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A First Course on Wavelets Steven G AFirst Courseon wavelets c 1996 by CRC Press LLC Studies in Advanced iMathematics Series Editor STEVEN G. KRAN7Z Washington University in St. L.auis Editorial Board R. Michael Beals Gerald B. Folland Rutgers University University of Washington Dennis de Turck William Helton University of Pennsylvania University of California at San Diego Ronald DeVore Norberto Salinas University of South Carolina University of Kansas Luwrence C. Evans Michael E. Taylor Universitywavelets of California at Berkeley University of North Carolina Titles included in the Series Steven R. Bell, The Cauchy Transform, Potential Theory, and Conformal Mapping John J. Benedetto, Harmonic Analysis and Applications John J. Benedetto and Michael W. Fmzier, Wavelets: Mathematics and Applications Albert Boggess, CR Manifolds and the Tangential Cauchy-Riemann Complex Goong Chen and Jiamin Zhou, Vibration and Damping in Distributed Systems, Vol. I : Analysis, Estimation, Attenuation, and Design. Vol. 2: WKB and Wave Methods, Visualization, and Experimentation Carl C. Cowen und Burbara D. MacCluer, Composition Operators on Spaces of Analytic Functions John P. D’Angelo, Several Complex Variables and the Geometry of Real Hypersurfaces Lawrence C. Evans and Ronald F. Gariepy, Measure Theory and Fine Properties of Functions Gerald B. Folfand, A Course in Abstract Harmonic Analysis Jest! Car&-Cuerva, Eugenio Herndndez9 Fernando Soria, und Jo&-Luis Torrea, Fourier Analysis and Partial Differential Equations Peter B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, 2nd Edition Alfred Gray, Differential Geometry and Its Applications with Mathematics, 2nd Edition Eugenio Herndndez and Guide Weiss, A First Course on Wavelets Steven G. Krantz, Partial Differential Equations and Complex Analysis Steven G. Kruntz, Real Analysis and Foundations Ciurk Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos John Ryan, Clifford Algebras in Analysis and Related Topics Xavier Saint Raymond, Elementary Introduction to the Theory of Pseudodifferential Oprators Robert Strichartz, A Guide to Distribution Theory and Fourier Transforms And& Unterberger and Harald Upmeier, Pseudodifferential Analysis on Symmetric Cones James S. Wafker, Fast Fourier Transforms, 2nd Edition Gilbert G. Wulter, Wavelets and Other Orthogonal Systems with Applications Kehe Zhu, An Introduction to Operator Algebras c 1996 by CRC Press LLC AFirst Courseon !ii!iY!VELETS EugenioHernhdez Universidad Autbnoma de Madrid GuidoWeiss Washington University in St. Louis 0c@C CRC PRESS Boca Raton London New York Washington, D.C. Library of Congress Cataloging-in-Publication Data Hemandez, Eugenio, 1954 A first course on wavelets I Eugenio Hemandez. Guide Weiss. p. cm. - (Studies in advanced mathematics) Includes bibliographical references and indexes. ISBN 0-8493-8274-2 (alk. paper) I. Wavelets (Mathematics). I. Weiss, Guido L., 1928- II Title. III. Series. QA403.3.H47 1996 5 15’.2433Ac20 96-27 I 11 CIP I- _I This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the pubhsher cannot assume responsibility for the validity of ali materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 3343 1. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe. Visit the CRC Press Web site at www.crcpress.com @ 1996 by CRC Press LLC No claim to original U.S. Government works International Standard Book Number 0-8493-8274-2 Library of Congress Card Number 96-27 I I 1 Printed in the United States of America 4567890 Printed on acid-free paper About the Authors Eugenio Hernandez graduated from the Universidad Complu- tense de Madrid in 1977 and obtained his Ph.D. degree from Washington University in St. Louis in 1981. He has been a member of the faculty of the Universidad Autonoma de Madrid since 1982 and is now Professor of Mathematics. Dur- ing the academic year 1987-88 Dr. Hernandez was a Fulbright Fellow and a member of the Mathematical Sciences Research Guido Weiss (left) and Eugenio Hernandez Institute in Berkeley, California. He was a visiting professor at Washington University in St. Louis in 1994-95. His research interests lie in the areas of the theory of interpolation of operators, weighted inequalities, and most recently, in the theory of wavelets. Guido Weiss obtained his undergraduate and graduate degrees from the University of Chicago, receiving his Ph.D. degree in 1956. He served on the faculty at DePaul University from 1955 to 1960, and joined the faculty of Washington University in 1960 where he is now the Elinor Anheuser Professor of Mathematics. During the past 35 years he has had leaves of absence that have allowed him to be visiting professor in several different institutions: the Sorbonne, the University of Geneva, the University of Paris in Orsay, the Mathematical Sciences Research Institute in Berkeley, California (in each case for an academic year). He also was visiting professor during semester academic leave at the Universidad de Buenos Aires, Peking University, Beijing Normal University, and the Universidad Autonoma de Madrid. His research involves a broad area of mathematical analysis, particularly harmonic analysis. Some of his work, especially his contributions to the atomic and molecular characterizations of certain function spaces (particularly the Hardy spaces), is closely related to the theory of wavelets, a subject that has commanded his attention during the last few years. He has been awarded several honors- among them the Chauvenet Prize and honorary degrees from Beijing Normal University, the University of Milano, and the University of Barcelona. © 1996 by CRC Press LLC To Barbara and Jody c 1996 by CRC Press LLC Contents 1BasesforL2(R) 1.1Preliminaries 1.2Orthonormalbasesgeneratedbyasinglefunction;theBalian- Lowtheorem 1.3SmoothprojectionsonL2(R) 1.4Localsineandcosinebasesandtheconstructionofsome wavelets 1.5Theunitaryfoldingoperatorsandthesmoothprojections 1.6Notesandreferences 2Multiresolutionanalysisandtheconstructionofwavelets 2.1Multiresolutionanalysis 2.2Constructionofwaveletsfromamultiresolutionanalysis 2.3Theconstructionofcompactlysupportedwavelets 2.4Betterestimatesforthesmoothnessofcompactlysupported wavelets 2.5Notesandreferences 3Band-limitedwavelets 3.1Orthonormality 3.2Completeness c 1996 by CRC Press LLC 3.3TheLemari´e-Meyerwaveletsrevisited 3.4Characterizationofsomeband-limitedwavelets 3.5Notesandreferences 4Otherconstructionsofwavelets 4.1Franklinwaveletsontherealline 4.2Splinewaveletsontherealline 4.3Orthonormalbasesofpiecewiselinearcontinuousfunctions forL2(T) 4.4Orthonormalbasesofperiodicsplines 4.5Periodizationofwaveletsdefinedontherealline 4.6Notesandreferences 5Representationoffunctionsbywavelets 5.1BasesforBanachspaces 5.2UnconditionalbasesforBanachspaces 5.3ConvergenceofwaveletexpansionsinLp(R) 5.4Pointwiseconvergenceofwaveletexpansions 5.5H1andBMOonR 5.6WaveletsasunconditionalbasesforH1(R)andLp(R)with 1<p<∞ 5.7Notesandreferences 6Characterizationsoffunctionspacesusingwavelets 6.1Waveletsandsamplingtheorems 6.2Littlewood-Paleytheory 6.3Necessarytools 6.4TheLebesguespacesLp(R)with1<p<∞ 6.5TheHardyspaceH1(R) 6.6TheSobolevspacesLp,s(R),1<p<∞,s=1,2,3,··· 6.7TheLipschitzspacesΛα(R),0<α<1,andtheZygmund classΛ∗(R) c 1996 by CRC Press LLC 6.8Notesandreferences 7Characterizationsinthetheoryofwavelets 7.1Thebasicequations 7.2Someapplicationsofthebasicequations 7.3ThecharacterizationofMRAwavelets 7.4Acharacterizationoflow-passfilters 7.5Acharacterizationofscalingfunctions 7.6Non-existenceofsmoothwaveletsinH2(R) 7.7Notesandreferences 8 Frames 8.1Thereconstructionformulaforframes 8.2TheBalian-Lowtheore mforframes 8.3Framesfromtranslationsanddilations 8.4SmoothframesforH2(R) 8.5Notesandreferences 9Discretetransformsandalgorithms 9.1 The discrete Fourier transform (DFT) and the fast Fourier transform(FFT) 9.2 The discrete cosine transform (DCT) and the fast cosine transform(FCT) 9.3Thediscreteversionofthelocalsineandcosinebases 9.4Decompositionandreconstructionalgorithmsforwavelets 9.5Waveletpackets 9.6Notesandreferences References c 1996 by CRC Press LLC Foreword by Yves Meyer Membre de l’Institut (Acad´emiedes Sciences) Foreign Honorary Member of the American Academy of Arts and Sciences Wavelet analysis can be defined as an alternative to the classical win- dowed Fourier analysis. In the latter case the goal is to measure the local frequency content of a signal, while in the wavelet case one is comparing several magnifications of this signal, with distinct resolutions. The building blocks of a windowed Fourier analysis are sines and cosines (waves) multi- plied by a sliding window. They are usually referred to as time-frequency
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