626 Operator Methods in Wavelets, Tilings, and Frames AMS Special Session Harmonic Analysis of Frames, Wavelets, and Tilings April 13–14, 2013 Boulder, Colorado Veronika Furst Keri A. Kornelson Eric S. Weber Editors AmericanMathematicalSociety Operator Methods in Wavelets, Tilings, and Frames AMS Special Session Harmonic Analysis of Frames, Wavelets, and Tilings April 13–14, 2013 Boulder, Colorado Veronika Furst Keri A. Kornelson Eric S. Weber Editors 626 Operator Methods in Wavelets, Tilings, and Frames AMS Special Session Harmonic Analysis of Frames, Wavelets, and Tilings April 13–14, 2013 Boulder, Colorado Veronika Furst Keri A. Kornelson Eric S. Weber Editors AmericanMathematicalSociety Providence,RhodeIsland EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor Michael Loss Kailash C. Misra Martin J. Strauss 2010 Mathematics Subject Classification. Primary 41Axx, 42Axx, 42Cxx, 43Axx, 46Cxx, 47Axx, 94Axx. Library of Congress Cataloging-in-Publication Data Operator methods in wavelets, tilings, and frames / Veronika Furst, Keri A. Kornelson, Eric S. Weber,editors. pagescm. –(Contemporarymathematics;volume626) “AMS Special Session on Harmonic Analysis of Frames, Wavelets, and Tilings, April 13-14, 2013,Boulder,Colorado.” Includesbibliographicalreferences. ISBN978-1-4704-1040-7(alk. paper) 1. Frames(Combinatorialanalysis)2. Wavelets(Mathematics)I.Furst,Veronika,1979-editor of compilation. II. Kornelson, Keri A., 1967- editor of compilation. III. Weber, Eric S., 1972- editorofcompilation. QA403.3.O642014 511(cid:2).6–dc23 2014009729 ContemporaryMathematicsISSN:0271-4132(print);ISSN:1098-3627(online) DOI:http://dx.doi.org/10.1090/conm/626 Copying and reprinting. Materialinthisbookmaybereproducedbyanymeansfor edu- cationaland scientific purposes without fee or permissionwith the exception ofreproduction by servicesthatcollectfeesfordeliveryofdocumentsandprovidedthatthecustomaryacknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercialuseofmaterialshouldbeaddressedtotheAcquisitionsDepartment,AmericanMath- ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can [email protected]. Excludedfromtheseprovisionsismaterialinarticlesforwhichtheauthorholdscopyright. In suchcases,requestsforpermissiontouseorreprintshouldbeaddresseddirectlytotheauthor(s). (Copyrightownershipisindicatedinthenoticeinthelowerright-handcornerofthefirstpageof eacharticle.) (cid:2)c 2014bytheAmericanMathematicalSociety. Allrightsreserved. TheAmericanMathematicalSocietyretainsallrights exceptthosegrantedtotheUnitedStatesGovernment. Copyrightofindividualarticlesmayreverttothepublicdomain28years afterpublication. ContacttheAMSforcopyrightstatusofindividualarticles. PrintedintheUnitedStatesofAmerica. (cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttp://www.ams.org/ 10987654321 191817161514 Contents Preface vii Participants ix Phase retrieval by vectors and projections Peter G. Casazza and Lindsey M. Woodland 1 Scalable frames and convex geometry Gitta Kutyniok, Kasso A. Okoudjou, and Friedrich Philipp 19 Dilations of frames, operator-valued measures and bounded linear maps Deguang Han, David R. Larson, Bei Liu, and Rui Liu 33 Images of the continuous wavelet transform Mahya Ghandehari and Keith F. Taylor 55 Decompositions of generalized wavelet representations Bradley Currey, Azita Mayeli, and Vignon Oussa 67 Exponential splines of complex order Peter Massopust 87 Local translations associated to spectral sets Dorin Ervin Dutkay and John Haussermann 107 Additive spectra of the 1 Cantor measure 4 Palle E. T. Jorgensen, Keri A. Kornelson, and Karen L. Shuman 121 Necessary density conditions for sampling and interpolation in de Branges spaces Sa’ud al-Sa’Di and Eric Weber 129 Dynamical sampling in hybrid shift invariant spaces Roza Aceska and Sui Tang 149 Dynamical sampling in infinite dimensions with and without a forcing term Jacqueline Davis 167 v 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 volumecontainspapersexpositingthethemeofoperatortheoreticmethodsinframe theory in four specific contexts: frame constructions, wavelet theory, tilings, and sampling theory. There are numerous constructions of frames, as there are numerous situations in which frame theory plays a central role, and each of these situations requires a framewithdifferentcharacteristics. ThepaperbyCasazzaandWoodlanddiscusses frame constructions, andassociatedprojections, which allow forthe reconstruction of an unknown vector using the magnitude of frame coefficients without the phase. The paper by Kutyniok, Okoudjou, and Phillips concerns frames which can be preconditioned via scalar multiplication to obtain a tight frame. The paper by Han, Larson, Liu and Liu approaches the idea of a frame in a generalized sense, in which the frame is given by a set of operators, not a set of vectors. Although the first wavelet was introduced by Haar in 1909, wavelet analysis officiallytookoffwiththepioneering workofDaubechies, Grossman, andMeyer in the 1980s. The main attractiveness of a wavelet is its simultaneous localization of a square-integrable function in both time and frequency. Its “zooming” capability isformalizedinthedefinitionofamultiresolutionanalysis. GhandehariandTaylor generalize the classical dilation and translation operators by considering a unitary representation of a locally compact group G and defining a wavelet to be a vector in the associated Hilbert space for which a reconstruction formula holds in a weak sense. Their focus is how the images of the corresponding continuous wavelet transform, as subspaces of L2(G), change and are related to one another, as a consequence of varying the wavelet. The paper by Currey, Mayeli, and Oussa also generalizes the wavelet representation of the subgroup of the ax+b group that is isomorphic tothe subgroup of unitary operators generatedby the classical dilation and translation. The authors replace the Hilbert space L2(Rn) by L2(N) for a simply connected, connected nilpotent Lie group N. They define a corresponding wavelet representation and analyze its direct integral decomposition, particularly fornon-commutativeN. InthepaperbyMassopust,exponentialsplinesofcomplex orderextendtheclassofexponentialB-splinesofordernforn∈Nandpolynomial B-splinesofcomplexorder. Thenewclassofsplinesdefinesmultiresolutionanalyses of L2(R) and corresponding wavelet bases. vii viii PREFACE The Fuglede conjecture from 1974 presents the connection that is often, but not always, present between sets that tile Rd by translation and the existence of spectral sets associated with the tiling. The conjecture is resolved for at least dimension 3 but not dimensions 2 and 1. The paper by Dutkay and Hausserman considers tiling sets in dimension 1. The authors present properties of unitary groups of local translations acting on subsets of the real line and draw connections totilings. TheFuglede conjecturecreatedincreasedinterestinthepresenceorlack ofFourier basesorFourier frameswithrespecttoavarietyofmeasures. Thepaper by Jorgensen, Kornelson, and Shuman presents spectra on a fractal measure space and gives structural information about the connections between different spectra on the same space. Sampling theory concerns the reconstruction of an unknown function from its known samples at certain points in its domain. This idea can be traced back to Cauchy, where the unknown function was a trigonometric polynomial, but in the moderncontext,samplingtheorycanbedescribedintermsofframes. Inthisform, the main problem is when a certain operator possesses a generalized inverse. The paper by al-Sa’di and Weber gives necessary conditions which guarantee that this operatordoespossessageneralizedinverse,wheretheunknownfunctionbelongsto a Hilbert space of entire functions. The papers by Aceska and Tang, and by Davis concern the variation on sampling theory in which some of the known samples of the unknown function are obtained after an operator acts upon the function. In the paper by Aceska and Tang, the space of functions is a hybrid shift invariant space, and the operator which acts in between successive sampling operations is a convolutionoperator. InthepaperbyDavis,thefunctionspaceofsquare-summable sequences, and the operator acting between sampling operations may involve a nonlinear forcing term. In both papers, the essential question is: When does the matrix representation for an operator possess an appropriate submatrix with a generalized inverse? This collection of papers covers a wide variety of topics, including: convex ge- ometry,directintegraldecompositions,Beurlingdensity,operator-valuedmeasures, splines, and more. These topics arise naturally in the study of frames, which again is the unifying theme in this volume. In nearly all of the papers, ideas and results from operator theory are the crucial tools in solving the problems in the study of frames. This volume will be of interest to researchers in frame theory, and also to those in approximation theory, representation theory, functional analysis, and harmonic analysis. Veronika Furst Keri Kornelson Eric Weber Participants SpeakersandtitlesfromtheAMSSpecialSession“HarmonicAnalysisofFrames, Wavelets, and Tilings” from the AMS Western Sectional Meeting, Boulder, CO, April 13–14, 2013. Marcin Bownik Existence of Frames with Prescribed Norms and Frame Operator Peter G. Casazza Fusion Frames for Wireless Sensor Networks Jacqueline Davis Dynamical Sampling Dorin Dutkay The Fuglede Conjecture in Dimension One Matthew Fickus Characterizing Completions of Finite Frames Deguang Han Spectrally Optimal Frames for Erasures John Haussermann Tiling Properties of Spectra of Measures John Jasper Spectra of Frame Operators with Prescribed Frame Norms Palle Jorgensen Tilings in Wavelet Theory: IFS Measures and Wavelet Packets Chun-Kit Lai Spectral Property of Cantor Measures with Consecutive Digits David R. Larson Frames, Dilations and Operator-Valued Measures Peter Massopust Exponential Splines with Complex Order Azita Mayeli Bracket Map for the Heisenberg Group and the Characterization of Cyclic Subspaces ix