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706

Frames and

AMS Special Session on Frames, and Gabor Systems

AMS Special Session on Frames, Harmonic Analysis, and

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; theory – Basic – 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 | – 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

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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 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 theory. In recent years, frame theory has found applications to problems in , 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 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 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, , 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 Effects Using Daubechies and Coiflet Tight Framelet Systems (16) Conditions on Shape Preserving of Stationary Polynomial Reproducing Subdivision Schemes (17) W -Markov Measures, Transfer Operators, Wavelets, and Multiresolutions As outlined above, this collection of papers covers a wide variety of topics. As such, this volume will be of interest to researchers in frame theory, as well as approximation theory, data science, representation theory, and functional and harmonic analysis.

Yeonhyang Kim Sivaram K. Narayan Gabriel Picioroaga Eric S. Weber Participants of the AMS Special Session “Frames, Wavelets and Gabor Systems”

Speakers and titles from the AMS Special Session “Frames, Wavelets and Gabor Systems” from the AMS Central Sectional Meeting, Fargo, ND, April 16–17, 2016. Roza Aceska Local sparsity and fusion frames Radu Balan The iterative and regularized least squares (IRLS) algorithm for phase retrieval Laura De Carli Stability theorems for systems of rect and sinc Peter G. Casazza Infinite dimensional phase retrieval Xuemei Chen The gap between NSP and RIP Cheng Cheng Spatially distributed sampling and reconstruction Matthew Fickus Equiangular tight frames from hyperovals John Isaac Haas Tight orthoplectic Grassmannian frames Bin Han Tight framelets and refinable structure Christopher Heil HRT versus the zero divisor conjecture John Jasper Tremain equiangular tight frames and strongly regular graphs Alex Iosevich On the Fuglede conjecture Azita Mayeli Sampling and interpolation on certain nilpotent lie groups Dustin G. Mixon The Voronoi means conjecture

ix x FRAMES, WAVELETS AND GABOR SYSTEMS

Kasso Okoudjou A new proof of the three-point HRT conjecture Vignon Oussa A classification of irreducible admissible groups in dimension three Judith A. Packer Wavelets and frames associated to representations of higher-rank graph algebras Armenak Petrosyan Frames and Bessel systems generated by the iterative actions of normal operators Darrin Speegle Sampling from a continuous frame to obtain a frame Participants of the AMS Special Session “Frames, Harmonic Analysis, and Operator Theory”

Speakers and titles from the AMS Special Session “Frames, Harmonic Analysis, and Operator Theory” from the AMS Central Sectional Meeting, Fargo, ND, April 16–17, 2016. Cyrus P. Aryana Self-adjoint Toeplitz operators associated with representing measures on doubly connected planar regions and their eigenvalues Marcin Bownik Improved bounds in Weaver and Feichtinger conjectures Laura De Carli Exponential bases on rectangles in Rd Peter G. Casazza Frame scalings a condition number approach Dorin Ervin Dutkay Fourier series on self-affine measures Jean-Pierre Gabardo Beurling density in weighted Fourier spaces John Herr Fourier series for singular measures Calvin F. Hotchkiss A fast for fractal approximations Joseph W. Iverson Group frames with several generators Palle E. T. Jorgensen Probability theory of infinite iterated function systems Ilya A. Krishtal On the spectral theory of operator polynomials with coefficients in a Banach algebra Chun-Kit Lai Non-spectral fractal measures with Fourier frames Shidon Li Sparsity-inducing dual frames and sparse signal recovery with coherent frames

xi xii FRAMES, HARMONIC ANALYSIS, AND OPERATOR THEORY

Richard G. Lynch Weaving Schauder frames Azita Mayeli Tiling and spectral sets in Zp × Zp Kasso Okoudjou On the boundedness of the bilinear Hilbert transform on modulation spaces Gabriel Picioroaga Fourier frames for the Cantor-4 set Sam L. Scholze Bridging frame erasures Keith F. Taylor Admissible groups in three dimensions Dongmian Zou Lipschitz properties for deep networks SELECTED PUBLISHED TITLES IN THIS SERIES

706 Yeonhyang Kim, Sivaram K. Narayan, Gabriel Picioroaga, and Eric S. Weber, Editors, Frames and Harmonic Analysis, 2018 702 Mark Grant, Gregory Lupton, and Lucile Vandembroucq, Editors, Topological Complexity and Related Topics, 2018 701 Joan-Carles Lario and V. Kumar Murty, Editors, Number Theory Related to Modular Curves, 2018 700 Alexandre Girouard, Dmitry Jakobson, Michael Levitin, Nilima Nigam, Iosif Polterovich, and Fr´ed´eric Rochon, Editors, Geometric and Computational Spectral Theory, 2017 699 Mark L. Agranovsky, Matania Ben-Artzi, Catherine B´en´eteau, Lavi Karp, Dmitry Khavinson, Simeon Reich, David Shoikhet, Gilbert Weinstein, and Lawrence Zalcman, Editors, Complex Analysis and Dynamical Systems VII, 2017 698 Alexander M. Blokh, Leonid A. Bunimovich, Paul H. Jung, Lex G. Oversteegen, and Yakov G. Sinai, Editors, Dynamical Systems, Ergodic Theory, and Probability: in Memory of Kolya Chernov, 2017 697 Fabrizio Broglia, Fran¸coise Delon, Max Dickmann, Danielle Gondard-Cozette, and Victoria Ann Powers, Editors, Ordered Algebraic Structures and Related Topics, 2017 696 Ara S. Basmajian, Yair N. Minsky, and Alan W. Reid, Editors, In the Tradition of Ahlfors–Bers, VII, 2017 695 Katrina Barron, Elizabeth Jurisich, Antun Milas, and Kailash Misra, Editors, Lie Algebras, Vertex Operator Algebras, and Related Topics, 2017 694 Manjul Bhargava, Robert Guralnick, Gerhard Hiss, Klaus Lux, and Pham Huu Tiep, Editors, Finite Simple Groups: Thirty Years of the Atlas and Beyond, 2017 693 Michael Cwikel and Mario Milman, Editors, Functional Analysis, Harmonic Analysis, and Image Processing, 2017 692 Anatole Katok, Yakov Pesin, and Federico Rodriguez Hertz, Editors, Modern Theory of Dynamical Systems, 2017 691 Farrell Brumley, Maria Paula G´omez Aparicio, and Alberto M´ınguez, Editors, Around Langlands Correspondences, 2017 690 Andr´es Eduardo Caicedo, James Cummings, Peter Koellner, and Paul B. Larson, Editors, Foundations of Mathematics, 2017 689 Erica Flapan, Allison Henrich, Aaron Kaestner, and Sam Nelson, Editors, Knots, Links, Spatial Graphs, and Algebraic Invariants, 2017 688 Jeffrey Bergen, Stefan Catoiu, and William Chin, Editors, Groups, Rings, Group Rings, and Hopf Algebras, 2017 687 FernandaBotelho,RaenaKing,andT.S.S.R.K.Rao,Editors, Problems and Recent Methods in Operator Theory, 2017 686 Alp Bassa, Alain Couvreur, and David Kohel, Editors, Arithmetic, Geometry, Cryptography and Coding Theory, 2017 685 Heather A. Harrington, Mohamed Omar, and Matthew Wright, Editors, Algebraic and Geometric Methods in Discrete Mathematics, 2017 684 Anna Beliakova and Aaron D. Lauda, Editors, Categorification in Geometry, Topology, and Physics, 2017 683 Anna Beliakova and Aaron D. Lauda, Editors, Categorification and Higher Representation Theory, 2017 682 Gregory Arone, Brenda Johnson, Pascal Lambrechts, Brian A. Munson, and Ismar Voli´c, Editors, Manifolds and K-Theory, 2017

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/conmseries/. CONM 706 rmsadHroi Analysis Harmonic and Frames

This volume contains the proceedings of the AMS Special Sessions on Frames, Wavelets and Gabor Systems and Frames, Harmonic Analysis, and Operator Theory, held from April 16-17, 2016, at North Dakota State University in Fargo, North Dakota. The papers appearing in this volume cover frame theory and applications in three spe- cific contexts: frame constructions and applications, Fourier and harmonic analysis, and wavelet theory. • i ta. Editors al., et Kim

ISBN 978-1-4704-3619-3

9 781470 436193 CONM/706