Λ ⊆ R is a set of stable sampling for .V (g), i.e., .Λ obeys a sampling inequality of the form (10), then the (lower) Beurling density of .Λ satisfies D − (Λ) := lim inf inf

.

r→∞ x∈R

#Λ ∩ [x, x + r] ≥ 1. r

(11)

Note that the cardinal sine function . sinπ πx x does not satisfy the assumptions of Theorems 5 and 6. Thus, for bandlimited functions one would have to modify some of the arguments slightly.

3.2 The Role of Totally Positive Functions In comparison to bandlimited functions, it is much harder to prove optimal sampling theorems for shift-invariant spaces. The above qualitative sampling theorem does not give much information about the sufficient sampling density and may lead to huge oversampling factors. Clearly, in view of the precise information of the classical sampling theorem, heavy oversampling may be of no use. This is where totally positive functions start to play an important role. In the last few years, it has become clear that totally positive functions form an important class of generators for which precise and optimal sampling theorems can be derived. We give three examples of such results and refer to the literature for the proofs and more details. The first result from [21] treats totally positive functions of “finite type,” namely, finitely many convolutions of one-sided exponentials. Theorem

7 Assume that g is a totally positive function whose Fourier transform is g(ξ ˆ ) = nj=1 (1 + 2π iδj ξ )−1 for .δj ∈ R. If .δ(Λ) < 1, then .Λ is a set of sampling for .V (g). In particular, for .α < 1, the lattice .αZ is a set of stable sampling for .V (g).

.

The next result from [22] covers totally positive generators whose factorization (5) contains a Gaussian factor. The proxy in these results is the average number of samples per unit interval measured with the Beurling density from (11). Theorem 8 Assume that g is totally positive with a Fourier transform of the form 2 n −1 for .γ > 0 and .δ ∈ R. g(ξ ˆ ) = e−γ ξ j j =1 (1 + 2π iδj ξ )

.

200

K. Gröchenig

If .Λ ⊆ R is separated and its lower Beurling density satisfies .D − (Λ) > 1, then .Λ is a set of sampling for .V (φ). The analogous theorem for bandlimited functions is already due to Beurling [6]. In view of the necessary density condition of (11), Theorem 8 is almost optimal. As with bandlimited functions, the case of critical density is subtle. A complete understanding of the sampling properties at the critical density is 2 currently missing, except for the Gaussian generator .e−ct . For Gaussians one can completely characterize those sets that are sampling and interpolating, i.e., .Λ satisfies a sampling inequality (10), and for every sequence .a = (aλ )λ∈Λ ∈ 𝓁2 (Λ), there exists an .f ∈ V (g), such that .f (λ) = aλ . It follows from Theorem 11 (and its counterpart for interpolation) that a complete interpolating = sequence must have Beurling density .D(Λ) = limr→∞ infx∈R #Λ∩[x,x+r] r = 1. The following result [5] treats this case for a limr→∞ supx∈R #Λ∩[x,x+r] r Gaussian generator. Theorem 9 Let .g(t) = e−ct for .c ∈ C with .a = Re c > 0. An increasing sequence 2 .Λ ⊂ R is a complete interpolating sequence for .V (g), if and only if .Λ is separated and there exists an enumeration .Λ = {λn }n∈Z , .λn = n + δn , .n ∈ Z, such that 2

(a) .supn∈N |δn | < ∞ and (b) there exists .N ≥ 1 and .δ > 0 such that 1 . sup n∈Z N

 n+N     1   δk  ≤ δ < .    2 k=n+1

With some additional work, one can deduce the sufficient density condition of Theorem 8 from Theorem 9 (for the special case of a Gaussian generator). Condition (b) is well known in the sampling theory of bandlimited functions; an important perturbation result of Avdonin [4] asserts that every sequence satisfying conditions (a) and (b) (in this case with .δ < 1/4) is completely interpolating for bandlimited functions. However, the characterization of all complete interpolating sequences for bandlimited functions is rather difficult and complicated [37]. It is therefore surprising that Avdonin’s condition yields a full characterization for the shift-invariant space with a Gaussian generator. These selected results may convince the reader that totally positive functions do indeed play a special role in sampling theory.

4 The Riemann Hypothesis and Totally Positive Functions Let .ζ (s) =

∞

−s n=1 n

for .s ∈ C, Re s > 1, be the Riemann zeta function and let

Totally Positive Functions and the Zeta Function

201

ξ(s) = 12 s(s − 1)π −s/2 Γ

s 

.

Ξ (s) = ξ

1 2

+ is

2



ζ (s).

(12) (13)

be the Riemann xi-functions (where .Γ is the usual gamma function). Then the functional equation for the Riemann zeta function is expressed by the symmetry ξ(s) = ξ(1 − s)

Ξ (s) = Ξ (−s)

and

.

(14)

for the xi-functions. The Riemann hypothesis conjectures that all nontrivial zeros of the zeta function lie on the critical line .1/2 + it. See the monographs [27, 28, 50], the two volumes about equivalents of the Riemann hypothesis [8, 9], or the survey articles [7, 12]. Expressed in terms of the xi-functions, the Riemann hypothesis states that .Ξ has only real zeros; in other words, .Ξ belongs to the Laguerre-Polya class. Thus, many investigations of the zeta function involve complex analysis related to the LaguerrePolya class. Schoenberg’s theorem immediately leads to the following equivalent condition for the Riemann hypothesis to hold. Theorem 10 The Riemann hypothesis holds, if and only if there exists a Polya frequency function .Λ, such that .

1 = Ξ (s)



∞ −∞

Λ(x)e−sx dx

for s ∈ C, |Re s| < t0 ,

(15)

where .1/2 + it0 is the first zero of the zeta function on the critical line. Let us make this statement a bit more explicit by taking the Fourier transform instead of the Laplace transform. Theorem 11 The Riemann hypothesis holds, if and only if Λ(x) =

.

1 2π



∞

1 −∞ ξ( 2

1 + τ)

e−ixτ dτ

(16)

is a Polya frequency function.

  The growth of .ξ in the complex plane is .|ξ(s)| = O eA|s| ln |s| , [50] and on the positive real line .

ln ξ(σ ) 12 σ log σ

σ > 1.

1 −|σ | log |σ |/2 decays super-exponentially. Since .ζ and thus Consequently, . ξ(σ ) ≤ Ce .ξ do not have any real zeros in the interval .[0, 1] and .ζ > 0 on .(1, ∞), the function

202

K. Gröchenig

ξ is therefore strictly positive on .R and .1/ξ is integrable. Thus, its Fourier transform is well defined. Using .s = 2π iτ , we can rewrite (15) as a Fourier transform. The inversion formula for the Fourier transform now yields

.

 Λ(x) =

∞

.

−∞ ∞

 =

−∞

1 e2π ixτ dτ Ξ (2π iτ ) 1 e2π ixτ dτ , ξ(1/2 − 2π τ )

which is (16). Using the symmetry of .Ξ , there is an alternative formulation of Theorem 10 with the restricted Laguerre-Polya class defined in (7). Since .Ξ is symmetric, it can be written as .Ξ (s) = Ξ1 (−s 2 ) for an entire function .Ξ1 of order .1/2. Furthermore, .Ξ has only real zeros, if and only if .Ξ1 has only negative zeros (with convergence exponent at most 1). The characterization of one-sided Polya frequency function s yields the following equivalence. Theorem 12 The Riemann hypothesis holds, if and only if there exists a Polya frequency function .Λ with support in .[0, ∞), such that .

1 = Ξ1 (s)



∞

Λ(x)e−sx dx

for s ∈ C, Re s > α ,

(17)

0

for some .α < 0. These equivalences seem to be new. Schoenberg’s name is not even mentioned in [8, 9] on equivalents of the Riemann hypothesis. One may speculate whether Schoenberg himself thought about the Riemann zeta function. He was the son-in-law of the eminent number theorist Edmund Landau, he collaborated with Polya, and he knew deeply the work of Fekete, Hamburger, Polya, and Schur about the Laguerre-Polya class of entire functions that remains influential in the study of the Riemann hypothesis. Yet, to my knowledge, he never mentioned any number theory in his work on totally positive functions and splines; by the same token, Schoenberg’s name is not mentioned in analytic number theory. Remarks It is interesting that the characterization of Theorem 11 is “orthogonal” to most research on .ζ and to the well-known criteria for the Riemann hypothesis. Theorem 11 requires only the values of .ζ on the real line to probe the secrets of .ζ in the critical strip. This fact is remarkable, but the price to pay is the added difficulty to extract any meaningful statements about .ξ on the critical strip from its restriction to .R. This seems much harder, if not impossible. To work with Theorem 11, one would need a viable expression for the FourierLaplace transform of .1/ξ , but there seems to be none. The 1-positivity in (1) says that .Λ ≥ 0, which is equivalent to the Fourier transform .Λˆ = 1/ξ(1/2 − ·) to be positive definite by Bochner’s theorem. Explicitly, we would need to know that, for all choices of .cj ∈ C, τj ∈ R, j = 1, . . . , n, and all .n ∈ N, we have

Totally Positive Functions and the Zeta Function

203

n

.

+ τj − τk )−1 ≥ 0. Not even this property of .1/ξ seems to be known. It is therefore unlikely that much is gained by Theorems 10–12. By contrast, the Fourier transform of .Ξ (x) on the critical line (!) was already known to Riemann (see [50, 2.16.1]) and is the starting point of a program to prove the Riemann hypothesis that goes back to Polya [40]. After important work of de Bruijn, Hejhal, and Newman, this line of thought has recently culminated in the resolution of the Newman conjecture by Rodgers and Tao [43]. 1 j,k=1 cj ck ξ( 2

4.1 Some Nontrivial Polya Frequency Functions Perhaps Schoenberg had the Riemann hypothesis in mind, when he investigated Polya frequency functions. The examples in [44, 46] of unusual totally positive functions smell of the zeta function. (i) The zero set .{0, −1, −2, . . . } with multiplicity one yields the entire function Ψ (s) = eγ s s

∞ 

.

(1 +

n=1

s −s/n , )e n

(18)

where .γ is the Eulerconstant. By a classical result .Ψ is the reciprocal of the ∞ Γ -function .Γ (s) = 0 x s−1 e−x dx. Consequently, the Laplace transform of −1 = Γ (s) is a totally positive function. Indeed, using the substitution .Ψ (s) −t in the definition of .Γ , one obtains .x = e .

 Γ (s) =

∞

−x

.

−∞

e−e e−sx dx

Re s > 0 .

(19)

Theorem 1 implies that Λ(x) = e−e

−x

.

is totally positive. By removing the pole of .Γ at 0, we obtain  sΓ (s) =

∞

.

−∞

Λ' (x)e−sx dx =



∞

−x

−∞

e−x e−e e−sx dx ,

−x

Res > −1 .

Consequently, .Λ1 (x) = e−x−e is a Polya frequency function. (ii) The zero set .Z with simple zeros yields .Ψ (s) = sinππ s . By Theorem 1, .1/Ψ is the Laplace transform of a totally positive function on a suitable strip of convergence. Schoenberg’s calculation yields the totally positive function Λ(x) =

.

1 . 1 + e−x

204

K. Gröchenig

(iii) Finally, the zero set .{−n2 : n ∈ N} yields the entire function Ψ (s) = s

∞ 

.

(1 +

n=1

√ 1√ s )=− −s sin π −s . 2 π n

The associated totally positive function is the Jacobi theta function  Λ(x) =

∞ j −j 2 x j =−∞ (−1) e

.

for x > 0 for x ≤ 0 .

0

All three functions show up prominently in the treatment of the functional equation of the zeta function: .Γ is contained in the definition of the xifunction, .sin is used in the formulation of the functional equation, and a Jacobi theta function is used in several proofs of the functional equation (Riemann’s original proof; see [50]).

5 More on Laguerre-Polya and Schoenberg In this section we outline some of the background results that go into proving Schoenberg’s characterization of totally positive functions (Theorem 1).

5.1 Intrinsic Characterization of Polya Frequency Functions The fundamental property of Polya frequency function s is their smoothing property or variation diminishing property. The relevance of smoothing properties for many applications is outlined in Schoenberg’s survey [47]. In this context the variation of a real-valued function on .R is measured either by the number of sign changes or by the number of real zeros. Formally, given .f : R → R, let v(f ) = sup #{n ∈ N : ∃xj ∈ R, x0 < x1 < · · · < xn with f (xj )f (xj +1 ) < 0} , (20) and let .N (f ) be the number of real zeros of f counted with multiplicity. Given a function .Λ, let .TΛ be the convolution operator .TΛ f = f ∗ Λ. Schoenberg’s second characterization of Polya frequency functions is as follows [45]. .

Theorem 13 Let .Λ be integrable and continuous. Then .Λ is variation diminishing, i.e., v(TΛ f ) ≤ v(f )

.

Totally Positive Functions and the Zeta Function

205

for all functions that are locally Riemann integrable, if and only if either .Λ or .−Λ is a Polya frequency function. This characterization is “intrinsic” in the sense that it uses only the properties of the matrices occurring in the definition (1) of total positivity. With a perturbation argument one can replace sign changes with zeros and obtain the following consequence. Corollary 14 Let .Λ be a Polya frequency function. Then for every real-valued polynomial p, the convolution .TΛ is zero-decreasing, i.e., N(TΛ p) ≤ N(p) .

.

5.2 Intrinsic Characterizations of the Laguerre-Polya Class There are several characterizations of the Laguerre-Polya class that require only their properties as entire functions. This is part of classical complex analysis and the results are due to Polya and Schur [39, 42] building on work of Laguerre, Hadamard, and many others. These results relate the properties of the zero set to properties of the power series expansion of an entire function. Before formulating of  a sequence j yields a equivalences, we note that every formal power series .F (s) ∼ ∞ a s j =0 j  j p(x) with .D = d . The differential differential operator .F (D)p(x) = ∞ a D j j =0 dx operator is well defined at least on polynomials, and the mapping .F I→ F (D) is an algebra homomorphism and thus provides a simple functional calculus.  βj j Theorem 15 Let .Ψ (s) = ∞ j =0 j ! s be an entire function. Then the following are equivalent: (i) .Ψ belongs to the Laguerre-Polya class. (ii) .Ψ can be approximated uniformly on compact sets by polynomials with only real zeros. n j n (iii) For all .n ∈ N, the polynomials .pn (x) = j =0 βj j x and .qn (x) = n n−j n have only real zeros. j =0 βj j x m (iv) If .p(x) = j =0 cj x j is a polynomial with only real, nonpositive zeros, then  the polynomial .q(x) = βj cj x j has only real zeros. ∞ γj j If, in addition, .Ψ (0) > 0 and . Ψ 1(s) = j =0 j ! s , then the following property is equivalent to (i) – (iv). 1 (v) The transform .p I→ Ψ (D) p is zero-decreasing, i.e., the polynomial .q(x) =  ∞ γj (j ) 1 j =0 j ! p (x) has at most as many real zeros as p (realΨ (D) p(x) = valued):  N

.

 1 p ≤ N (p) . Ψ (D)

206

K. Gröchenig

Applying condition (iv) to the polynomials .x n−1 (1+x)2 , one obtains a necessary condition on the Taylor coefficients of a function in the Laguerre-Polya class, namely, the so-called Turan inequalities.  βj j Corollary 16 If .Ψ (s) = ∞ j =0 j ! s belongs to the Laguerre-Polya class, then βn2 − βn−1 βn+1 ≥ 0

.

for all n ∈ N

 Applying condition (v) to polynomials of the form .p(x) = ( nk=1 ak x k )2 and working out .Ψ (D)−1 p, one obtains the following necessary condition for the Laguerre-Polya class [39, p. 235]. Corollary 17 Assume that .Ψ belongs to the Laguerre-Polya class, .Ψ (0) > 0, ∞ γj j Ψ (s) /≡ eas+b and .1/Ψ has the Taylor expansion . Ψ 1(s) = j =0 j ! s . Then for  every .n ∈ N, the .n × n Hankel matrix . γj +k j,k=0,...,n−1 is positive definite (and thus invertible).

.

However, the positivity of the Hankel matrices is not sufficient for .Ψ to be in the Laguerre-Polya class, as was proved already by Hamburger [24]. Theorem 15 and its corollaries are all contained in the seminal papers of Polya and Schur [39, 42] from 1914 and 1915 and have inspired a century of exciting mathematics. Each of the equivalent conditions in Theorem 15 is a point of departure for the study of the Riemann hypothesis. No list can do justice to all contributions between 1914 and 2020, so let us mention only a few directions whose origin is in Polya’s work. Further references and more detailed history can be found in the cited articles. (a) Condition (iii) applied to the Riemann function .Ξ yields an important equivalence of the Riemann hypothesis. The polynomials in condition (iii) are nowadays called Jensen polynomials. In modern language (iii) says that “the Jensen polynomials for the Riemann function .Ξ (s) must be hyperbolic.” Significant recent progress on this equivalence is reported in [20]. (b) The relations between the Jensen polynomials, the multiplier sequences of condition (iv), and the Turan inequalities and their generalizations have been studied in depth by Craven, Csordas, and Varga [13–15] who found many additional equivalences to the Riemann hypothesis. A particular highlight is √ their proof that .Ξ , or rather the Taylor coefficients of .Ξ ( s), satisfy the Turan inequalities [16], thereby resolving a 60- year-old conjecture going back to Polya. (c) Finally, let us mention that total positivity enters the investigation of the Laguerre-Polya class in yet another way. An entire function belongs to the restricted Laguerre-Polya class defined by (7), if and only if the sequence of its Taylor coefficients .(an ) is a Polya frequency sequence [1]. This means that the infinite upper triangular Toeplitz matrix A with entries .Aj k = ak−j , if .k ≥ j and .Aj k = 0, if .k < j has only nonnegative minors. This aspect of total positivity has been used in [31, 36] for the investigation of the zeta function.

Totally Positive Functions and the Zeta Function

207

5.3 From Total Positivity to the Laguerre-Polya Class By comparing the two intrinsic characterizations in Theorems 13 and 15, one may guess that the respective conditions on zero diminishing must play the decisive role in the proof of Theorem 1(i). To give the gist of this argument, we cannot do better than repeat Schoenberg’s beautiful argument. First, since .Λ is assumed to be a Polya frequency function, .Λ must decay exponentially [46, Lemma 2]; therefore, its moments of all orders exist. Let  μn =

.

R

x n Λ(x) dx

be the nth moment. By expanding the exponential .e−sx = express the Laplace transform of .Λ as a power series  .

R

e−sx Λ(x) dx =

∞  (−1)j j =0

j!

∞

j =1

μj s j := F (s) .

(−s)j j!

x j , we

(21)

Since .Λ /≡ 0 and .Λ ≥ 0, we have .F (0) > 0, and its reciprocal also possesses a power series expansion around 0 with a positive radius of convergence: ∞

Ψ (s) =

.

 βj 1 sj . = j! F (s) j =0

Next, we consider the convolution of .Λ with a polynomial p of degree n and relate it to the moments of .Λ:  q(x) = (Λ ∗ p)(x) =

.

=

n  (−1)j j =0

j!

R

p(x − t)Λ(t) dt =

  n (−t)j R

j =0

j!

 p(j ) (x) Λ(t) dt

μj p(j ) (x) = F (D)p(x) .

By Corollary 14 the number of real zeros of q (counted with multiplicity) does not exceed the number of real zeros of p, N(q) = N(F (D)p) ≤ N(p) .

.

(22)

Using the functional calculus, we can invert .F (D) and recover p from .q = Λ ∗ p via ∞

 βj 1 q(x) = Ψ (D)q(x) = q (j ) (x) . .p(x) = F (D) j! j =0

208

K. Gröchenig

For the monomial .q(x) = x n , we obtain the polynomial qn (x) = Ψ (D)x = n

.

n  j =0

  n n−j βj x j

of degree n. Since .x n = F (D)qn , (22) implies the count of zeros (with multiplicities): n = N(x n ) ≤ N(qn ) ≤ n .

.

For every n, .qn therefore has only real zeros. This is precisely condition (iii) of Theorem 15, and we conclude that .Ψ is in the Laguerre-Polya class.

6 Summary Schoenberg’s characterization of totally positive functions implies a condition equivalent to the Riemann hypothesis. The characterization is interesting in itself because it involves only the values of the Riemann zeta function on the real axis. To the best of our knowledge, the characterization of the Laguerre-Polya class by means of totally positive functions has not yet been tested on the Riemann zeta function. In sampling theory totally positive generators may be used as a substitute of the cardinal sine function . sinπ πx x . At this time, these are the only generators in the theory of shift-invariant spaces that are amenable to optimal sampling theorems.

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Sampling via the Banach Gelfand Triple Hans G. Feichtinger

1 Introduction It is a widely recognized fact that sampling is highly relevant in our technological world. For many years we are now using CD players storing the (quantized) information of recorded sound signals at the rate of .44,100 (= .22 32 52 72 ) samples per second. The streaming of music is based on a reduced data set, which is obtained with the help of local FFTs (fast Fourier transforms), making use of the so-called masking effect. It applies to any human auditory system and thus only removes sound information which can hardly be noticed. Parallel to this development, which is influencing our daily life and has brought us to the digital age, a mathematical theory of sampling has been established, based on the pioneering work of Paul Butzer, Abdul Jerri, and Rowland Higgins, to mention at least three of the pioneers in this area. Sampling theory has become a field of its own, with a conference series (named SamPTA) and a corresponding journal, which originally appeared under the name STSIP and meanwhile continued as SaSiDa. Starting from the famous Shannon sampling theorem, which can be formulated as an orthogonal expansion of a band-limited function in the Hilbert space of band limited functions in the Hilbert space . L2 (Rd ), ‖ · ‖2 , it was natural to extend this setting to other Banach spaces of band-limited functions, among them particularly the classical .Lp -spaces (first for .1 < p < ∞) and then to other spaces, such as weighted or mixed-norm versions of these spaces. This requires a replacement of the generating function, which is typically the standard .SINC function, by some band-limited function with good decay properties.

H. G. Feichtinger () Faculty of Mathematics, University of Vienna, Vienna, Austria ARI (Acoustic Research Institute, OEAW), Vienna, Austria e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. D. Casey et al. (eds.), Sampling, Approximation, and Signal Analysis, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-031-41130-4_10

211

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In this chapter we will not pursue another branch of development, which is concerned with the reconstruction of such functions (but also functions in so-called shift-invariant or spline-type spaces) from irregular sampling values, assuming that these are available over a sufficiently dense subset of the domain of such a function. The ideal sufficient condition is of course the standard Nyquist criterion, which indicates that there should be a good chance of (in fact iterative) recovery from irregular samples, if the maximal sampling gap (on .G = R) is controlled by the inverse of the length of the spectrum .[−a/2, a/2] ⊂ R. Also in this context, Wiener amalgam spaces play an important role. The estimates derived with these tools guarantee a certain rate of convergence for certain iterative algorithms, which depends mostly on the geometric properties of the sampling set, but imply uniform rates of convergence for all the functions in a certain functions space, for all bandlimited functions with a given spectrum. Moreover, these estimates are uniform over large classes of function spaces (e.g., they do not depend on the parameter .p > 1). The focus of this note, however, will be more on the use of nonstandard mathematical methods and function spaces, which have first proved to be useful in the mathematical discussion of the irregular sampling problem and later on in the discussion of Gabor analysis, i.e., the reconstruction of an STFT (short-time Fourier transform) from samples, typically along some lattice .Λ, or the expansion of a given function (or distribution) using so-called Gaborseries expansions, i.e., as (unconditionally) convergent multiple sums of the form . λ∈Λ cλ gλ , where .gλ = π(λ)g are time-frequency shifted versions of a given Gabor atom g, again typically some Gauss function or a compactly supported bump function. Here .λ = (t, ω) is a Rd , and .π(λ) point in the phase space .Rd ×   = Mω Tt is the corresponding (unitary) time-frequency shift on the Hilbert space . L2 (Rd ), ‖ · ‖2 . Even if one is only interested in the study of Gabor  analysis (up to  the study of Gabor multipliers) in the context of the Hilbert space . L2 (Rd ), ‖ · ‖2 , it turned out that it is more or less unavoidable (if one does not want to restrict the attention to functions in the much smaller Schwartz space .S(Rd ) of rapidly decreasing functions) to make use of a space know as Segal algebra . S0 (Rd ), ‖ · ‖S0 (see [19], or [51]) or .(M 1 (Rd ), ‖ · ‖M 1 ) (see [45]). This Banach space has many desirable properties, among others isometric invariance under the Fourier transform and under time-frequency shifts (TF shifts). This allows to extend (via duality) many operations even to the dual space .(S0' (Rd ), ‖ · ‖S0' ). Together with the Hilbert space  2 d  . L (R ), ‖ · ‖2 , these spaces constitute the so-called Banach Gelfand triple, which seems to be the correct tool for the discussion of many classical and modern problems of mathematical analysis. In recent papers, various alternative approaches and the usefulness of this setting for the treatment of classical questions has been established, also introducing the name mild distributions for the members ' d .σ ∈ S (R ). 0 It is the purpose of this note to demonstrate that this setting can also be used to provide simple, yet mathematically correct derivations of many results which are needed in the context of engineering applications. Unfortunately, these topics (such

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as translation-invariant systems or sampling theorems) are either formulated simply in a heuristic way (thus leaving the students in these fields with the impression that they just have to blindly follow the mystifying explanations of the authors) or by relating to the nontrivial idea of tempered distributions as introduced by Laurent Schwartz more than 60 years ago. But since the “Dirac function” is often introduced in an informal way, e.g., through the so-called sifting property, most arguments involving Dirac measures or Dirac combs remain a bit vague, even if they come with indications that statements made are correct in the context of distribution theory. In fact, this often gives the impression that “only mathematicians see a problem, but they also know how to pedantically fix it with a highly complicated machinery.” In this sense the author hopes to provide with this note some new sight on the topic, focussing on the underlying methods. Since only basic arguments from functional analysis are used (which appears to be unavoidable, since function spaces with respect to continuous variables are typically infinite-dimensional!), the statements and their proofs should be also accessible to mathematically interested students of engineering schools. Some of the material has been classroom tested at ETH Zürich or TU Munich in the last 5 years, but of course, such a short note can only provide an outline of parts of a material of such an integrated course (available from the author or via the Internet).

1.1 The Goal Let us give a short description of the goal of this article. It is supposed to provide nothing else than a mathematically correct, yet not too complicated description of the Shannon sampling theorem. It should work not just for “functions of finite energy” (i.e., .f ∈ L2 (Rd )), where the expansion of band-limited functions can be seen as one of the many (important) examples of an orthogonal expansion in a Hilbert space. This nice mathematical (in fact functional analytic) aspect comes however with a downside due to the poor decay of the classical .SINC function. Hence, there is no chance to use it, not even to recover band-limited functions in 1 d .L (R ), if one wants to have norm convergence with respect to the natural .‖ · ‖1 . The case is even worse if one wants to have signal expansions in weighted .Lp -spaces, but already for a long time engineers and mathematicians have realized that this problem can be overcome easily by allowing a bit of oversampling. Then versions of the Shannon sampling theorem can be obtained which work for much more general function spaces. The corresponding statement should explain how band-limited functions f (i.e., they have to have a Fourier transform .fˆ, and this .fˆ has to have compact support) can be recovered from the (regular) samples, as long as some Nyquist density criterion is satisfied. We will demonstrate that the flexible framework of “mild distributions” allows to pursue the engineering approach in a mathematically precise manner: sampling

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corresponds to multiplication with a Dirac comb; hence, the sampled function has a (generalized) Fourier transform which can be shown to be a periodized version of .fˆ. Hence, .fˆ can be recovered by pointwise multiplication with a suitable plateau-type function (which eliminates the unwanted repetitions of .fˆ, while preserving the original version). Engineers call this step a low-pass filtering. Taking another (inverse) Fourier transform, this simple procedure can be described as the Shannon series expansion of the band-limited function f , using the samples of f as coefficients.

1.2 Pictorial Illustration The idea (which is made precise by the approach provided in this note) can be simply illustrated via Fig. 1. The subfigures can be described as follows (and of course the MATLAB experiment used works in the finite discrete setting): 1. 2. 3. 4.

The left upper frame shows the complex-valued, smooth signal. The right upper frame describes its Fourier transform. The left lower frame describes the sampled signal. The right lower frame provides its Fourier transform.

The pictures demonstrate two things: the sampled signal has a periodic Fourier transform.1 That sampling occurs over a lattice .Λ which is fine enough, corresponding to the fact that the periodization of .fˆ along the “orthogonal” lattice .Λ⊥ is coarse enough and avoids aliasing, i.e., overlap of the translated copies of the Fourier transform which leads to a misinterpretation of frequencies on the time side. In the situation illustrated in Fig. 1, it is clear that pointwise multiplication with some smooth plateau-type function recovers the original well-concentrated Fourier transform of the given signal. Since pointwise multiplication on the FT side corresponds to (semidiscrete) convolution on the time side, the Shannon-like reconstruction formula arises.

1 In the finite-dimensional setting this can be reduced to the consideration that a polynomial of order n with nonzero coefficients only for multiples of say 3, i.e., of the form .p(x) = a0 + a3 x 3 + ..a3k x 3k , can be written as a polynomial .q(y), with .y = x 3 , of order .k = n/3. Since the DFT can be reinterpreted as the transition from the coefficients .a = [a0 , . . . , an−1 ] of a polynomial of order n to the values of that polynomial .p(x) over the set .Zn of unit roots of order, we find three copies of .Zk , i.e., a periodic repetition.

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band-limited signal: max.fr = 14 0.3 0.1 0.2 0.05

0.1

0

0 −0.1

−0.05 −0.2 −0.1 −200

−100

0

100

−200

200

−100

0

100

200

cutting out in a smooth way

sampled signal: sampling rate 10 0.03

0.1

0.02 0.05

0.01 0

0

−0.01 −0.05 −0.02 −0.1

−200

−100

0

100

−200

200

−100

0

100

200

Fig. 1 If there is a bit of oversampling, i.e., the Dirac comb used for sampling is a bit finer than the minimal requirement (Nyquist criterion), one has more freedom in the choice of . g , filtering out the basic period of .fˆ from its periodic repetition. A smooth trapezoidal-like function shows better concentration on the time side and thus grants much better locality of the reconstruction compared to the classical .SINC window

1.3 Historical Aspects It is a long-standing insight (first within the engineering community and later described in great detail in various mathematical papers and books, including the pioneering papers of P. Butzer, R. Higgins, and A. Jerri) that there is a close connection between Poisson’s formula and the description of the regular sampling process as a multiplication operation by a Dirac comb, i.e., (in the case of the real line) a sum of Dirac measures located at the lattice points .αk, k ∈ Z. Even if one has doubts concerning the realization of a point measurement in a real-world experiment, it is clear that the knowledge of the values of a function (which is after all continuous) at the sampling points provides exactly the same information as the pointwise product, where exactly those point values arise as (unique) coefficients of the corresponding Dirac measures. In fact f·

.

a

=f ·

 k∈Zd

δak =

 k∈Zd

f (ak)δak .

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Sometimes the proofs just make use of Poisson’s formula in combination with the inequalities of Hölder and Minkowski; in some other cases, the Dirac comb is explicitly treated as an unbounded measure, or better, as a tempered distribution, because it appears that in the traditional setting it is the only way to apply the Fourier transform to such an object (as it is obviously not a bounded measure and no other form of a generalized Fourier transform is widely known). We also should mention [48], [49] in this respect. Poisson’s formula also plays a prominent role in [10], [12], and [11]. The famous “Five Stories” [47] is clearly a standard reference to this topic, as is Abdul J. Jerri’s survey article [52] from 1977, published under the title The Shannon sampling theorem - its various extensions and applications: a tutorial review.

2 Mild Distribution Let us go straight to a description of what we consider as the natural tool for the treatment of the (regular and irregular) sampling problem: ' 2 d  Thed Banach Gelfand triple .(S0 , L , S0 )(R ) is based on the Segal algebra . S0 (R ), ‖ · ‖S0 , a Banach space of continuous and absolutely Riemann integrable test functions on .Rd , which has been introduced already 40 years ago in [19] by the author. It plays a crucial role in many questions of Gabor analysis, as has first been demonstrated in the articles [29] and [34]. In [45] it appears as the modulation space .(M 1 (Rd ), ‖ · ‖M 1 ). Note that we have .L2 (Rd ) = M 2 (Rd ) and .S0' (Rd ) = M ∞ (Rd ). The recent survey article [51] summarizes various and provides a comprehensive list of equivalent characterizations of properties d . S0 (R ), ‖ · ‖S0 .   The Banach Gelfand triple (BGT) consists of three spaces: . S0 (Rd ), ‖ · ‖S0 of (called Feichtinger’s algebra according to [64]), the Hilbert space test2 functions  d d . L (R ), ‖ · ‖2 (which can be viewed as the completion of .S0 (R ) with respect to the usual .L2 -norm, expressed via Riemann integrals), and the dual space, the space .(S0' (Rd ), ‖ · ‖S0' ), which is also known as the space of mild distributions. Equivalently, this ambient Banach space can be introduced  of mild distributions  as a kind of a completion of the space . Cb (Rd ), ‖ · ‖∞ (of bounded, continuous functions on .Rd , endowed with the sup-norm), by suitable equivalence classes of Cauchy sequences in a Lighthill spirit (see [25]). In order to quickly compare our scenario with the more well-known setting of Schwartz distributions, let us recall the chain of continuous, dense embeddings (where we consider pointwise convergence in dual spaces, e.g., .w ∗-convergence in ' d .S (R )): 0     S(Rd ) ͨ→ S0 (Rd ), ‖ · ‖S0 ͨ→ L2 (Rd ), ‖ · ‖2 ͨ→ (S0' (Rd ), ‖ · ‖S0' ) ͨ→ S' (Rd ).

.

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We note that the spaces constituting the Banach Gelfand triple .(S0 , L2, S0' )(Rd ) can be introduced without any reference to the theory of tempered distributions and that the natural setting for these spaces is the realm of LCA groups (e.g., by pursuing a Lighthill-like sequential approach to mild distributions, as outlined in [25]). Even compared with the simplified approach to the Schwartz-Bruhat distributions offered by Osborne [62], it is much less technically involved. In fact, all three are just Banach spaces with a simple norm. Consequently, no involved theory of topological vector spaces is required in order to deal with this triple. Although the BGT appears implicitly in [34], a systematic treatment has only be given in [15]; see also the master thesis of S. Bannert [4].

  2.1 The Segal Algebra S0 (Rd ), ‖ · ‖S0   Let us first recall that the standard definition of . S0 (Rd ), ‖ · ‖S0 makes use of the STFT (short-time Fourier transform), with respect to the Gaussian window, i.e., using .g0 (t) := exp(−π |t|2 ) on .Rd (see [45], Lemma 1.5.1). We use the standard symbols for translation and modulation operators:2 Definition 1 Tx f (t) = f (t − x),

.

My f (t) = e2π iyt f (t),

x, y, t ∈ Rd .

The modulation operator is a multiplication by some pure frequency or character, given by .χs (t) = e2π ist . Our normalization of the Fourier transform is this one: fˆ(ω) =





.

Rd

f (t)χs (t)dt =

Rd

f (t)e−2π iωt dt =

 Rd

[M−ω f ](t)dt.

(1)

The Gauss function is a very convenient Gabor atom or window function because it is invariant under the Fourier transform. In general, the short-time Fourier transform (STFT) is defined with the help of the time and frequency shift (or modulation) operators, respectively: Definition 2 The short-time Fourier transform (STFT) of a function .f ∈ L2 (Rd ) with respect to a window3 .g ∈ L2 (Rd ) is defined for .λ = (x, ω) ∈ Rd ×  Rd via

yt denotes the scalar product in .Rd word “window” is used with the understanding that the multiplication of the signal f to be analyzed by the typically compactly supported function g localizes the function. Often g is assumed to be symmetric and real-valued, centered at zero, may be a plateau function, or a Bspline. For this reason the STFT is also often called the sliding window Fourier transform. 2 where 3 The

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 Vg f (x, ω) =

.

Rd

f (t)g(t − x)e−2π it·ω dt

 =

Rd

f (t)Mω Tx g(t)dt = 〈f, Mω Tx g〉.

Sometimes it is easier to describe the combined operator .π(λ) := Mω Tx for .λ = (x, ω) ∈ Rd ×  Rd as a time-frequency shift. Hence, for .λ in phase space ones has Vg f (x, ω) = Vg f (λ) = 〈f, π(λ)g〉,

.

λ ∈ Rd ×  Rd .

Once the STFT is well defined, one observes that it defines for    any normalized  window with .‖g‖2 = 1 an isometry from . L2 (Rd ), ‖ · ‖2 into . L2 (R2d ), ‖ · ‖2 , by the so-called Moyal formula (see [45], Chap. 3). This, in turn, implies a natural inversion formula (in such a case the adjoint mapping is the inverse on the range of the isometric embedding).   Now the most elegant definition of . S0 (Rd ), ‖ · ‖S0 reads as follows: Definition 3 S0 (Rd ) := {f ∈ L2 (Rd ) | Vg0 f ∈ L1 (R2d )},

.

endowed with the norm ‖f ‖S0 (Rd ) := ‖Vg0 f ‖L1 (R2d ) .

.

  1 (Rd ). Formula It is continuously embedded into . L2 (Rd ), ‖ · ‖2 and hence into .Lloc (3.10) in [45], which tells us that Vg f (x, ω) = e−2π ixω V g (fˆ)(ω, −x),

.

(x, ω) ∈ Rd ×  Rd ,

(2)

implies immediately that the Fourier transform (which can be actually realized in this setting via absolutely convergent Riemann integrals, applied to continuous  functions) is an isometric automorphism of . S0 (Rd ), ‖ · ‖S0 because it is a wellknown classical fact that .F(g0 ) = g0 . This also implies that the usual Fourier inversion formula is valid in the pointwise sense (and in fact also requires only Riemann integration).   It is also easy to show that translation is isometric on . S0 (Rd ), ‖ · ‖S0 and that translation is strongly continuous, i.e., for any .f ∈ S0 G one has ‖f − Tx f ‖S0 → 0 for x → 0.

.

Thus, it is a homogeneous Banach space in the sense of Y. Katznelson [54]; hence, L1(Rd ) ∗ S0 (Rd ) ⊆ S0 (Rd ),

.

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combined with the corresponding norm estimates. In fact it is continuously embed ded into . L1(Rd ), ‖ · ‖1 and thus a Segal algebra in the sense of Reiter (see [64]).   Consequently, . S0 (Rd ), ‖ · ‖S0 is also a Banach algebra with respect to both convolution and pointwise multiplication. Obviously these two types of abstract multiplications change their roles under the Fourier transform (see [19] or [51]). The fact that it is a Banach algebra under also follows  pointwise multiplication  easily from an alternative description of . S0 (Rd ), ‖ · ‖S0 . In fact, this was the way how the space was introduced in [19] as Wiener amalgam space of the form 1 1 d .(W (FL , 𝓁 )(R ), ‖ · ‖ W (FL1,𝓁1 ) ). For this description we need a bounded partition of unity in the Fourier algebra 1 1 d d .FL (R ) := {fˆ | f ∈ L (R )}, endowed with the norm .‖fˆ‖ FL1 (Rd ) := ‖f ‖L1(Rd ) ,  ∈ L1(Rd ) such i.e., simply a continuous function .ψ with compact support and .ψ that   . Tk ψ(x) = ψ(x − k) ≡ 1, x ∈ Rd . (3) k∈Zd

k∈Zd

  Any such family will be called a (regular) BUPU in . FL1 (Rd ), ‖ · ‖FL1 , a bounded (in the Fourier algebra) uniform (with respect to the size of the supports of its members) partition of unity (of size .γ if .supp(ψ) ⊆ Bγ (0)). The Wiener amalgam space .W (FL1, 𝓁1 )(Rd ) is defined as W (FL1, 𝓁1 )(Rd ) := {f ∈ FL1 (Rd ) | ‖f ‖W (FL1,𝓁1 ) :=



.

‖f · Tk ψ‖FL1 < ∞}.

k∈Zd

(4) Lemma 1 For any BUPU .(Tk ψ)k∈Zd as above the Wiener amalgam space W (FL1, 𝓁1 ), endowed with the norm .f I→ ‖f ‖W (FL1,𝓁1 ) coincides with the   Segal algebra . S0 (Rd ), ‖ · ‖S0 and the corresponding norms are equivalent. Consequently, different BUPUs define the same space and equivalent norms.   Note that also in the definition of . S0 (Rd ), ‖ · ‖S0 using .g0 , the specific window can be replaced by any nonzero .g ∈ S(Rd ), in fact even by any nonzero element of d .S0 (R ), defining the same Banach space and also with equivalence of norms. .

Remark 1 The general Wiener amalgam spaces .W (B, C) require the use of a local component .(B, ‖ · ‖B ) and a global component .(C, ‖ · ‖C ) (which can be substituted by a sequence space over .Zd ). For our discussion spaces of the form p ∞ p q 1 ∞ d d d d .W (L , 𝓁 )(R ), .W (C0 , 𝓁 )(R ),.W (FL , 𝓁 )(R ), or .W (M, 𝓁 )(R ), known in the literature as the space of translation-bounded Radon measures are the most relevant ones. They are also relevant in the context of transformable measures, following the work of L. Argabright and J. Gil de Lamadrid (see [1] and [2], or [60] or [57] for more recent papers). Convolution relations and important equivalent norms are described in [20] and [46]. Essentially they follow an “amalgamation” of local and global properties (e.g., convolution relations among sequence spaces over d .Z ).

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W (C0 , 𝓁1 )(Rd ) is one of the Segal algebras mentioned in Reiter’s book as “Wiener’s algebra” (see [64]), and it is the pre-dual of .W (M, 𝓁∞ )(Rd ). The minimality of .W (C0 , 𝓁1 )(G) shown in [17] was a motivation for [19], replacing the pointwise .C0 (Rd )-module property by the less restrictive pointwise multiplication by elements of the Fourier algebra .FL1 (Rd ). .

Our definition shows that .S0 (Rd ) is a member of the family of modulation spaces and is therefore denoted by .(M 1 (Rd ), ‖ · ‖M 1 ) in the Gröchenig’s book [45], where the reader finds details. The characterization as Wiener amalgam space describes the elements of .S0 (Rd) as absolutely  convergent series of functions which belong to Wiener’s algebra4 . A(T), ‖ · ‖A of absolutely convergent Fourier series (by viewing each function .f · Tk ψ as a periodic function). One of the crucial properties of .S0 (Rd ) is the fact, which was indeed to main motivation for [19], that it is the smallest among all nontrivial Banach spaces which allow an isometric action of TF shifts, i.e. Banach spaces .(B, ‖ · ‖B ) which satisfy ‖Mω Tx f ‖B = ‖f ‖B ,

.

x, ω ∈ Rd , f ∈ B.

(5)

The classical .Lp -spaces as well as many other function spaces (such as Orlicz or Lorentz spaces) satisfy this property. This implies a continuous embedding of the form     S0 (Rd ), ‖ · ‖S0 ͨ→ Lp (Rd ), ‖ · ‖p ,

.

1 ≤ p < ∞.

(6)

Remark 2 The most elegant way to introduce the biggest space with property 5, the space of mild distributions, is to make use of duality of Banach spaces. Alternatively, the Banach space of mild distributions could also be described (without any reference to .L2 (Rd ) or Lebesgue integration) as a kind of completion of .S0 (Rd ). This has been pointed out in [25] providing a sequential approach to mild distributions, which is comparable to the Lighthill method for the description of tempered distributions (see [58]). However, it has to be said that the verification of the fact that the functional analytic approach using topological vector space theory and the approach via equivalence classes of Cauchy sequences (defined appropriately in each of these contexts) are in equivalent is much easier for the case of .(S0' (Rd ), ‖ · ‖S0' ) compared to the Schwartz setting. Remark 3 Although the Banach Gelfand triple arose as a natural concept in Gabor analysis (see [29, 30, 34]), one can say that there are many other application areas, where this Banach Gelfand triple appears to be more natural and flexible compared to other tools (like vector measures for generalized stochastic processes; see [28], a summary of the PhD thesis [50]), or quasi-measures for the theory of multipliers, as used in the work of Gaudry; see [42], [56], and so on). Typical applications in

4 The reader has to be aware that this name, also quite popular in the literature, may cause confusion with the Wiener algebra .W (C0 , 𝓁1 )(Rd ) mentioned earlier.

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the context of classical Fourier analysis, e.g., in the connection with summability methods, are given in [24]. Remark 4 The inclusion .S(Rd ) ͨ→ S0 (Rd ) provides a large number of smooth functions in .S0 (Rd ). For .d = 1 the condition .f, f ' , f '' ∈ L1 (R) implies .f ∈ S0 (R). But the elements of .S0 (Rd ) do not have to satisfy any differentiability property. For .d = 1 the triangular function (the piecewise linear function .Δ, with .Δ(±1) = 0, Δ(0) = 1) is compactly supported with .F(Δ) ∈ L1 (R); hence, .Δ ∈ S0 (R). As a consequence, any piecewise linear function h with .h(k) = dk for some sequence  1 .(dk )k∈Z ∈ 𝓁 (Z), or equivalently .h = d k∈Z k Tk Δ, belongs to .S0 (R). By dilation one can obtain similar functions with nodes at .aZ, for some .a > 0. A rich variety of examples of functions in .S0 (Rd ), including practically all the classical summability kernels is provided in the paper [32]. Further sufficient conditions are given in [44], where the validity of Poisson’s formula is discussed in great detail. In fact, all the technical conditions given there which ensure the validity of Poisson’s formula in the form (in the form of (14) below) are technical assumptions on the decay of f and .fˆ in terms of weighted .Lp -spaces and are sufficient conditions for membership in .S0 (Rd ). For example, a sufficient condition is the membership of f and .fˆ in a weighted version of .L2 (Rd ), as long as the weight is strong enough (see Proposition 12.1.6 in [45]). In the language of Shubin classes, this result appears as Theorem 3.3 in [59]. Remark 5 An important property of .S0 (Rd ) (described again just for the case .d = 1 here) is the following: Given .f ∈ S0 (R), then the piecewise linear interpolation .PLa (f ) of f with notes at any of the lattices .aZ belongs to .S0 (R), and moreover, one has according to [30] convergence of these piecewise linear interpolants in the norm of .S0 (R) (Corollary 2.3) as .a → 0. In the approximation theory literature, the mapping .PLa is called the Schoenberg operator. This result is the basis for the verification of the fact that the Fourier transform .fˆ of any .f ∈ S0 (R) can be approximated in the norm of .S0 (R) with the help of an FFT routine (as one may expect); see [53].

3 The Space of Mild Distributions The paper [25] describes the space .(S0' (Rd ), ‖ · ‖S0' ) as a kind of completion of the   space . Cb (Rd ), ‖ · ‖∞ of all continuous, bounded complex-valued functions with respect to uniform convergence of the corresponding STFTs (Short-Time Fourier transforms) over compact subsets of phase space. Thus, in a way, it is the biggest space of “objects” which have a bounded (and continuous) STFT. On the other hand, this can be taken as the characterization of .S0' (Rd ) inside of the space .S' (Rd ) of all tempered distributions, as we have. Lemma 2 A tempered distribution .σ ∈ S' (Rd ) belongs to .S0' (Rd ) if and only if the short-time Fourier transform (STFT) .Vg σ ∈ Cb (R2d ), and the .sup-norm of .Vg σ

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defines a natural norm, equivalent to the norm defined by duality which can be realized as follows:5  σ (f ) =

.

Rd × Rd

Vg σ (λ)Vg f (λ)dλ,

The space .(S0' (Rd ), ‖ · ‖S0' ) of mild distributions satisfies quite a few interesting properties which make it a very useful and simple tool for many applications: Proposition 1 1. .(S0' (Rd ), ‖ · ‖S0' ) is (isometrically) invariant under the Fourier transform, but also under time-frequency shifts.   2. Any of the Banach spaces . Lp (Rd ), ‖ · ‖p , for .1 ≤ p ≤ ∞, allows the following chain of continuous embeddings:     S0 (Rd ), ‖ · ‖S0 ͨ→ Lp (Rd ), ‖ · ‖p ͨ→ (S0' (Rd ), ‖ · ‖S0' ).

.

For .p < ∞ the first embedding is dense, while the second one gives a .w ∗-dense embedding in any case. 3. S0 (Rd ) · (S0 (Rd ) ∗ S0' (Rd )) ⊂ S0 (Rd )

.

and equivalently S0 (Rd ) ∗ (S0 (Rd ) · S0' (Rd )) ⊂ S0 (Rd ).

.

The last statement can be used to show that .S0 (Rd ) (considered as a subspace of S0' (Rd )) is .w ∗ -dense in .S0' (Rd ). To make clear how norm convergence (i.e. uniform convergence in the STFT-domain) is related to .w ∗ -convergence in the dual space ' d .S (R ), let us recall 0 .

Lemma 3 A (bounded) sequence .(σn )n≥1 in .(S0' (Rd ), ‖ · ‖S0' ) is convergent to .σ0 ∈ S0' (Rd ) in the .w ∗-sense, meaning that .

lim σn (f ) = σ0 (f ),

n→∞

∀f ∈ S0

if and only if we have Vg σ n (λ) → Vg σ 0 (λ),

.

5 which

uniformly over compact subsets of Rd ×  Rd .

can be realized via Riemannian integrals

(7)

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The extended actions (e.g., of TF shifts, or the Fourier transform) turn out to be the unique .w ∗−w ∗-continuous6 extensions of the classical definitions for ordinary functions, in particular the one given for .f ∈ S0 (Rd ) by the Riemann integral (1), while the extension, which could be characterized via a regularized version of the input, can be, more elegantly, characterized by the simple duality relation  σ (f ) = σ (fˆ),

f ∈ S0 (Rd ).

.

(8)

For an alternative description of the (extended) Fourier transform, very much in the spirit of equivalence classes of mild Cauchy sequences (ECmiCS in [25]), let us recall the two versions of the dilation operator, valid for any .ρ > 0: Definition 4 Stρ g(x) = ρ −d g(x/ρ)

.

and

Dρ h(x) = h(ρx).

(9)

It is well known that the Fourier transform turns convolution into pointwise multiplication (and vice versa) and intertwines these two versions of the dilation operator, which in addition are adjoint to each other, i.e., F ◦ Stρ = Dρ ◦ F,

.

or

 St ρ f = Dρ f,

and

〈Stρ g, h〉L2 = 〈g, Dρ h〉L2 . (10)

  The first operation acts isometrically on . L1(Rd ), ‖ · ‖1 and is compatible with convolution, while the second one preserves the sup-norm and is compatible with pointwise multiplication. In other words, ‖Stρ g‖1 = ‖g‖1 , g ∈ L1(Rd ),

.

Stρ (g ∗ f ) = Stρ g ∗ Stρ f,

(11)

Dρ (g · f ) = Dρ g · Dρ f.

(12)

and ‖Dρ h‖∞ = ‖h‖∞ , h ∈ C0 (Rd ),

.

Using the Wiener amalgam characterization, it is not difficult to control the norm of both dilation operators (they are not uniformly bounded anymore) on   d . S0 (R ), ‖ · ‖S0 . a Dirac net, respectively a bounded approximate For .ρ → 0 the first one creates  identity for . L1(Rd ), ‖ · ‖1 whenever . g (0) = 1, while .Dρ h has a similar role   for . FL1 (Rd ), ‖ · ‖FL1 with respect to pointwise multiplication if .h(0) = 1, d .h ∈ S0 (R ). Another way to characterize this unique extension of the Fourier transform from ' d d .S0 (R ) to .S (R ) is to set (it is well defined!) 0 6 I.e.,

any .w ∗-convergent net .(σα )α∈I is mapped to a .w ∗-convergent net with the correct limit.

224

H. G. Feichtinger

 σ = lim F(Dρ h · (Stρ g ∗ σ )),

.

ρ→0

(13)

where we use the fact that .(Stρ g)ρ→0 in .S0 (Rd ), with .‖Stρ g‖1 =  is da Dirac sequence  ‖g‖1 for .ρ > 0 (unbounded in . S0 (R ), ‖ · ‖S0 , but bounded in . L1(Rd ), ‖ · ‖1 and thus in the operator norm the spaces constituting the BGT) and .Dρ h describes a usual summability kernel, with .h(0) = 1.) For technical details, the reader may consult Theorem 3 of [27]. Their adjoints, which are of the same form, are thus providing .w ∗-approximations to the identity. In combination with Proposition 1, this also shows that .S0 (Rd ) is .w ∗-dense in .S0' (Rd ).   The space .S0' (Rd ) contains not only any of the spaces . Lp (Rd ), ‖ · ‖p for .p ≥ 1, but also the space of bounded measures, which can be described   as .(Mb (Rd ), ‖ · ‖Mb ) = C0' (Rd ), ‖ · ‖C0' (see [23]). In fact, from the dense     embedding . S0 (Rd ), ‖ · ‖S0 ͨ→ C0 (Rd ), ‖ · ‖∞ , we derive .(Mb (Rd ), ‖ · ‖Mb ) ͨ→ (S0' (Rd ), ‖ · ‖S0' ). A more refined consideration, based on the dense embedding of     Segal algebras, namely, . S0 (Rd ), ‖ · ‖S0 ͨ→ W (C0 , 𝓁1 )(Rd ), ‖ · ‖W , implies that translation-bounded measures can also be understood as mild distributions; hence, ∞ 1 ' ' d d .W (M, 𝓁 )(R ) = W (C0 , 𝓁 ) ͨ→ S (R ). 0   Note that Wiener’s algebra (in the terminology of [63]) . W (C0 , 𝓁1 )(Rd ), ‖ · ‖W can be characterized as the minimal Segal algebra (in Reiter’s sense, see [64]) or even homogeneous Banach space (in the sense of Katznelson [54]) with   the additional property of being a pointwise Banach module over . C0 (Rd ), ‖ · ‖∞ (see [17]). In the context of sampling theory, it is clear that the Dirac comb . Λ plays an important role. It is given as .

Λ

:=



δλ .

λ∈Λ

For the additive group .G = Rd (which we discuss in this note), the (discrete and cocompact) subgroups of interest are of the form .Λ = A(Zd ), for some nonsingular .d × d-matrix .A. By using the generic notation, it will be clear how to formulate (and even prove) results in the context of general LCA (locally compact Abelian) groups, and therefore, we prefer the abstract and simple notation. It is easy to show that . Λ ∈ W (M, 𝓁∞ )(Rd ) ⊂ S0' (Rd ), and hence, the (distributional) Fourier transform is well defined. In fact, the validity of Poisson’s formula in the form  .

f (λ) = CΛ



fˆ(λ⊥ ),

f ∈ S0 (Rd ),

(14)

λ⊥ ∈Λ⊥

λ∈Λ

can be recast into the more compact form of the key identity for sampling theory: .

 Λ = F(

Λ)

= CΛ

Λ⊥ .

(15)

Sampling via the Banach Gelfand Triple

225

Remark 6 While the compact version of this last result appears to coincide with the corresponding familiar result in the context of tempered distributions, one has to mention there that the validity of (15) in the sense of mild distributions is only a reformulation of the validity of (14) for any .f ∈ S0 (Rd ). Since .S(Rd ) ⊆ S0 (Rd ) is a strict inclusion (members of .S0 (Rd ) need not even be differentiable nor do they have to satisfy any decay rate), it is clear that requiring the validity of Poisson’s formula only for Schwartz functions is a much easier, though in fact (by way of proofs using Wiener amalgams) it is an equivalent claim.

3.1 Sampling and Dirac Combs We will focus our attention on the sampling problem for the regular case, i.e., the sampling along lattices .Λ ⊲ Rd . So as to avoid technical complications we limit our discussion to translation-invariant Banach spaces of mild distributions, because in this setting the basic analytic and algebraic structure of the problems discussed become clear. On the other hand, the presentation allows to extend the setting to a much wider variety of spaces. The model problem will be the discussion of the sampling problem for bandlimited functions in .Lp (Rd ), for .1 ≤ p < ∞. More generally, we will work in the following setting: Definition 5 A Banach space .(B, ‖ · ‖B ) is called a homogeneous Banach space of mild distributions (HBMD) if we have: 1. .(B, ‖ · ‖B ) ͨ→ (S0' (Rd ), ‖ · ‖S0' ). 2. Translations are isometric and act strongly continuous on .(B, ‖ · ‖B ), i.e., ‖Tx f ‖B = ‖f ‖B

.

and

lim ‖Tx f − f ‖B = 0,

x→0

f ∈ B.

Remark 7 It is easy to verify that this definition is (only slightly) more general than the concept introduced in the book of Katznelson [54] and resumed in the recent paper [26] in the setting of general LCA groups. In fact, Katznelson assumes that .(B, ‖ · ‖B ) is continuously embedded into the topological vector space of all locally 1 (Rd ), but due to the translation invariance of the norm, this integrable functions .Lloc implies the continuous embedding .B ͨ→ W (L1 , 𝓁∞ )(Rd ). Of course, any Segal algebra  1 d (whichis by definition continuously embedded as a dense subspace into . L (R ), ‖ · ‖1 ) belongs to this family. The assumptions also imply that .(B, ‖ · ‖B )   is a Banach module over . L1(Rd ), ‖ · ‖1 under convolution (see [26]), with ‖g ∗ f ‖B ≤ ‖g‖L1 ‖f ‖B,

.

g ∈ L1(Rd ), f ∈ B.

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Since in the context of .(S0' (Rd ), ‖ · ‖S0' ) both the Fourier transform of .σ ∈ S0' (Rd ) is well defined as well as the support of a mild distributions (in the usual way), we can also define the spectrum of .f ∈ B for any such HBMD: Definition 6 Given .σ ∈ S0' , the spectrum of .σ is defined as .

spec(f ) = supp(fˆ).

Remark 8 For .h ∈ L∞ (Rd ) ⊂ S0' (Rd ) the definition of spectrum (as the comple ment of the cospectrum of the corresponding annihilating ideal in . L1(Rd ), ‖ · ‖1 ) as given in [64], Chap. 7, is equivalent, even if it looks quite differently from a technical point of view. One can say that .s0 ∈ Rd does not belong to .spec(h) if and only if there is some small neighborhood .Bδ (s0 ) such that any function 1 d d .g ∈ L (R ) (or equivalently .g ∈ S0 (R )) with .supp( g ) ⊂ Bδ (s0 ) satisfies  .σh (g) = g(x)h(x)dx = 0. d R An interesting property of .S0' (Rd ) which distinguishes the space of mild distributions drastically from the Schwartz space is the following characterization of mild distributions supported on a lattice. Proposition 2 Given any discrete lattice .Λ ⊲ Rd we have: A mild distribution .σ ∈ S0' (Rd ) satisfies .supp(σ ) ⊂ Λ if and only if there exists a bounded sequence .(cλ )λ∈Λ ∈ 𝓁∞ (Λ) such that σ =



.

cλ δλ .

λ∈Λ

Moreover, for fixed .Λ, the norms .‖σ ‖S0' and .‖(cλ )λ∈Λ ‖𝓁∞ (Λ) are equivalent. Equivalently, we can characterize these mild distributions as the set of (tempered) distributions arising by applying the sampling operator (multiplication by the Dirac comb . Λ ) by continuous, bounded complex-valued functions on .Rd : {h ·

.

Λ |h

∈ Cb (Rd )},

with equivalence of the norms in .S0' (Rd ) and the corresponding quotient norm of the sampling operator . Cb (Rd ), ‖ · ‖∞ → (S0' (Rd ), ‖ · ‖S0' ). In the spirit of a series of papers by Jens Fischer (and R. Stens), [36], [37], [38], [35] (see [39], [40]), we can thus use the Dirac combs to define both sampling and periodization operators, which correspond to each other under the Fourier transform, thus providing multidimensional versions of the results in those papers: Definition 7 Let us fix any co-compact discrete lattice .Λ ⊲ Rd . Then the pointwise multiplication operator f I→ ⊥⊥⊥(f ) =

.

Λ

·f =

 λ∈Λ

f (λ)δλ

Sampling via the Banach Gelfand Triple

227

and the corresponding periodization operator f I→ ∆∆∆(f ) :=

.

Λ

∗f =



Tλ f

λ∈Λ

are well defined on .S0 (Rd ), with range in .S0' (Rd ). Starting from these two definitions, the generally valid convolution theorem (pointwise multiplication is turned into convolution), in conjunction with formula (15), gives for .f, g ∈ S0 (Rd ): F ( ⊥⊥⊥f ) = ∆∆∆(Ff )

and.

.

F ( ∆∆∆g) = ⊥⊥⊥(Fg)

(16) (17)

As the symbols . ∆∆∆ and . ⊥⊥⊥ represent operators, one can reformulate the above formulas (we talk about .T = B = 1) as F ◦ ∆∆∆ = ⊥⊥⊥ ◦ F,

.

F ◦ ⊥⊥⊥ = ∆∆∆ ◦ F.

(18)

  The properties of . S0 (Rd ), ‖ · ‖S0 imply immediately several relevant facts for these two operators: Lemma 4 1. The sampling operator I ⊥⊥⊥Λ(f ) defines a surjective mapping from    1.f → d . S0 (R ), ‖ · ‖S0 onto . 𝓁 (Λ), ‖ · ‖1 . 2. Correspondingly, I ∆∆∆Λ(f ) defines a surjective  the periodization  operator .f → mapping from . S0 (Rd ), ‖ · ‖S0 onto the subspace the space of .Λ-periodic functions .A(Λ) which have an absolutely convergent Fourier series expansion of the form f =



.

cλ⊥ χλ⊥ ,

λ⊥ ∈Λ⊥

with



|cλ⊥ | < ∞.

λ⊥ ∈Λ⊥

  Proof The restriction property for . S0 (Rd ), ‖ · ‖S0 has been established already early on (see [19], or more recently [51]). The second characterization follows directly by taking the (distributional) Fourier transform of the first one, recalling the fact that pointwise multiplication (also by elements with .S0' (Rd )) goes into convolution, and vice versa. For later reference let us state this last statement separately: Lemma 5 Given .σ ∈ S0' (Rd ) and .f ∈ S0 , we have F(f · σ ) = fˆ ∗  σ,

.

(19)

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H. G. Feichtinger

and F(f ∗ σ ) = fˆ ·  σ.

(20)

.

The statements are obvious for the case  of a regular mild distribution, i.e., .σ = σg for some .g ∈ S0 (Rd ), via .σg (f ) = Rd f (x)g(x)dx, and is validated by taking ' ∗ d ∗ ∗ .w -limits, taking into account that the Fourier transform on .S (R ) is also .w −w 0 continuous. As a special case of these statements, one can choose .σ = Λ or . Λ⊥ . The above results show that the Fourier transform intertwines the two operators, i.e., .

F−1 ◦ ∆∆∆Λ ◦ F = ⊥⊥⊥Λ⊥

respectively (or equivalently) .

F−1 ◦ ⊥⊥⊥Λ ◦ F = ∆∆∆Λ⊥ .

3.2 Characterizing Generators of BUPUs In this subsection we show how the property of defining a BUPU (even in the case of non-compact support, e.g., for Schwartz functions, or even functions .ψ ∈ S0 (Rd )) of a function .ψ for a given lattice .Λ can be characterized by the interpolatory  over the dual lattice. Such a situation is well known for the indicator property of .ψ function of .[−1/2, 1/2) (the symmetric fundamental domain of .Z ⊲ R), which corresponds to the well-known .SINC function on the Fourier transform side. Strictly speaking, this case is not covered by the following lemma (because .SINC ∈ / L1 (R) implies that .SINC ∈ / S0 (R)). We write alternating . Zd or for the standard Dirac comb over .Zd ⊲ Rd . Lemma 6 Let .ψ ∈ S0 (Rd ) be given, and then .ψ defines a regular UCPU (a uniformly concentrated partition of unity) i.e., satisfies the condition  .

Tk ψ(x) ≡ 1,

or

Zd

∗ψ = 1

(21)

k∈Zd

if and only if 7 (k) = δ0,k , k ∈ Zd ψ

.

7 Here

we use Kronecker’s delta function.

or

Zd

 = δ0 . ·ψ

(22)

Sampling via the Banach Gelfand Triple

229

Proof This statement is a simple consequence of the fact that the Poisson formula, valid in the form   . f (k) = fˆ(n), f ∈ S0 (Rd ), k∈Zd

n∈Zd

' d can be described as .F( Zd ) = Zd , (in the context of .S0 (R )) and that convolution products are mapped into pointwise convolution and vice versa, so we simply have

F(

.

∗ f ) = F(

· fˆ,

) · F(f ) =

combined with the well-known formula  . ·f = f (k)δk ,

f ∈ S0 (Rd ),

k ∈ Zd .

k∈Zd

By applying the transformation to general lattices, we obtain the following result (first written in terms of abstract groups): Theorem 1 A function .ϕ ∈ S0 (Rd ) satisfies . Λ ∗ϕ ≡ 1, i.e., determines a regular, uniformly concentrated partition of unity, if and only if one has (for a suitable constant .CΛ > 0): .

Λ⊥

· ϕ = CΛ δ0 ,

i.e., if and only if  ϕ (λ⊥ ) = CΛ δ0,λ⊥ ,

.

λ⊥ ∈ Λ⊥ .

Proof The argument is the same as for .Λ = Zd , once it has been established that for any .Λ ⊲ Rd on has .F( Λ ) = CΛ Λ⊥ .

4 Band-Limited Functions Now we are able to provide a simple characterization of band-limited functions in such a homogeneous Banach space of mild distributions: d , the (closed) subspace of .Ω-bandDefinition 8 Given a closed subset .Ω ⊂ R limited elements in .(B, ‖ · ‖B ) is defined as follows: B Ω := {f ∈ B, supp(fˆ) ⊆ Ω}.

.

230

H. G. Feichtinger

Lemma 7 For any homogeneous Banach space .(B, ‖ · ‖B ) of mild distributions and any compact set .Ω ⊂ Rd , one has .B Ω ⊂ W (FL1 , 𝓁∞ )(Rd ). Moreover, there exists a constant .CΩ such that ‖f ‖∞ ≤ ‖f ‖W (FL1 ,𝓁∞ ) ≤ CΩ ‖f ‖B ,

.

f ∈ B Ω.

Proof Due to the continuous embedding .(B, ‖ · ‖B ) ͨ→ (S0' (Rd ), ‖ · ‖S0' ), it is sufficient to verify the condition for .(B, ‖ · ‖B ) = (S0' (Rd ), ‖ · ‖S0' ). Given a compact set .Ω, we can find some .h ∈ Cc (Rd ) with . h ∈ L1(Rd ), with the property that .h(ω) ≡ 1 on .Ω. Such a function belongs to .FL1 (Rd ) and has compact support and thus belongs to .S0 (Rd ). It is easy to obtain such a function, e.g., as a convolution product of the indicator function of a somewhat larger set ' .Ω = Ω + Bγ (0) with the indicator function of some small set, e.g., of .Bγ /2 (0). Since such a convolution product belongs to .L2 (Rd ) ∗ L2 (Rd ) ⊂ FL1 (Rd ) (by Plancherel’s theorem) and even satisfies .supp(h) ⊂ Ω + B2γ (0). As a consequence we have derive that .f ∈ B Ω satisfies .fˆ · h = fˆ, orequivalently-for .g = F−1 (h) ∈ S0 (Rd ) = W (FL1, 𝓁1 )(Rd ): .f = g ∗ f for any Ω .f ∈ B . But the usual convolution relations for Wiener amalgam spaces (see [20]) imply f = g ∗ f ∈ W (FL1, 𝓁1 ) ∗ W (FL∞ , 𝓁1 ) ⊂ W (FL1 , 𝓁∞ ) ⊂ Cb (Rd ),

.

(23)

together with the corresponding norm estimates, i.e., up to some universal constant (depending on the choice of the concrete norm on each of these spaces): ‖f ‖W (FL1 ,𝓁∞ ) ≤ C‖g‖S0 ‖f ‖B .

.

(24)

As a consequence, for any lattice .Λ ⊲ Rd the product . ⊥⊥⊥Λ(f ) = f · Λ is well defined, and moreover, .f I→ ⊥⊥⊥Λ(f ) defines a bounded linear operator on Ω (endowed with the norm of .(B, ‖ · ‖ )) into .S ' (Rd ). .B B 0

5 Wiener Amalgam Spaces and Sampling For the description of the (regular) sampling problem, the use of Wiener amalgam spaces and especially their natural behavior with respect to convolution and pointwise multiplication (or the Fourier transform) play a significant role. These spaces allow to discuss many questions concerning sampling by a set of arguments which is quite similar for a variety of settings. This author has used them mostly in connection with the irregular sampling problem for band-limited functions or functions belonging to spline-type spaces (also known as shift-invariant spaces, such as spaces of cubic splines). These methods can be used in the Euclidean case (one-dimensional or multidimensional), but also in the more general setting of

Sampling via the Banach Gelfand Triple

231

LCA groups. They can be used which are isometrically invariant  for function spaces  under translations (such as . Lp (Rd ), ‖ · ‖p ) or weighted versions of such spaces. They are not only suitable for the verification that the standard iterative methods for irregular sampling problems are actually convergent (at a geometric rate), but also for a qualitative estimate of the behavior of such reconstruction methods (including the regular case) in terms of jitter or truncation errors. For the irregular sampling problem for band-limited functions, such considerations have been laid out in the publications such as [31].

5.1 Robustness of Minimal Norm Interpolation Another situation where the use of Wiener amalgam spaces has proven to be very natural and powerful is the question of minimal norm interpolation of square summable data in Sobolev spaces, which will be discussed next. The situation can be described as follows: Whereas the reconstruction of bandlimited functions on .Rd is a perfect one, there is of course some freedom in the choice of a given smooth function, if only the samples along some lattice of the form d .aZ , for .a > 0, are given. This applies in particular to the case of Sobolev spaces d .Hs (R ) for .s > d/2, because in this case Sobolev’s embedding can be proved in a simple way, making use of the Hausdorff-Young principle for Wiener amalgam spaces established in [18]. In fact, for .s ≥ 0 we can define .Hs (Rd ) as the inverse image under the Fourier-Plancherel transform of a weighted .L2 -space .L2vs (Rd ) with (submultiplicative weight) .vs (y) = (1 + |y|2 )s/2 . The natural norm is thus .‖f ‖Hs (Rd ) = ‖fˆ · vs ‖L2 (Rd ) = ‖fˆ‖L2 (Rd ) . In fact, it is a Hilbert space with a natural scalar vs product related to this norm. Lemma 8 For .s > d/2, we have the following amalgam version of the Sobolev embedding theorem: 

   (Hs (Rd ), ‖ · ‖Hs ͨ→ W (C0 , 𝓁2 )(Rd ), ‖ · ‖W (C0 ,𝓁2 )(Rd ) ͨ→ C0 (Rd ), ‖ · ‖∞ .

.

Proof Since L2vs (Rd ) = W (L2 , 𝓁2vs ) ͨ→ W (L2 , 𝓁1 ) = W (FL2 , 𝓁1 )

.

via the Cauchy-Schwarz inequality, using the fact that the sequence .(1/vs (k))k∈Zd belongs to .𝓁2 (Zd ) if .s > d/2, and Plancherel’s theorem, we find that by the Hausdorff-Young principle provided in [18] (as obviously .1 ≤ 2): Hs (Rd ) = F−1 (L2vs (Rd )) ⊂ W (FL1 , 𝓁2 )(Rd ) ⊂ W (C0 , 𝓁2 )(Rd ).

.

232

H. G. Feichtinger

Based on this embedding result, we can apply the various reconstruction formulas to data obtained by sampling a function in a Sobolev space along some lattice. In fact, for any .Λ ⊲ Rd we have the following: The mapping .f I→ (f (λ))λ∈Λ is a bounded linear mapping from .W (C0 , 𝓁2 )(Rd ) into .𝓁2 (Λ). Consequently, the combined mapping 

f I→

.

f (λ)Tλ g =

λ∈Λ



f (λ)δλ ∗ g

λ∈Λ

defines a bounded linear mapping from .W (C0 , 𝓁2 ) into .W (M, 𝓁2 ) ∗ W (C0 , 𝓁1 ) ⊂ W (C0 , 𝓁2 ), for any .g ∈ W (C0 , 𝓁1 ). Thus, it defines a bounded operator on 2 d .W (C0 , 𝓁 )(R ), with a corresponding norm estimate. For .Λ = aZd one can show that for any given .s > d/2, the space .

  h ∈ Hs (Rd ) | h(ak) = 0 ∀k ∈ Zd

 is a closed subspace of the (reproducing kernel) Hilbert space .(Hs (Rd ), ‖ · ‖Hs , and therefore, the interpolation problem for a given sequence of the form d .(f (ak))k∈Zd , with .f ∈ Hs0 (R ), has a minimal norm solution for any .s ≥ s0 > d/2 and .a > 0. In fact, for fixed parameters it can be shown (see [33]) that the reconstruction can be obtained by means of a suitable function L ϕs,a ∈ Hs (Rd ) ⊂ S0 (Rd ) ⊂ W (C0 , 𝓁1 )(Rd )

.

by the linear operator Qa,s (f ) := (f ·

.

L a ) ∗ ϕs,a ,

L is the unique Lagrange interpolator in the closed linear span of translates where .ϕs,a of the function characterizing the reproducing kernel for .Hs (Rd ), i.e., satisfying L d .ϕs,a (ak) = δ0,k , k ∈ Z . It is one of the main results of [33] that the solution of this problem depends continuously on these parameters.

Theorem 2 Given .s0 > d/2, then for any .f ∈ Hs0 (Rd ), the minimal norm interpolation of the data .(f (ak))k∈Zd with the parameters .(s, a) depends continuously (in the norm of .Hs0 (Rd ) and thus in .L2 ∩ C0 (Rd )) on the domain .[s0 , ∞) × (0, ∞). The tools used are a combination of properties of Wiener amalgam spaces, of shiftinvariant (respectively, spline-type spaces, as also described in [21]) and Wiener’s inversion theorem.

Sampling via the Banach Gelfand Triple

233

6 The Shannon Sampling Theorem While the classical Shannon sampling theorem starts from a Hilbert space setting and allows to push the sampling rate to its maximum (the well-known Nyquist rate), i.e., with our normalization of the Fourier transform, this corresponds to the sampling at the lattice points of .Zd if .spec(f ) ⊂ Ω := [−1/2, 1/2]d , it has some serious drawbacks if we consider it from a practical point of view. First of all, we have to note that .SINC∈ / L1(Rd ), and thus, one cannot hope for norm convergence of the partial sums in . L1(Rd ), ‖ · ‖1 if the given band-limited function is in .L1(Rd ) (and hence in .L2 (Rd ), due to the band-limitedness). In fact, any band-limited function .f ∈ L1(Rd ) belongs to the smaller space .S0 (Rd ), because we can find some .g ∈ S0 (Rd ) (e.g., some multidimensional De la Vallee Poussin kernel) such that . g (s) ≡ 1 on .Ω, and thus, .fˆ ·  g = fˆ whenever .supp(fˆ) ⊂ Ω, or .f = f ∗ g, and thus, by the convolution relations for Wiener amalgams, one has f ∈ L1(Rd ) ∗ S0 (Rd ) = W (L1 , 𝓁1 ) ∗ W (FL1, 𝓁1 ) ⊆ W (FL1, 𝓁1 ) = S0 (Rd ).

.

Hence, we have for band-limited functions f ∈ W (FL1, 𝓁1 )(Rd ) ⊂ W (FL1 , 𝓁2 )(Rd ) ͨ→ L2 (Rd )

.

and even with a norm estimate ‖f ‖2 ≤ CΩ,g ‖f ‖1 ,

.

f ∈ L1(Rd ), spec f ⊆ Ω.

For the formulation, we recall once more the notion of the orthogonal subgroup Λ⊥ ⊲ Rd :

.

d | χ (λ) ≡ 1, λ ∈ Λ}. Λ⊥ := {χ ∈ R

.

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d , and some lattice .Λ ⊲ Rd with the Theorem 3 Given a compact set .Ω ⊂ R ⊥ property that the .Λ -translates of .Ω are pairwise disjoint, then one can recover any 1 d .f ∈ L (R ) with .supp(fˆ) ⊂ Ω from the .Λ-samples of f by the series expansion f (t) =



.

f (λ)g(t − λ),

(26)

λ∈Λ

  with convergence in . S0 (Rd ), ‖ · ‖S0 , in particular absolutely for every .t ∈ Rd and   uniformly (in . C0 (Rd ), ‖ · ‖∞ ), for any .g ∈ S0 (Rd ) with . g (ω) ≡ 1 for .ω ∈ Ω and .supp( g ) ∩ λ⊥ + Ω = ∅ for any .λ⊥ ∈ Λ⊥ , .λ⊥ /= 0. Proof Any band-limited function in .L1(Rd ) belongs to .S0 (Rd ) ⊂ W (C0 , 𝓁1 )(Rd ) and for any fixed, compact set .Ω there exists .CΩ > 0 such that

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H. G. Feichtinger

f ∈ L1(Rd ), supp(fˆ) ⊂ Ω.

‖f ‖S0 ≤ CΩ ‖f ‖L1 ,

.

(27)

This also implies that .(f (λ))λ∈Λ ∈ 𝓁1 (Λ) and altogether  .

' |f (λ)| ≤ CΩ ‖f ‖L1 .

(28)

λ∈Λ

The task is to reconstruct f from . Λ ·f = same information on the Fourier transform side: F(

.

Λ

· f ) = CΛ



λ∈Λ f (λ)δλ ,

or equivalently, the

∗ fˆ ∈ S0' (Rd ).

Λ⊥

Then the properties of g resp. . g imply  g · [CΛ

.

Λ⊥

∗ fˆ] = fˆ.

Taking the inverse Fourier transform and recalling the identity .δλ ∗ g = Tλ g, we get  f = F−1 [F(f )] = F−1 ( g) ∗ (

.

Λ

· f) =

 λ∈Λ

 f (λ)δλ ∗ g =



f (λ)Tλ g,

λ∈Λ

  with absolute convergence in . S0 (Rd ), ‖ · ‖S0 . Remark 9 The above statement only grants (unconditional) convergence in the .w ∗sense inside of .S0' (Rd ). Thus, it guarantees at least that a complete recovery from the sampling values is possible. This leaves the question of convergence (which depends on more specific properties of the space .(B, ‖ · ‖B )) open.   For illustration let us discuss specifically the case .(B, ‖ · ‖B ) = Lp (Rd ), ‖ · ‖p , with .1 ≤ p < ∞. The restriction to the case .p < ∞ is due to the fact that there is only a chance for norm convergence in .(B, ‖ · ‖B ) if the  compactly supported  functions are dense in .(B, ‖ · ‖B ). This would be the case . C0 (Rd ), ‖ · ‖∞ (which   can be treated in a similar way), but not for . L∞ (Rd ), ‖ · ‖∞ . lemma goes back to the study of the cardinal series in  pThe following  . L (R), ‖ · ‖p given already by Gosselin in [43].   Lemma 9 Any band-limited function in . Lp (Rd ), ‖ · ‖p , with .1 ≤ p < ∞ can be recovered by means of the unconditionally norm convergent sum (26), which is unconditionally convergent in the norm of .W (C0 , 𝓁p )(Rd ). Proof Let us only sketch the proof; a detailed discussion of such results is given in [21] and [22]. First, we note that band-limited functions in .Lp (Rd ) belong to .W (C0 , 𝓁p )(Rd ) (with equivalence of norms), because such functions can be reproduced by convolution with functions in Wiener’s algebra .W (C0 , 𝓁1 )(Rd ). Hence, the sequence of

Sampling via the Banach Gelfand Triple

235

p d sampling values belongs to .𝓁p (Λ), or equivalently, .f · Λ ∈ W (M, 𝓁 )(R ), again with the corresponding norm estimate. Following the illustration in Fig. 1, we can recover the band-limited function .f ∈ Lp (Rd ) by convolution with a suitable band-pass filter function .g ∈ W (C0 , 𝓁1 )(Rd ):

f = f ∗ g = (f ·

.

Λ) ∗ g

∈ W (M, 𝓁p ) ∗ W (C0 , 𝓁1 ) ⊂ W (C0 , 𝓁p )(Rd ),

 which also implies that we have unconditional norm convergence in . Lp (Rd ), .‖ · ‖p .

7 Connections to Kantorovich Operators In this final section of our contribution, we would like to point to a couple of results which have been obtained already a few years ago and which have received considerable attention in recent years under the terminology of Kantorovich operators (see [5, 13, 16, 55, 61]). Earlier papers use the terms quasi-interpolation operators or generalized sampling operators (see [6, 8, 9, 65]). Basically, such operators can be understood as generalizations of the simple concept of piecewise linear interpolation of a (continuous) function based on its regular samples, producing (over .R) spline functions of order 1 (degree 2) which interpolate exactly over the grid .αZ. Obviously these operators approximate a uniformly continuous function in the sense of the sup-norm, as .a → 0. If one wants to use spline functions of higher order and follows the same rule, i.e., uses an approximation of the form Qϕ,a f (t) :=



.

f (ak)Tak ϕ

(29)

k∈Zd

for some function .ϕ (typically a B-spline, with some degree of smoothness and compact support), one can hope to approximate f also in other function spaces requiring higher-order smoothness, such as Sobolev spaces. As a benefit, the approximating function .Qϕ,a f is a smooth function and approximation is also valid for suitable Sobolev spaces. As a downside, .Qϕ,a f is not any more an interpolating function, hence the name quasi-interpolation. The proof is based on the shrinking support and the fact that the set of translates of .ϕ forms a partition of unity, a so-called BUPU. If one starts with a function which is not continuous by itself, e.g., a generic function .f ∈ Lp (Rd ), with .1 ≤ p ≤ ∞, then the application of such an operator is not meaningful (since the lattice .aZd is just a set of measure zero), and therefore, one has to carry out some smoothing before applying the generalized sampling operator.

236

H. G. Feichtinger

Wavelet theory provides a large variety of examples for such a situation. In fact, using the idea of an MRA (multi-resolution analysis), one describes the orthogonal projection onto a dilated version of the scale space .V0 (also called shift-invariant space or spline type space) in the form of such a quasi-interpolation operator. In such cases, .Tak ϕ forms a Riesz basic sequence for its closed linear span (see [14]) and the biorthogonal family is of the form .(Tak  ϕ ). The orthogonal projection is given by Pϕ,a f =



.

(f ∗  ϕ )(ak)Tak ϕ = Qϕ,a (f ∗  ϕ ),

f ∈ Lp (Rd ).

(30)

k∈Zd

The typical Kantorovich sampling operator (as it is treated in a series of papers, mostly by G. Vinti and his coauthors) is of a similar form, with different ingredients. Instead of the Riesz basic sequence assumption, the family of building blocks is obtained by dilating a given family .Tk ϕ which forms a partition of unity (again B-splines are a perfect example), but uses for the smoothing mostly the classical regularization by convolution with a Dirac sequence, obtained from the indicator function of a cube or parallel-epiped in .Rd . Typically such operators are of the form Kφ,ϕ,a =



.

(f ∗ φa )(ak)Tak ϕ,

(31)

k∈Zd

for suitable normalized and typically compactly supported pairs of functions .(ϕ, φ). Again, Wiener amalgam spaces allow us to understand the properties of such operators at a quite general level. The first step, namely, convolution by .φ (say), improves local properties of the function f while preserving the global behavior. The quasi-interpolation operators can be described as a sampling operator (i.e., essentially a pointwise multiplier with a Dirac comb . a ), followed by a convolution operator. We thus have (to give an example) Kφ,ϕ,a = [(f ∗ φ) ·

.

a ] ∗ ϕ.

(32)

This allows to make use of the following simple Wiener amalgam estimates, which are easy to use, as both convolutions and pointwise products can be taken coordinate-wise, first for the local and then for the global components: Whenever ∞ 1 d .φ ∈ W (L , 𝓁 )(R ), in particular, for bounded functions with compact support, like the boxcar function, or .φ in Wiener’s algebra .W (C0 , 𝓁1 )(Rd ), or even better 1 d d .φ ∈ S0 (R ) and .ϕ ∈ W (C0 , 𝓁 )(R ), we have f ∗ φ ∈ Lp ∗ W (L∞ , 𝓁1 ) ⊂ W (L1 , 𝓁p ) ∗ W (L∞ , 𝓁1 ) ⊆ W (C0 , 𝓁p )(⊂ Lp )

.

and thus after pointwise multiplication (f ∗ φ) ·

.

a

∈ W (C0 , 𝓁p ) · W (M, 𝓁∞ ) ⊂ W (M, 𝓁p ).

Sampling via the Banach Gelfand Triple

237

Using once more the convolution properties, we then derive that .f I→ Kφ,ϕ,a defines a bounded linear operator from .Lp (Rd ) into .W (C0 , 𝓁p )(Rd ), since [(f ∗ φ) ·

a] ∗ ϕ

.

∈ W (M, 𝓁p ) ∗ W (C0 , 𝓁1 ) ⊂ W (C0 , 𝓁p ).

For these simple situations the convolution relations have been already obtained in the early work of Fournier and Stewart [41] or Busby and Smith [7]. This author has used such estimates in two papers published in 1992 in the proceedings of a conference on function spaces ([21] and [22]), explaining the possible beneficial role of the use of Wiener amalgam spaces for sampling theory. Among others, such an approach allows us to give an easy qualitative analysis of jitter errors arising in the context of sampling theory. One of the main results (Theorem 8 in [22]) provides a proof for the convergence of Kantorovich operators   (they do not appear by this name at this place) in . Lp (Rd ), ‖ · ‖p . The proof only  (0) = d φ(t)dt = 1 and that it belongs assumes (using the above notation) that .φ R to .W (C0 , 𝓁1 )(Rd ), while .ϕ ∈ W (C0 , 𝓁1 )(Rd ) should satisfy the partition of unity property. In fact, the result was inspired by a corresponding result in the book of Aubin [3], which deals with  the case .p = 2. In the proof given in [22], the dilation invariance of the space . Lp (Rd ), ‖ · ‖p plays a crucial role. We leave it to the reader to go back to this paper in order to investigate details in that paper. Let us just cite Theorem 8 of that paper. It makes use of the natural notation .ρΨ for the compressed partition of unity, obtained by rescaling the elements of .Ψ. Using dilation operators we have the identity ρ d Stρ (Tk ψ) = D1/ρ (Tk ψ) = Tρk D1/ρ ψ.

.

(33)

 Theorem 4 Let .h ∈ W (C0 , 𝓁1 )(Rd ) with . Rd h(x)dx = 1 and some .ψ with compact support such that .Ψ := (Tk ψ)k∈Zd creates a BUPU  be given. Then  we have strong convergence of the Kantorovich operators on . Lp (Rd ), ‖ · ‖p , for .1 ≤ p < ∞ .

lim ‖f − SpρΨ (Stρ h ∗ f )‖Lp (Rd ) → 0,

ρ→0

f ∈ Lp (Rd ).

(34)

  7.1 Kantorovich Operators on S0 (Rd ), ‖ · ‖S0 Finally, we want   to show that convergence results for Kantorovich operators for S0 (Rd ), ‖ · ‖S0 can be easily obtained from a result derived by Feichtinger and Kaiblinger formulated in [30]. This paper is in fact an important cornerstone for the approximation of continuous problems (such as the approximate computation of the Fourier transform of a function in .S0 (Rd ) via the FFT algorithm, or the constructive

.

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realization of an approximate dual Gabor atom using finite-dimensional algorithms, as given in [53]) through constructive, finite-dimensional computations. As a warm-up, let us recall the definition of a quasi-interpolation operator (which will be mostly applied in a situation where the function .ψ generates a BUPU in 1 d .FL (R ) with respect to a given lattice .Λ. The symbol .SpΨ is mostly used in the context of irregular sampling and is a cornerstone to the introduction of convolution for bounded measures (in a functional analytic fashion) in [23], which is then further explored in [26]. Definition 9 Given a lattice .Λ ⊲ Rd , we define the quasi-interpolation or spline approximation operators by SpΨ(f ) = Qψ,Λ (f ) = (

.

Λ

· f) ∗ ψ =

⊥⊥⊥Λ(f )

∗ ψ.

(35)

Lemma 10 For any .ψ ∈ W (C0 , 𝓁1 )(Rd ) and any .Λ ⊲ Rd , the series defining .Qψ,Λ   is uniformly convergent and the operator .Qψ,Λ is bounded on . Cb (Rd ), ‖ · ‖∞ , with ‖Qψ,Λ (f )‖∞ ≤ CΛ ‖ψ‖W (C0 ,𝓁1 ) ‖f ‖∞ =: Cψ,Λ ‖f ‖∞

.

f ∈ Cb (Rd ).

(36)

Proof This estimate is an easy consequence of well-established facts concerning Wiener amalgam spaces and the corresponding convolution relations, which can be summarized in one line as the inclusion (combined with the corresponding norm estimates): First of all,  we observe that for any lattice .Λ ⊲ Rd the corresponding Dirac comb . Λ = λ∈Λ δλ is a translation-bounded Radon measure, i.e., belongs to ∞ d .W (M, 𝓁 )(R ). In fact, the estimate is given by (e.g.) the number .

sup #{λ∈Λ | λ ∈ BR (y)}, y∈Rd

i.e., an estimate on the number of lattice points inside of a ball of radius .R > 0 (for some fixed R, e.g., for .R = 1), independent from the center .y ∈ Rd . Since we have .( Λ · f ) ∗ ψ ∈ [W (M, 𝓁∞ ) · Cb ] ∗ W (C0 , 𝓁1 ) ⊂ W (M, 𝓁∞ ) ∗ W (C0 , 𝓁1 ) ⊂ W (C0 , 𝓁∞ ) = Cb , the estimate follows from the usual “coordinatewise convolution relations” established in [20]. Remark 10 Specializing to .1 ∈ Cb (Rd ) we note that for .ψ ∈ W (C0 , 𝓁1 )(Rd ), the partition of unity condition for the family .(ψλ )λ∈Λ is equivalent to the assumption that .Qψ,Λ (1) = 1 in this case. The Kantorovich sampling operators can be considered as a concatenation of a convolution operator (with a dilated boxcar function, i.e., of the indicator function of a parallel-epiped), combined with a quasi-interpolation operator. To fix notations let us write:

Sampling via the Banach Gelfand Triple

239

Definition 10 Kg,ψ (f ) = Qψ (g ∗ f ).

(37)

.

with a given .ρ > 0 we set Kg,ψ ρ (f ) = Qψ,ρZd (Stρ g ∗ f ) = [(Stρ g ∗ f ) ·

ρZd ] ∗ D1/ρ ψ.

.

(38)

Theorem 5 (i) For any .ψ ∈W (C0 , 𝓁1 )(Rd ) and .g ∈ L1(Rd ), the operator .Kg,ψ is a bounded operator on . S0 (Rd ), ‖ · ‖S0 . (ii) If in addition . g (0) = 1, and .(Tn ψn )n∈Zd defines a BUPU for some .ψ ∈ FL1 (Rd ) ∩ Cc (Rd ) ⊂ S0 (Rd ), then one has .

lim ‖Kg,ψ ρ (f ) − f ‖S0 = 0,

ρ→0

f ∈ S0 (Rd ).

(39)

Proof The first claim follows from the estimate ‖Kg,ψ (f )‖S0 ≤ ‖Qψ (g ∗ f )‖S0 ≤ |‖Qψ |‖S0 →S0 ‖g‖1 ‖f ‖S0 ,

.

f ∈ S0 (Rd ). (40)

The convergence result from the following estimate: g,ψ

.‖Kρ

(f ) − f ‖S0 ≤ ‖Qψ,ρZd (Stρ (g) ∗ f − f )‖S0 + ‖Qψ,ρZd (f ) − f ‖S0 ,

f ∈ S0 (Rd ).

(41) According to the main result of [30], the second term converges to 0 for .ρ → 0. The first term tends to zero for .ρ → 0 because  the family of  quasi-interpolation operators .Qψ,ρZd is uniformly bounded on . S0 (Rd ), ‖ · ‖S0 by Lemma 4.5 of 8 and the usual bounded approximate identities of the form .(St g) ρ ρ→0 in [30],  1 d . L (R ), ‖ · ‖1 act as expected on any Segal algebra, here in particular on   d . S0 (R ), ‖ · ‖S0 (for the convergence result for homogeneous Banach spaces, see [26]), using the convergence .limρ→0 ‖Stρ g ∗ f − f ‖S0 = 0.

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Part III

Frame Theory

A Survey of Fusion Frames in Hilbert Spaces L. Köhldorfer, P. Balazs, P. Casazza, S. Heineken, C. Hollomey, P. Morillas, and M. Shamsabadi

1 Introduction Hilbert space frames .(ϕi )i∈I are (possibly redundant) sequences of vectors in a Hilbert space satisfying a Parseval inequality (see Definition 1). This is a generalization of orthogonal expansions. This subject was introduced in the setting of nonharmonic Fourier series in 1952 in [51]. The topic then became dormant for 30 years until it was brought back to life in [47] in the setting of data processing. After this, the subject advanced rapidly and soon became one of the most active areas of research in applied mathematics. Frames were originally used in signal and image processing and later in sampling theory, data compression, time-frequency analysis, coding theory, and Fourier series. Today, there are everincreasing applications of frames to problems in pure and applied mathematics, computer science, engineering, medicine, and physics with new applications arising regularly (see [40, 42, 88] and the “Further Study” section). Redundancy plays a fundamental role in applications of frame theory. First, it gives much greater flexibility in the design of families of vectors for a particular problem not possible

L. Köhldorfer () · P. Balazs · C. Hollomey · M. Shamsabadi Acoustics Research Institute, Austrian Academy of Sciences, Vienna, Austria e-mail: [email protected]; [email protected]; [email protected]; [email protected] P. Casazza (retired) University of Missouri, Department of Mathematics, Columbia, Missouri S. Heineken Instituto de Investigaciones Matemáticas Luis A. Santaló (UBA-CONICET), Buenos Aires, Argentina e-mail: [email protected] P. Morillas Instituto de Matemáticas Aplicada San Luis (UNSL-CONICET), San Luis, Argentina © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. D. Casey et al. (eds.), Sampling, Approximation, and Signal Analysis, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-031-41130-4_11

245

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for linearly independent sets of vectors. In particular, the advantages of redundancy was shown in phase reconstruction, where we need to determine a vector from the absolute value of its frame coefficients [5, 6]. Second, we can mitigate the effects of noise in a signal and get robustness to erasures by spreading information over a wider range of vectors. For an introduction to frame theory, we recommend [40, 42, 88]. One can think of a frame as a family of weighted one-dimensional subspaces of a Hilbert space. As a consequence, frame theory has its limitations. In particular, they do not work well in distributed processing where one has to project a signal onto multidimensional subspaces. This requires fusion frames for the applications which include wireless sensor networks [60], visual and hearing systems [78], geophones in geophysics [46], distributed processing [41], packet encoding [12, 15, 32], parallel processing [30], and much more. Fusion frames also arise in various theoretical applications including the Kadison-Singer problem [29] and optimal subspace packings [27, 28]. All of these problems require working with a countable family of subspaces of a Hilbert space with positive weights and satisfying a type of Parseval inequality (see Definition 2). Fusion frames were introduced in [39] where they were called “frames of subspaces.” In a follow-up paper [41], the name was changed to “fusion frames” so they would not be confused with “frames in subspaces”. Because of their many applications, fusion frame theory has developed very rapidly over the years (see also [40]). Here, we give a new survey on this topic.

1.1 Motivation The motivation for this chapter is to present the fundamentals of fusion frame theory with an emphasis on results from frame theory which hold, or do not hold, for fusion frames. The idea is to help future researchers deal with the delicate differences between frames and fusion frames accurately. As such, proofs for fusion frames will be as self-contained as possible. Let us mention that there are very nice articles [39, 41]; we can recommend unrestricted; still we think that there is room for this survey. Finite frame and fusion frame theory became more and more important recently. It became apparent that there is a necessity for the implementation of fusion frame approaches. There are, of course, algorithms, but we provide a reproducible research approach—integrating the codes in the open-source Toolbox LTFAT [76, 85]. Fusion frames have a natural connection to domain decomposition methods and can be used for the solution of operator equations [75]. In [10] a direct fusion frame formulation is used for that. Here it can be observed that to use the more general definition of the coefficient spaces—not restricting to the Hilbert direct sum of the given subspaces—has advantages; see below. Here we adopt this approach, also used in [3]. Many concepts of classical frame theory have been generalized to the setting of fusion frames. Having a proper notion of dual fusion frames permits us to have different reconstruction strategies. Also, one wants from a proper definition to yield duality results similar to those known for classical frames. The duality concept for fusion frames is more elaborate but therefore also more interesting than the

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standard frame setting. This is why we decided to put some focus on this problem and summarized results from [57–59]. For this we have also chosen to make properties of operators between Hilbert spaces explicit and present the proofs of some results that were often only considered implicitly. So, in summary, we provide a new survey on fusion frame theory, aiming to provide an easier entry point into this theory for mathematicians.

1.2 Structure This chapter is structured as follows: Sect. 1 introduces and motivates this chapter. In particular, we present a list of differences between frame and fusion frame theory (which we present throughout the chapter) in the next subsection. Section 2 introduces notation, general preliminary results, and some basic facts on frame theory, which we apply on many occasions throughout these notes. Section 3 presents the notion of fusion frames, Bessel fusion sequences and their canonically associated frame-related operators. The reader familiar with frame theory will recognize many parallels between fusion frames and frames in this section. We also add a subsection on operators between Hilbert direct sums, which serves as a preparation for later and leads to a better understanding of many operators occurring in this chapter. In Sect. 4, we present other notions in fusion frame theory including generalizations of orthonormal bases, Riesz bases, exact frames, and minimal sequences to the fusion frame setting. In Sect. 5, we study fusion frames under the action of bounded operators, where we present a number of differences between frame and fusion frame theory. In Sect. 6, we present the theory of dual fusion frames. Duality theory probably provides the greatest contrast between frames and fusion frames. We first show the reader how the concept of dual fusion frame arrived. We study properties, present results, and provide examples of this concept. We discuss particular cases and other approaches. In Sect. 7 we focus on fusion frames in finite dimensional Hilbert spaces. In Sect. 8 we present thoughts about concrete implementations in the LTFAT toolbox [85] and a link to a reproducible research page. Finally, in Sect. 9 we list a collection of topics on fusion frames for further study, which are not presented in this chapter.

1.3 “There Be Dragons”: Traps to Be Avoided When Going from Frame Theory to Fusion Frame Theory In the following, we give an overview of the differences between properties of frames and properties of fusion frames presented in this chapter:

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• Existence of Parseval fusion frames: It is known that for every .m ≥ n ∈ N, there is a m-element Parseval frame for any n-dimensional Hilbert space [33] (and the vectors may even be equal norm). In the fusion frame setting, this is not true −1/2 (see Remark 1). In particular, applying .SV to the fusion frame does not yield a Parseval fusion frame in general (see Sect. 5). Currently, there are no necessary and sufficient conditions known for the existence of Parseval fusion frames for arbitrary dimensions of the subspaces. • Coefficient space: In contrast to frame theory, there is no natural coefficient space. It can either be chosen to be the Hilbert direct sum of the defining subspaces, which is the standard approach in fusion frame theory. But then the coefficient space is different for each fusion system, and a naive combination of analysis and synthesis is not possible anymore. Or it can be chosen to be the Hilbert direct sum of copies of the whole Hilbert space—the approach taken in this survey—but then the analysis operator associated to a fusion frame V maps 2 2 .H bijectively onto the subspace .KV and not the full coefficient space .K H (see Theorem 14). • Exact systems are not necessarily Riesz; see Proposition 5 or [39]. • Applying a bounded surjective operator on a fusion frame does not give a fusion frame, in general. Even for orthonormal fusion bases this is not true; see Sect. 5. • Duality: – Duality for fusion frames cannot be defined via .DW CV = IH as in the frame case, if one wants fusion frame duality to be compatible with frame duality; see Sect. 6.9. – For the dual fusion frame definition, an extra operator Q has to be inserted between .CW and .DV ; see Definition 5 and the extensive discussion in Sect. 6. – Duals of fusion Riesz bases are in general not unique; see Sect. 6.8. – The fusion frame operator of the canonical dual can differ from the inverse of the fusion frame operator; see Sect. 6.8. – The canonical dual of the canonical dual of a fusion frame V is not V in general; see Sect. 6.8. For a full treatment of fusion frame duality, see Sect. 6. • The frame operator of an orthonormal fusion basis .(Vi )i∈I is not necessarily the identity. This is only the case, if the associated weights are uniformly 1. In fact, the definition of an orthonormal fusion basis is independent of weights, whereas we have .SV 1 = IH for .V 1 = (Vi , 1)i∈I ; see Theorem 15.

2 Preliminaries Throughout this notes, .H is always a separable Hilbert space and all considered index sets are countable. For a closed subspace V of .H, .πV denotes the orthogonal projection onto V . The set of positive integers .{1, 2, 3, . . . } is denoted by .N and .δij denotes the Kronecker delta. The cardinality of a set J is denoted by .|J |. The

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domain, kernel, and range of an operator T is denoted by .dom(T ), .N(T ), and R(T ), respectively. .IX denotes the identity operator on a given space X. The set of bounded operators between two normed spaces X and Y is denoted by .B(X, Y ) and we set .B(X) := B(X, X). For an operator .U ∈ B(H1 , H2 ) (.H1 , .H2 Hilbert spaces) with closed range .R(U ) = R(U ), its associated pseudo-inverse [44] is denoted by † † ∈ B(H , H ), satisfying .U . The pseudo-inverse of U is the unique operator .U 2 1 the three relations

.

N(U † ) = R(U )⊥ , R(U † ) = N(U )⊥ , U U † x = x

.

(x ∈ R(U )).

(1)

Moreover, .U U † = πR(U) and .U † U = πR(U† ) . We also note that if U has a closed range, then so does .U ∗ and it holds .(U ∗ )† = (U † )∗ . On .R(U ) we explicitly have † ∗ ∗ −1 . In case U is bounded and invertible, it holds that .U † = U −1 . .U = U (U U ) We refer to [44] for more details on the pseudo-inverse of a closed range operator between Hilbert spaces. The following important preliminary results will be applied without further reference throughout this manuscript: Theorem 1 (Density Principle [54]) Let X and Y be Banach spaces and let V be a dense subspace of X. If .T : V −→ Y is bounded, then there exists a unique bounded extension .T˜ : X −→ Y of T satisfying .‖T˜ ‖ = ‖T ‖. Theorem 2 (Neumann’s Theorem [44]) Let X be a Banach space. If .T ∈ B(X) satisfies .‖IX − T ‖ < 1, then T is invertible. Theorem 3 (Bounded Inverse Theorem [44]) Any bounded and bijective operator between two Banach spaces has a bounded inverse. Theorem 4 (Uniform Boundedness Principle [44]) Let X be a Banach space and Y be a normed space. Suppose that .(Tn )n∈N is a family of bounded operators .Tn : X −→ Y and assume that .(Tn )n∈N converges pointwise (as .n −→ ∞) to some operator .T : X −→ Y , .T x := limn−→∞ Tn x. Then T defines a bounded linear operator and we have ‖T ‖ ≤ lim inf ‖Tn ‖ ≤ sup ‖Tn ‖ < ∞.

.

n∈N

n∈N

2.1 Frame Theory We begin with a review of frame theory and present a collection of those definitions and results which are the most important ones for this manuscript. In the later sections, we will apply these results mostly without further reference. We refer to [44] for the missing proofs and more details on frame theory.

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Definition 1 Let .ϕ = (ϕi )i∈I be a countable family of vectors in .H. Then .ϕ is called: • A frame for .H, if there exist positive constants .Aϕ ≤ Bϕ < ∞, called frame bounds, such that  2 .Aϕ ‖f ‖ ≤ |〈f, ϕi 〉|2 ≤ Bϕ ‖f ‖2 (∀f ∈ H) (2) i∈I

• A frame sequence, if it is a frame for .span(ϕi )i∈I • An .Aϕ -tight frame (sequence), or simply tight frame (sequence), if it is a frame (sequence), whose frame bounds .Aϕ and .Bϕ in (2) can be chosen to be equal • A Parseval frame (sequence), if it is a 1-tight frame (sequence) • An exact frame, if it ceases to be a frame whenever one of the frame vectors .ϕi is removed from .ϕ • A Bessel sequence for .H, if the right but not necessarily the left inequality in (2) is satisfied for some prescribed .Bϕ > 0. In this case we call .Bϕ a Bessel bound • A complete sequence, if .span(ϕi )i∈I = H • A Riesz basis, if it is a complete sequence for which there exist constants .0 < αϕ ≤ βϕ < ∞, called Riesz constants, such that

αϕ



.

i∈I

2       |ci |2 ≤  ci ϕi  ≤ βϕ |ci |2   i∈I

(3)

i∈I

for all finite scalar sequences .(ci )i∈I ∈ 𝓁00 (I ) • a minimal sequence, if for each .i ∈ I , .ϕi ∈ / span(ϕk )k∈I,k/=i . To any sequence .ϕ = (ϕi )i∈I of vectors in .H, we may associate the following canonical frame-related operators [11]: 2 • The operator  synthesis  ϕ ) ⊆ 𝓁 (I ) −→  .Dϕ : dom(D  H, where .dom(Dϕ ) = 2 (ci )i∈I ∈ 𝓁 (I ) : i∈I ci ϕi ∈ H and .Dϕ (ci )i∈I = i∈I ci ϕi . • The analysis operator .Cϕ : dom(Cϕ ) ⊆ H −→ 𝓁2 (I ), where .dom(Cϕ ) = {f ∈ H : (〈f, ϕi 〉)i∈I ∈ 𝓁2 (I )} and .Cϕ f = (〈f, ϕi 〉)i∈I .  • The ) ⊆ H −→ H, where .dom(Sϕ ) = f ∈ H :  frame operator.Sϕ : dom(Sϕ i∈I 〈f, ϕi 〉ϕi ∈ H and .Sϕ f = i∈I 〈f, ϕi 〉ϕi .

It can be shown that .ϕ is a Besselsequence with Bessel bound .Bϕ if and only if dom(Dϕ ) = 𝓁2 (I ) and .‖Dϕ ‖ ≤ Bϕ . In this case, .Dϕ∗ = Cϕ ∈ B(H, 𝓁2 (I )). If .ϕ is a frame, then .Sϕ is a bounded, self-adjoint, positive, and invertible operator, yielding the possibility of frame reconstruction via .IH = Sϕ Sϕ−1 = Sϕ−1 Sϕ , i.e.,

.

f =

.

  〈f, Sϕ−1 ϕi 〉ϕi = 〈f, ϕi 〉Sϕ−1 ϕi i∈I

i∈I

(∀f ∈ H).

(4)

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The family .( ϕi )i∈I := (Sϕ−1 ϕi )i∈I in (4) is again a frame (called the canonical dual frame). More generally, a frame .(ψi )i∈I , which satisfies f =

.

  〈f, ϕi 〉ψi = 〈f, ψi 〉ϕi i∈I

(∀f ∈ H),

(5)

i∈I

is called a dual frame of .ϕ. This means that frames yield (possibly nonunique and redundant) series expansions of elements in a separable Hilbert space similar to orthonormal bases, whereas in stark contrast to the latter, the frame vectors are not necessarily orthogonal and may be linearly dependent. Moreover, frames and Bessel sequences can often be characterized by properties of the frame-related operators. For instance, a frame is .Aϕ -tight if  and only if .Sϕ = Aϕ · IH . In this case, frame reconstruction reduces to .f = A−1 ϕ i∈I 〈f, ϕi 〉ϕi . Further results in this direction are given below: Theorem 5 ([43]) Let .ϕ = (ϕi )i∈I be a countable family of vectors in .H. Then the following are equivalent: (i) .ϕ is a Bessel sequence for .H. (ii) The synthesis operator .Dϕ is well-defined on .𝓁2 (I ) and bounded. (iii) The analysis operator .Cϕ is well-defined on .H and bounded. Theorem 6 ([43]) Let .ϕ = (ϕi )i∈I be a countable family of vectors in .H. Then the following are equivalent: (i) .ϕ is a frame (resp. Riesz basis) for .H. (ii) The synthesis operator .Dϕ is well-defined on .𝓁2 (I ), bounded, and surjective (resp. well-defined on .𝓁2 (I ), bounded, and bijective). (iii) The analysis operator .Cϕ is well-defined on .H, bounded, and injective and has a closed range (resp. well-defined on .H, bounded, and bijective). In particular, the latter result implies that every Riesz basis is a frame. Below we give other equivalent conditions for a frame being a Riesz basis. Recall that two families .(ϕi )i∈I and .(ψi )i∈I of vectors with the same index set are called biorthogonal, if .〈ϕi , ψj 〉 = δij for all .i, j ∈ I . Theorem 7 ([44]) Let .ϕ = (ϕi )i∈I be a frame for .H. Then the following are equivalent: (i) (ii) (iii) (iv) (iii) (iii)

ϕ is a Riesz basis for .H. ϕ is minimal. .ϕ is exact. .ϕ has a biorthogonal sequence. −1 .(ϕi )i∈I and .(Sϕ ϕi )i∈I are biorthogonal. The canonical dual frame .(Sϕ−1 ϕi )i∈I is the unique dual frame of .ϕ. . .

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Bessel sequences, frames, and Riesz bases can be characterized in terms of operator perturbations of orthonormal bases; see below. Theorem 8 ([44]) Let .H be a separable Hilbert space. Then the following hold: (i) The Bessel sequences for .H are precisely the families of the form .(U ei )i∈I , where .U ∈ B(H) and .(ei )i∈I is an orthonormal basis for .H. (ii) The frames for .H are precisely the families of the form .(U ei )i∈I , where .U ∈ B(H) is surjective and .(ei )i∈I is an orthonormal basis for .H. (iii) The Riesz bases for .H are precisely the families of the form .(U ei )i∈I , where .U ∈ B(H) is bijective and .(ei )i∈I is an orthonormal basis for .H. This allows us to formulate a result about which operators preserve certain properties: Corollary 1 ([44]) Let .H be a separable Hilbert space. Then the following hold: (i) If .(ϕi )i∈I is a Bessel sequence for .H and .U : H −→ H is bounded, then .(U ϕi )i∈I is a Bessel sequence for .H as well. (ii) If .(ϕi )i∈I is a frame for .H and .U : H −→ H is bounded and surjective, then .(U ϕi )i∈I is a frame for .H as well. (iii) If .(ϕi )i∈I is a Riesz basis for .H and .U : H −→ H is bounded and bijective, then .(U ϕi )i∈I is a Riesz basis for .H as well. We conclude this section with the following characterization of all dual frames of a given frame .ϕ, which stems from a characterization of all bounded left inverses of the analysis operator .Cϕ (analogous to Lemma 17) and an application of Theorem 8. Theorem 9 ([44]) Let .ϕ = (ϕi )i∈I be a frame for .H. Then the dual frames of .ϕ are precisely the families (ψi )i∈I =

.

Sϕ−1 ϕi

+ hi −

 k∈I

〈Sϕ−1 ϕi , ϕk 〉hk

,

(6)

i∈I

where .(hi )i∈I is a Bessel sequence for .H. By the above result and Theorem 7, a frame, which is not a Riesz basis, has infinitely many  dual frames. In case .ϕ is a Riesz basis, we see that the remainder terms .hi − k∈I 〈Sϕ−1 ϕi , ϕk 〉hk in (6) vanish, since .(ϕi )i∈I and .(Sϕ−1 ϕi )i∈I are biorthogonal in this case.

3 Fusion Frames In this section, we present the basic definitions and results regarding fusion frames, Bessel fusion sequences, and their relation to the fusion frame-related operators. Because this is a survey on this topic, we will give full proofs. The reader will

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notice many parallels between frames and fusion frames in this section. However, as we have already mentioned and will see later, fusion frame theory raises a lot of unexpected and interesting questions, which are nonexistent for Hilbert space frames. At first, we motivate the concept of fusion frames by generalizing the frame definition. We rewrite the terms .|〈f, ϕi 〉|2 from the frame inequalities (2) as follows: |〈f, ϕi 〉|2 = 〈f, ϕi 〉〈ϕi , f 〉  = 〈f, ϕi 〉ϕi , f

  ϕi ϕi 2 ,f . = ‖ϕi ‖ f, ‖ϕi ‖ ‖ϕi ‖

.

(7)

  If we set .Vi := span{ϕi } = span{ϕi }, then the singleton . ‖ϕi ‖−1 · ϕi is an orthonormal basis for .Vi . In particular, the orthogonal projection .πVi of .H onto  the one-dimensional subspace .Vi is given by .πVi f = f, ‖ϕϕii ‖ into (7) and setting .vi := ‖ϕi ‖ yields

ϕi ‖ϕi ‖ .

Inserting that

|〈f, ϕi 〉|2 = vi2 〈πVi f, f 〉

.

= vi2 〈πV2i f, f 〉 = vi2 〈πVi f, πVi f 〉 = vi2 ‖πVi f ‖2 . This allows us to rewrite the frame inequalities to [41, Proposition 2.14] Aϕ ‖f ‖2 ≤



.

vi2 ‖πVi f ‖2 ≤ Bϕ ‖f ‖2

(∀f ∈ H).

i∈I

Therefore, we see that frames can also be viewed as weighted sequences of 1dimensional closed subspaces satisfying the inequalities (2). Admitting arbitrary weights—in the sense of weighted frames [8]—and, in particular, arbitrary closed subspaces leads to the definition of a fusion frame and related notions [39]: Definition 2 Let .V = (Vi , vi )i∈I be a countable family of closed subspaces .Vi of H and weights .vi > 0. Then V is called:

.

• A fusion frame for .H, if there exist positive constants .0 < AV ≤ BV < ∞, called fusion frame bounds, such that AV ‖f ‖2 ≤



.

vi2 ‖πVi f ‖2 ≤ BV ‖f ‖2

(∀f ∈ H)

i∈I

• A fusion frame sequence, if it is a fusion frame for .H := span(Vi )i∈I

(8)

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• An .AV -tight fusion frame (sequence), or simply tight fusion frame (sequence), if it is a fusion frame (sequence), whose fusion frame bounds .AV and .BV in (8) can be chosen to be equal • A Parseval fusion frame (sequence), if it is a 1-tight fusion frame (sequence). • An exact fusion frame, if it is a fusion frame, that ceases to be a fusion frame whenever one component .(Vk , vk ) is removed from V • A Bessel fusion sequence for .H, if the right but not necessarily the left inequality in (8) is satisfied for some .BV > 0. In this case we call .BV a Bessel fusion bound Remark 1 (a) For any Bessel fusion sequence .V = (Vi , vi )i∈I (and hence any fusion frame), we will always implicitly assume (without loss of generality) that .Vi /= {0} for all .i ∈ I , because a zero space .Vi = {0} does not contribute to the sum in (8) anyway. This is done to avoid case distinctions in some of the theoretical results. (b) As a first glimpse at the differences between frames and fusion frames, consider the following: It is known that for every .m, n ∈ N, there is a m-element Parseval frame for any n-dimensional Hilbert space [33] (and the vectors may even be equal norm). The situation for fusion frames is much more complicated. For example, there is no Parseval fusion frame .V = (Vi , vi )2i=1 for .R3 consisting of two 2-dimensional subspaces for any choice of weights. This is true since, in this case, there must exist a unit vector f contained in the 1-dimensional space 2 2 2 2 2 2 2 .V1 ∩ V2 and so .v ‖πV1 f ‖ + v ‖πV2 f ‖ = (v + v )‖f ‖ , while a unit vector 1 2 1 2 2 2 2 2 2 2 2 .f ∈ V1 \ V2 will satisfy .v ‖πV1 f ‖ + v ‖πV2 f ‖ < (v + v )‖f ‖ , and thus, 1 2 1 2 we cannot have .SV = IH (see Corollary 4). Currently, there are no necessary and sufficient conditions known for the existence of Parseval fusion frames for arbitrary dimensions of the subspaces. (c) A converse to our motivation for fusion frames via frames is the following: Assume that .V = (Vi , vi )i∈I is a fusion frame such that .dim(Vi ) = 1 for all .i ∈ I . Then, for every i, there exists a suitable vector .ϕi , such that .Vi = span{ϕi } = span{ϕi } and .πVi f = 〈f, ‖ϕi ‖−1 · ϕi 〉‖ϕi ‖−1 · ϕi . It is now easy to see that V being a fusion frame is equivalent to .(vi ‖ϕi ‖−1 · ϕi ) being a frame. (d) If .V = (Vi , vi )i∈I is a fusion frame for .H, then the family .(Vi )i∈I of subspaces .Vi is complete, i.e., it holds .span(Vi )i∈I = H. To see this, assume, for the contrary, that there exists a nonzero .g ∈ (span(Vi )i∈I )⊥ . Then this g would violate the lower fusion frame inequality in (8). A different motivation for fusion frames comes from the following problem:

•

? Goal

Let .(Vi )i∈I be a family of closed subspaces of .H and assume that for every .i ∈ I , (ϕij )j ∈Ji is a frame for .Vi . Give necessary and sufficient conditions on the subspaces .Vi , such that the collection .(ϕij )i∈I,j ∈Ji of all local frames (including all resulting repetitions of the vectors .ϕij ) forms a global frame for the whole Hilbert space .H. .

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The following result shows that the notion of a fusion frame is perfectly fitting for this task. Theorem 10 ([41]) Let .(Vi )i∈I be a family of closed subspaces of .H and .(vi )i∈I be a family of weights. For every .i ∈ I , let .(ϕij )j ∈Ji be a frame for .Vi with frame bounds .Ai and .Bi , and suppose that .0 < A = infi∈I Ai ≤ supi∈I Bi = B < ∞. Then the following are equivalent: (i) .(Vi , vi )i∈I is a fusion frame for .H. (ii) .(vi ϕij )i∈I,j ∈Ji is a frame for .H. In particular, if .(Vi , vi )i∈I is a fusion frame with bounds .AV ≤ BV , then (vi ϕij )i∈I,j ∈Ji is a frame for .H with frame bounds .AAV ≤ BBV . Conversely, if .(vi ϕij )i∈I,j ∈Ji is a frame for .H with bounds .Avϕ ≤ Bvϕ , then .(Vi , vi )i∈I is a fusion A B frame with fusion frame bounds . Bvϕ ≤ Avϕ . .

Proof Since .(ϕij )j ∈Ji is a frame for .Vi , we formally see that A



.

vi2 ‖πVi f ‖2 ≤

i∈I



vi2 Ai ‖πVi f ‖2

i∈I

≤

 i∈I

≤



vi2



|〈πVi f, ϕij 〉|2 =

j ∈Ji

vi2 Bi ‖πVi f ‖2 ≤ B

i∈I





|〈f, vi ϕij 〉|2

i∈I j ∈Ji

vi2 ‖πVi f ‖2 .

i∈I

Thus, if .(Vi , vi )i∈I is a fusion frame for .H with bounds .AV ≤ BV , then we obtain AAV ‖f ‖2 ≤



.

|〈f, vi fij 〉|2 ≤ BBV ‖f ‖2 .

i∈I j ∈Ji

Conversely, if .(vi ϕij )i∈I,j ∈Ji is a frame for .H with frame bounds .Avϕ ≤ Bvϕ , then .

 Avϕ Bvϕ ‖f ‖2 . ‖f ‖2 ≤ vi2 ‖πVi f ‖2 ≤ A B i∈I

This completes the proof.

⨆ ⨅

The special case .A = Ai = K = Bi = B, for every .i ∈ I , in the latter result yields the following corollary. Corollary 2 ([41]) Let .(Vi )i∈I be a family of closed subspaces of .H and .(vi )i∈I be a family of weights. For every .i ∈ I , let .(ϕij )j ∈Ji be a Parseval frame for .Vi . Then, for every .K > 0, the following are equivalent: (i) .(Vi , vi )i∈I is a K-tight fusion frame for .H. (ii) .(vi ϕij )i∈I,j ∈Ji is a K-tight frame for .H.

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This—corresponding to the approach in many applications that one considers a fusion frame with local frames for each of its subspaces—leads to the following definition; see also Sects. 6.3 and 8: Definition 3 ([41]) Let .V = (Vi , vi )i∈I be a fusion frame (Bessel fusion sequence), and let .ϕ (i) := (ϕij )j ∈Ji be a frame for .Vi for .i ∈ I with frame bounds .Ai and .Bi . Assume that .A = inf Ai > 0 and .B = sup Bi < ∞. Then (i) ) .(Vi , vi , ϕ i∈I is called a fusion frame system (Bessel fusion system). In the following, we collect a few observations on the weights .vi associated to a Bessel fusion sequence. For that, the following notion will be useful: We call a family .(vi )i∈I of weights semi-normalized (see also [7]), if there exist positive constants .m ≤ M such that .m ≤ vi ≤ M for all .i ∈ I . Lemma 1 ([59]) Let .V = √ (Vi , vi )i∈I be a Bessel fusion sequence with Bessel fusion bound .BV . Then .vi ≤ BV for all .i ∈ I . Proof Since we always assume .Vi /= {0} for a Bessel fusion sequence .(Vi , vi )i∈I , we may choose some nonzero .f ∈ Vi and see that 2  2  2  vi2 ‖f ‖2 = vi2 πVi f  ≤ vi πVi f  ≤ BV ‖f ‖2

.

i∈I

for every .i ∈ I . Dividing by .‖f ‖2 yields the result.

⨆ ⨅

By Lemma 1, the weights .vi associated to a fusion frame .V = (Vi , vi )i∈I are uniformly bounded from above by the square root of its upper frame bound. However, .(vi )i∈I needs not to be semi-normalized in general. An easy counterexample is the Parseval fusion frame .(H, 2−n )∞ n=1 for .H. Lemma 2 Let .(wi )i∈I be a semi-normalized family of weights with respect to the constants .m ≤ M. Then .(Vi , vi )i∈I is a fusion frame for .H if and only if .(Vi , vi wi )i∈I is a fusion frame for .H. In particular, .(Vi , vi )i∈I is a Bessel fusion sequence for .H if and only if .(Vi , vi wi )i∈I is a Bessel fusion sequence for .H. Proof Similar to the proof of Theorem 10, the statement follows from m2



.

i∈I

vi2 ‖πVi f ‖2 ≤

 i∈I

vi2 wi2 ‖πVi f ‖2 ≤ M 2



vi2 ‖πVi f ‖2

(∀f ∈ H).

i∈I

⨆ ⨅ Next, we introduce the fusion frame-related operators and characterize Bessel fusion sequences and fusion frames with them. The canonical representation space for frames is the sequence space .𝓁2 (I ). In the theory of fusion frames, we consider generalizations of .𝓁2 (I ). For a family .(Vi )i∈I of Hilbert spaces (e.g., .Vi closed subspaces of .H), the Hilbert direct sum   . i∈I ⊕Vi 𝓁2 , also called the direct sum of the Hilbert spaces .Vi [23, 45], is defined by

A Survey of Fusion Frames in Hilbert Spaces

 .

⊕Vi

257

 = (fi )i∈I ∈ (Vi )i∈I :

i∈I



 ‖fi ‖ < ∞ . 2

i∈I

𝓁2

It is easy to verify that for .(fi )i∈I , (gi )i∈I ∈

 i∈I

⊕Vi

〈(fi )i∈I , (gi )i∈I 〉(i∈I ⊕Vi ) 2 :=

.



𝓁

 𝓁2

, the operation

〈fi , gi 〉

i∈I

  defines an inner product on . i∈I ⊕Vi 𝓁2 (see also [45]), yielding the norm ‖(fi )i∈I ‖2(

.

i∈I

⊕Vi )𝓁2

=



‖fi ‖2 .

(9)

i∈I

2 Moreover,  adapting  any elementary completeness proof for .𝓁 (I ) yields that the space . i∈I ⊕Vi 𝓁2 is complete with respect to the norm (9), i.e., a Hilbert space   (see, e.g., [62]). Note that . i∈I ⊕C 𝓁2 = 𝓁2 (I ). In analogy to the dense inclusion 00 2 .𝓁 (I ) ⊆ 𝓁 (I ), the space of finite vector sequences

 .

= {(fi )i∈I ∈ (Vi )i∈I : fi /= 0 for only finitely many i}

⊕Vi

i∈I

𝓁00

  is a dense subspace of . i∈I ⊕Vi 𝓁2 . For the remainder of this manuscript, we abbreviate



  2 00 .KV := ⊕Vi and KV := ⊕Vi i∈I

i∈I

𝓁2

𝓁00

and analogously K2H :=



.

i∈I

⊕H

and 𝓁2

K00 H :=

 i∈I

⊕H

. 𝓁00

We will continue our discussion on Hilbert direct sums in Sect. 3.1. For now, we proceed with the definition and basic properties of the fusion frame-related operators. Let .V = (Vi , vi )i∈I be an arbitrary weighted sequence of closed subspaces in .H. The following canonical operators associated to V play a central role in fusion frame theory: • The synthesis operator .DV : dom(DV ) ⊆ K2H −→ H, where .dom(DV ) =     (fi )i∈I ∈ K2H : i∈I vi πVi fi converges and .DV (fi )i∈I = i∈I vi πVi fi

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 • The analysis operator .CV : dom(CV ) ⊆ H −→ K2H , where .dom(CV ) = f ∈  H : (vi πVi f )i∈I ∈ K2H and .CV f = (vi πVi f )i∈I  • The fusion frame operator .SV : dom(SV ) ⊆ H −→ H, where .dom(SV ) = f ∈    H : i∈I vi2 πVi f converges and .SV f = i∈I vi2 πVi f The synthesis operator is always densely defined, since .K00 H ⊆ dom(DV ) is dense in .K2H , and we always have .DV∗ = CV (see below for the bounded case). On the other hand, one can show [84] that the analysis operator .CV , in general, is not necessarily densely defined. If .CV is densely defined (or, equivalently, .DV is closable), then .DV ⊆ CV∗ , and in this case, .DV = CV∗ . Moreover, one can show that always .SV = DV CV and that .SV is symmetric on its domain. We refer to [84] for more details on these (possibly unbounded) operators associated to general subspace sequences. For our purpose it is sufficient to treat the bounded case only (compare with Theorem 11). Lemma 3 ([39]) Let .V = (Vi , vi )i∈I be a weighted sequence of closed subspaces in .H. If .DV ∈ B(K2H , H) then .DV∗ = CV ; if .CV ∈ B(H, K2H ), then .CV∗ = DV ; and either of the two latter cases implies that .SV = DV CV ∈ B(H) is self-adjoint. Proof For .g ∈ H and .(fi )i∈I ∈ K2H , we have  ∗  DV g, (fi )i∈I K2 = g, DV (fi )i∈I H  〈g, vi πVi fi 〉 =

.

i∈I

=



〈vi πVi g, fi 〉

i∈I

 = (vi πVi g)i∈I , (fi )i∈I K2 H   = CV g, (fi )i∈I K2 = g, CV∗ (fi )i∈I K2 . H

This implies the first two statements and the rest is clear.

H

⨆ ⨅

Theorem 11 ([39]) Let .V = (Vi , vi )i∈I be a weighted sequence of closed subspaces of .H. Then the following are equivalent: (i) V is a Bessel fusion sequence with Bessel fusion bound .BV . 2 (ii) The synthesis operator √.DV associated to V is a bounded operator from .KH into .H with .‖DV ‖ ≤ BV . (iii) The analysis operator √ .CV associated to V is a bounded operator from .H into 2 .K H with .‖CV ‖ ≤ BV . (iv) The fusion frame operator .SV associated to V is bounded on .H with .‖SV ‖ ≤ BV . Proof (i) .⇒ (ii) Assume that V is a Bessel fusion sequence with Bessel fusion bound .BV . Then for any finite vector sequence .h = (hi )i∈J ∈ K00 H , we have

A Survey of Fusion Frames in Hilbert Spaces

259

 2  2          . vi πVi hi  = sup  vi πVi hi , g     ‖g‖=1  i∈J

i∈J

 2     = sup  vi 〈hi , πVi g〉  ‖g‖=1  i∈J

≤ sup

‖g‖=1

≤ sup

‖g‖=1



2 vi ‖πVi g‖‖hi ‖

i∈J



vi2 ‖πVi g‖2

i∈J

 i∈J

‖hi ‖2

≤ BV ‖h‖2K2 . H

√ 00 of This implies that .DV is bounded on .K00 H by . BV . Since .KH is a dense subspace √ 2 2 .K , .DV extends to a bounded operator .DV : K −→ H with .‖DV ‖ ≤ B . V H H (ii) .⇒ (iii) .⇒ (iv) This follows from Lemma 3. (iv) .⇒ (i) If .SV is bounded with .‖SV ‖ ≤ BV , then for all .f ∈ H we have  .

vi2 ‖πVi f ‖2 = 〈SV f, f 〉 ≤ ‖SV f ‖‖f ‖ ≤ ‖SV ‖‖f ‖2 ≤ BV ‖f ‖2 ,

i∈I

⨆ ⨅

i.e., V is a Bessel fusion sequence with Bessel fusion bound .BV .

Remark 2 The proof of Theorem 11 reveals that for any Bessel fusion sequence  V = (Vi , vi )i∈I , the sum . i∈I vi2 πVi fi converges unconditionally in .H for all 2 .(fi )i∈I ∈ K . H .

By Theorem 11, the fusion frame operator .SV is bounded on .H, whenever .V = (Vi , vi )i∈I is a Bessel fusion sequence. If, in addition, V is a fusion frame, then .SV is invertible and thus yields the possibility of perfect reconstruction; see Theorem 12 below.

In analogy to frame theory, fusion frames enable perfect reconstruction without the assumption that the subspaces .Vi are orthogonal. This can be seen as the most important advantage of fusion frames in comparison with orthogonal subspace decompositions.

Theorem 12 ([39]) Let .V = (Vi , vi )i∈I be a fusion frame with fusion frame bounds AV ≤ BV . Then the fusion frame operator .SV is bounded, self-adjoint, positive, and invertible with

.

AV ≤ ‖SV ‖ ≤ BV

.

and

BV−1 ≤ ‖SV−1 ‖ ≤ A−1 V .

(10)

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In particular f =



.

vi2 πVi SV−1 f =

i∈I



vi2 SV−1 πVi f

(∀f ∈ H)

(11)

i∈I

where both series converge unconditionally. Proof If V is a fusion frame, then V in particular is a Bessel fusion sequence. Hence, Theorem 11 and Lemma 3 imply that .SV is bounded and self-adjoint. Moreover, by the definition of .SV , we may rewrite the fusion frame inequalities (8) to 〈AV f, f 〉 ≤ 〈SV f, f 〉 ≤ 〈BV f, f 〉

.

(∀f ∈ H).

(12)

Thus, .SV is positive. Furthermore, by manipulating (12), we obtain

  BV − AV f, f 0 ≤ (IH − BV−1 SV )f, f ≤ BV

.

(∀f ∈ H).

and thus, since .IH − BV−1 SV is self-adjoint, ‖IH − BV−1 SV ‖ = sup 〈(IH − BV−1 SV )f, f 〉 ≤

.

‖f ‖=1

BV − AV < 1. BV

Now, an application of Neumann’s theorem yields that .BV−1 SV is invertible; hence, −1 −1 .SV is invertible too. In particular, for all .f ∈ H, we have .f = SV S V f = SV SV f , which implies (11) with unconditional convergence (see Remark 2). Finally, the left-hand side of (10) follows immediately from (12). To show the right-hand side of (10), observe that we have .‖SV−1 ‖ ≥ ‖SV ‖−1 ≥ BV−1 and that .AV ‖SV−1 f ‖2 ≤ ⨆ ⨅ 〈SV SV−1 f, SV−1 f 〉 = 〈f, SV−1 f 〉 ≤ ‖SV−1 ‖‖f ‖2 implies .‖SV−1 ‖ ≤ A−1 V . So, for any fusion frame V , we have the possibility of perfect reconstruction of any .f ∈ H via (11), where the subspaces .Vi need not be orthogonal and can have a nontrivial intersection. In particular, we can combine fusion frame reconstruction with local frame reconstruction in a fusion frame system. More precisely, given a fusion frame system .(Vi , vi , ϕ (i) )i∈I with associated global frame .vϕ, we obtain several possibilities to perform perfect reconstruction via the frame vectors .vi ϕij (see Definition 3), which demonstrates that fusion frames are tremendously useful for distributed processing tasks [41]. Lemma 4 ([41]) Let .V = (Vi , vi , ϕ (i) )i∈I be a fusion frame system in .H with global frame .vϕ. If .(ϕijd )j ∈Ji is a dual of .ϕ (i) for every .i ∈ I , then: (i) .(SV−1 vi ϕijd )i∈I,j ∈Ji is a dual frame of the global frame .vϕ. In particular

A Survey of Fusion Frames in Hilbert Spaces

f =

.

261

 f, vi ϕij SV−1 vi ϕijd

(∀f ∈ H).

i∈I j ∈Ji

(ii) .SV = Dvϕ d Cvϕ = Dvϕ Cvϕ d , where .vϕ d = (vi ϕijd )i∈I,j ∈Ji . Proof (i) For every .f ∈ H it holds .πVi f = 

f =

.

 i∈I

vi2 SV−1 πVi f =

 f, ϕij ϕijd , and hence,

 f, vi ϕij SV−1 vi ϕijd . i∈I j ∈Ji

i∈I

(ii) Similarly, we see that SV f =



.

vi2 πVi f =

 f, vi ϕij vi ϕijd , i∈I j ∈Ji

i∈I

where we note that .vϕ d is indeed a frame for .H by Theorem 10.

⨆ ⨅

In view of applications, knowing the action of the inverse fusion frame operator SV−1 on elements .f ∈ H is of utmost importance. If one prefers to avoid the inversion of .SV , approximation procedures such as the fusion frame algorithm (see Lemma 22) can be performed. However, for tight fusion frames, the computation of −1 .S V becomes fairly simple, and thus, fusion frame reconstruction can be computed with ease; see (13) and (14). First, we note the following reformulation of the inequalities (12). .

Corollary 3 ([41]) Let .V = (Vi , vi )i∈I be a weighted sequence of closed subspaces of .H and let .0 < AV ≤ BV < ∞. Then the following are equivalent: (i) V is a fusion frame with fusion frame bounds .AV ≤ BV . (ii) .AV · IH ≤ SV ≤ BV · IH (in the sense of positive operators). Corollary 4 ([41]) Let .V = (Vi , vi )i∈I be a weighted sequence of closed subspaces of .H. Then the following hold. (i) V is an .AV -tight fusion frame if and only if .SV = AV · IH . (ii) V is a Parseval fusion frame if and only if .SV = IH . In particular, for an .AV -tight fusion frame we have f =

.

1  2 vi πVi f AV

(∀f ∈ H),

(13)

i∈I

and for a Parseval fusion frame f =



.

i∈I

vi2 πVi f

(∀f ∈ H).

(14)

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L. Köhldorfer et al.

Similar to Theorem 11, we can characterize fusion frames in terms of their associated canonical operators. The crucial observation for this characterization is ‖CV f ‖2K2 =



.

H

vi2 ‖πVi f ‖2 .

(15)

i∈I

Theorem 13 ([41]) Let .V = (Vi , vi )i∈I be a weighted sequence of closed subspaces of .H. Then the following are equivalent: (i) V is a fusion frame for .H. (ii) The synthesis operator .DV is well-defined on .K2H , bounded, and surjective. (iii) The analysis operator .CV is well-defined on .H, bounded, and injective and has a closed range. Proof The equivalence (ii) .⇔ (iii) is generally true for a bounded operator between Hilbert spaces and its adjoint [45]. All conditions imply boundedness, as we will see in the following. (i) .⇒ (ii) If V is a fusion frame, then .DV ∈ B(K2H , H) by Theorem 11. If .DV was not surjective, then there would exist some nonzero .h ∈ R(DV )⊥ = N(DV∗ ) = N(CV ). However, inserting this h into Eq. (15) and the lower fusion frame inequality (8) would lead to a contradiction. (iii) .⇒ (i) If (ii) and (iii) hold, then .CV is bounded below on .H [45] and also bounded on .H (by Lemma 3). This means that the fusion frame inequalities are satisfied for all .f ∈ H. ⨆ ⨅ As we have seen above, the frame-related operators of fusion frames share many properties with the frame-related operators associated to frames (see Theorem 6). Analogously as in classical frame theory, there are—of course—other characterizations of weighted families of closed subspaces being a fusion frame via these operators. For instance, the following result gives a characterization of fusion frames and their frame bounds via their synthesis operator. Proposition 1 ([58]) Let .V = (Vi , vi )i∈I be a weighted family of closed subspaces in .H. Then V is a fusion frame for .H with bounds .AV ≤ BV if and only if the following two conditions are satisfied: (i) .(Vi )i∈I is complete, i.e., .span(Vi )i∈I = H. (ii) The synthesis operator .DV is well-defined on .K2H and AV ‖(fi )i∈I ‖2 ≤ ‖DV (fi )i∈I ‖2 ≤ BV ‖(fi )i∈I ‖2

.

(∀(fi )i∈I ∈ N(DV )⊥ ). (16)

Proof First assume that V is a fusion frame with fusion frame bounds .AV ≤ BV . By Remark 1 (d), .(Vi )i∈I is complete and Theorem 13 implies that the upper inequality in (16) is satisfied and that .R(CV ) is closed, since .R(DV ) is closed. The latter implies that .N(DV )⊥ = R(CV ) = R(CV ), i.e., .N(DV )⊥ consists all of families of the form .(vi πVi f )i∈I , where .f ∈ H. Now, for arbitrary .f ∈ H, we see that

A Survey of Fusion Frames in Hilbert Spaces

 .

263

2 vi2 ‖πVi f ‖2

= |〈SV f, f 〉|2 ≤ ‖SV f ‖2 ‖f ‖2 ≤ ‖SV f ‖2

i∈I

1  2 vi ‖πVi f ‖2 . AV i∈I

This implies that AV



.

vi2 ‖πVi f ‖2 ≤ ‖SV f ‖2 = ‖DV (vi πVi f )i∈I ‖2 ,

i∈I

as desired. Conversely, assume that (i) and (ii) are satisfied. Since .K2H = N(DV )⊕N(DV )⊥ , (ii) implies that the synthesis operator .DV is norm-bounded on .K2H by .BV . Thus, by Theorem 11, V is a Bessel fusion sequence with Bessel fusion bound .BV and it only remains to show the lower fusion frame inequality with respect to .AV . We first show that .R(DV ) = H. By the validity of (i) and .span(Vi )i∈I ⊆ R(DV ) ⊆ H, the claim follows if we show that .R(DV ) is closed. To this end, let .(gn )∞ n=1 be ∞ , where .(h )∞ a Cauchy sequence in .R(DV ). Note that .(gn )∞ = (D h ) V n n=1 n n=1 n=1 is a sequence in .N(DV )⊥ , and the lower inequality in (16) implies that .(hn )∞ n=1 is a Cauchy sequence in .N(DV )⊥ . Since .N(DV )⊥ is a closed subspace of .K2H , ∞ converges to some .h ∈ N(D )⊥ . By continuity of .D , this means that .(hn ) V V n=1 ∞ .(gn ) n=1 converges to .DV h in .R(DV ). Thus, .R(DV ) is closed and .R(DV ) = H follows. Note that due to Theorem 13 we can already deduce that V is a fusion frame. In order to establish that .AV is a lower fusion frame bound, recall from (1) that .DV† DV = πN(DV )⊥ and .DV DV† = πR(DV ) = IH . Hence, (ii) implies that for any .(fi )i∈I ∈ K2H we can estimate A‖DV† DV (fi )i∈I ‖2 ≤ ‖DV DV† DV (fi )i∈I ‖2 = ‖DV (fi )i∈I ‖2 .

.

Since also .N(DV† ) = R(DV )⊥ = {0}, the latter implies that .‖DV† ‖2 ≤ A−1 V . In † particular, this yields .‖(DV∗ )† ‖2 = ‖CV† ‖2 ≤ A−1 . Since .C CV is the orthogonal V V projection onto .R(CV† ) = R((DV† )∗ ) = N(DV† )⊥ = R(DV ) = H, we finally see that for arbitrary .f ∈ H it holds ‖f ‖2 = ‖CV† CV f ‖2 ≤

.

1  2 1 ‖CV f ‖2 = vi ‖πVi f ‖2 , AV AV i∈I

i.e., .AV is a lower fusion frame bound for V .

⨆ ⨅

3.1 Operators Between Hilbert Direct Sums Before we proceed with the next section, we provide an excursion into the realm of linear operators between Hilbert direct sums and their matrix representations. The

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L. Köhldorfer et al.

results presented here mainly serve as a preparation for several proof details in the subsequent sections. Thus, the reader who is only interested in the main results for fusion frames may skip this subsection.   Instead of considering spaces of the type .K2V = i∈I ⊕Vi 𝓁2 , where each .Vi is a closed subspace of a given Hilbert space .H, we may consider the slightly more general case, where .Vi are arbitrary Hilbert spaces, and set  .

i∈I



⊕Vi

:= (fi )i∈I : fi ∈ Vi ,



 ‖fi ‖2Vi