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Frames and Harmonic Analysis 706 Frames and Harmonic Analysis AMS Special Session on Frames, Wavelets and Gabor Systems AMS Special Session on Frames, Harmonic Analysis, and Operator Theory April 16–17, 2016 North Dakota State University, Fargo, ND Yeonhyang Kim Sivaram K. Narayan Gabriel Picioroaga Eric S. Weber Editors Frames and Harmonic Analysis AMS Special Session on Frames, Wavelets and Gabor Systems AMS Special Session on Frames, Harmonic Analysis, and Operator Theory April 16–17, 2016 North Dakota State University, Fargo, ND Yeonhyang Kim Sivaram K. Narayan Gabriel Picioroaga Eric S. Weber Editors 706 Frames and Harmonic Analysis AMS Special Session on Frames, Wavelets and Gabor Systems AMS Special Session on Frames, Harmonic Analysis, and Operator Theory April 16–17, 2016 North Dakota State University, Fargo, ND Yeonhyang Kim Sivaram K. Narayan Gabriel Picioroaga Eric S. Weber Editors EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor Michael Loss Kailash Misra Catherine Yan 2010 Mathematics Subject Classification. Primary 15Axx, 41Axx, 42Axx, 42Cxx, 43Axx, 46Cxx, 47Axx, 94Axx. Library of Congress Cataloging-in-Publication Data Names: Kim, Yeonhyang, 1972– editor. | Narayan, Sivaram K., 1954– editor. | Picioroaga, Gabriel, 1973– editor. | Weber, Eric S., 1972– editor. Title: Frames and harmonic analysis: AMS special sessions on frames, wavelets, and Gabor sys- tems and frames, harmonic analysis, and operator theory, April 16–17, 2016, North Dakota State University, Fargo, North Dakota / Yeonhyang Kim, Sivaram K. Narayan, Gabriel Pi- cioroaga, Eric S. Weber, editors. Description: Providence, Rhode Island: American Mathematical Society, [2018] | Series: Contem- porary mathematics; volume 706 Identifiers: LCCN 2017044766 | ISBN 9781470436193 (alk. paper) Subjects: LCSH: Frames (Vector analysis) | Harmonic analysis. | Wavelets (Mathematics) | Gabor transforms. | AMS: Linear and multilinear algebra; matrix theory – Basic linear algebra – Basic linear algebra. msc | Approximations and expansions – Approximations and expansions – Approximations and expansions. msc | Harmonic analysis on Euclidean spaces – Harmonic analysis in one variable – Harmonic analysis in one variable. msc | Harmonic analysis on Euclidean spaces – Nontrigonometric harmonic analysis – Nontrigonometric harmonic analysis. msc | Abstract harmonic analysis – Abstract harmonic analysis – Abstract harmonic analysis. msc | Functional analysis – Inner product spaces and their generalizations, Hilbert spaces – Inner product spaces and their generalizations, Hilbert spaces. msc | Operator theory – General theory of linear operators – General theory of linear operators. msc | Information and communication, circuits – Communication, information – Communication, information. msc Classification: LCC QA433 .F727 2018 | DDC 515/.63–dc23 LC record available at https://lccn.loc.gov/2017044766 DOI: http://dx.doi.org/10.1090/conm/706 Color graphic policy. Any graphics created in color will be rendered in grayscale for the printed version unless color printing is authorized by the Publisher. In general, color graphics will appear in color in the online version. Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center. For more information, please visit www.ams.org/publications/pubpermissions. Send requests for translation rights and licensed reprints to [email protected]. c 2018 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 232221201918 Contents Preface vii Participants of the AMS Special Session “Frames, Wavelets and Gabor Systems” ix Participants of the AMS Special Session “Frames, Harmonic Analysis, and Operator Theory” xi Constructions of biangular tight frames and their relationships with equiangular tight frames Jameson Cahill, Peter G. Casazza, John I. Haas, and Janet Tremain 1 Phase retrieval by hyperplanes Sara Botelho-Andrade, Peter G. Casazza, Desai Cheng, John Haas, Tin T. Tran, Janet C. Tremain, and Zhiqiang Xu 21 Tight and full spark Chebyshev frames with real entries and worst-case coherence analysis David Ellis, Eric Hayashi, and Shidong Li 33 Fusion frames and distributed sparsity Roza Aceska, Jean-Luc Bouchot, and Shidong Li 47 The Kadison-Singer problem Marcin Bownik 63 Spectral properties of an operator polynomial with coefficients in a Banach algebra Anatoly G. Baskakov and Ilya A. Krishtal 93 The Kaczmarz algorithm, row action methods, and statistical learning algorithms Xuemei Chen 115 Lipschitz properties for deep convolutional networks Radu Balan, Maneesh Singh, and Dongmian Zou 129 Invertibility of graph translation and support of Laplacian Fiedler vectors Matthew Begue´ and Kasso A. Okoudjou 153 Weighted convolution inequalities and Beurling density Jean-Pierre Gabardo 175 v vi CONTENTS p-Riesz bases in quasi shift invariant spaces Laura De Carli and Pierluigi Vellucci 201 On spectral sets of integers Dorin Ervin Dutkay and Isabelle Kraus 215 Spectral fractal measures associated to IFS’s consisting of three contraction mappings Ian Long 235 A matrix characterization of boundary representations of positive matrices in the Hardy space John E. Herr, Palle E. T. Jorgensen, and Eric S. Weber 255 Gibbs effects using Daubechies and Coiflet tight framelet systems Mutaz Mohammad and En-Bing Lin 271 Conditions on shape preserving of stationary polynomial reproducing subdivision schemes Yeon Hyang Kim 283 W -Markov measures, transfer operators, wavelets and multiresolutions Daniel Alpay, Palle Jorgensen, and Izchak Lewkowicz 293 Preface Frames were first introduced by Duffin and Schaeffer in 1952 in the context of nonharmonic Fourier series but have enjoyed widespread interest in recent years, particularly as a unifying concept. Indeed, mathematicians with backgrounds as diverse as classical and modern harmonic analysis, Banach space theory, operator algebras, and complex analysis have recently worked in frame theory. The present volume contains papers expositing frame theory and applications in three specific contexts: frame constructions and applications, Fourier and harmonic analysis, and wavelet theory. In recent years, frame theory has found applications to problems in computer science, data science, engineering, and physics. Many of these applications involve frames in finite-dimensional spaces; one focus of finite frame theory is the construc- tion of tight frames with desired properties such as equiangular tight frames. Other types of frames discussed in these papers include scalable frames, full-spark frames, and fusion frames. (1) Constructions of Biangular Tight Frames and Their Relationships with Equiangular Tight Frames (2) Phase Retrieval by Hyperplanes (3) Tight and Full Spark Chebyshev Frames with Real Entries and Worst- Case Coherence Analysis (4) Fusion Frames and Distributed Sparsity Historically there exists a strong connection between operator theory and frame theory. The recent solution of the Kadison-Singer problem is a further illustration of this connection. Modern connections are being formed between frame theory and machine learning. (5) The Kadison-Singer Problem (6) Spectral Properties of an Operator Polynomial with Coefficients in a Banach Algebra (7) Kaczmarz Algorithm, Row Action Methods, and Statistical Learning Al- gorithms (8) Lipschitz Properties for Deep Convolutional Networks There also exists a strong connection between frame theory and harmonic anal- ysis. This is seen in the context of classical Fourier analysis and shift invariant spaces, including in new settings such as on graphs. (9) Invertibility of Graph Translation and Support of Laplacian Fiedler Vec- tors (10) Weighted Convolution Inequalities and Beurling Density (11) p-Riesz Bases in Quasi Shift Invariant Spaces vii viii PREFACE This connection between frame theory and harmonic analysis also occurs in the context of spectral measures–those measures which possess an orthogonal basis of exponentials, or, more generally, those measures which possess a harmonic analysis in terms of boundary functions for elements in the Hardy space of the unit disc. (12) On Spectral Sets of Integers (13) Spectral Fractal Measures Associated to IFS’s Consisting of Three Con- traction Mappings (14) A Matrix Characterization of Boundary Representations of Positive Ma- trices in the Hardy Space The (modern) developments of wavelet theory and frame theory are intertwined, particularly in the construction of frames for function spaces. Both have a wide range of practical applications in numerical analysis, signal processing, and image processing. Papers in this volume study the Gibbs phenomenon for wavelet frames, subdivision schemes, and the connection between Markov chains and wavelets. (15) Gibbs
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