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

358 Pages·2018·3.24 MB·English

by  Kim

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,EricS.,1972–editor.
Title: Framesandharmonic analysis: AMSspecialsessionsonframes,wavelets,andGaborsys-
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,EricS.Weber,editors.
Description: Providence,RhodeIsland: AmericanMathematicalSociety,[2018]|Series: Contem-
porarymathematics;volume706
Identifiers: LCCN2017044766|ISBN9781470436193(alk. paper)
Subjects: LCSH:Frames(Vectoranalysis)|Harmonicanalysis. |Wavelets(Mathematics)|Gabor
transforms. | AMS: Linear and multilinear algebra; matrix theory – Basic linear algebra –
Basiclinearalgebra. msc|Approximationsandexpansions–Approximationsandexpansions
–Approximations and expansions. msc|Harmonicanalysis onEuclidean spaces–Harmonic
analysis in one variable – Harmonic analysis in one variable. msc | Harmonic analysis on
Euclideanspaces–Nontrigonometricharmonicanalysis–Nontrigonometricharmonicanalysis.
msc|Abstractharmonicanalysis–Abstractharmonicanalysis–Abstractharmonicanalysis.
msc | Functional analysis – Inner product spaces and their generalizations, Hilbert spaces
– Inner product spaces and their generalizations, Hilbert spaces. msc | Operator theory –
Generaltheoryoflinearoperators–Generaltheoryoflinearoperators. msc|Informationand
communication,circuits–Communication,information–Communication,information. msc
Classification: LCCQA433.F7272018|DDC515/.63–dc23
LCrecordavailableathttps://lccn.loc.gov/2017044766
DOI:http://dx.doi.org/10.1090/conm/706
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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 Begu´e 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: frameconstructionsandapplications,Fourierandharmonicanalysis,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
framesinfinite-dimensional spaces; onefocusoffiniteframetheoryistheconstruc-
tionoftightframeswithdesiredpropertiessuchasequiangulartightframes. Other
typesofframesdiscussedinthesepapersincludescalableframes,full-sparkframes,
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
Historicallythereexistsastrongconnectionbetweenoperatortheoryandframe
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
Therealsoexistsastrongconnectionbetweenframetheoryandharmonicanal-
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
Thisconnectionbetweenframetheoryandharmonicanalysisalsooccursinthe
context of spectral measures–those measures which possess an orthogonal basis of
exponentials, or, more generally, thosemeasures whichpossess aharmonic 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)developmentsofwavelettheoryandframetheoryareintertwined,
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. PapersinthisvolumestudytheGibbsphenomenonforwaveletframes,
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-MarkovMeasures, TransferOperators,Wavelets,andMultiresolutions
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”
SpeakersandtitlesfromtheAMSSpecialSession“Frames,WaveletsandGabor
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