Wavelets: a Primer

Wavelets A Primer Wavelets A Primer Christian Blatter Departement Mathematik ETH Zentrum Zurich, Switzerland Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an informa business AN A K PETERS BOOK First published 1998 by Ak Peters, Ltd. Published 2018 by CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 1998 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works ISBN 13: 978-1-56881-195-6 (pbk) ISBN 13: 978-1-56881-095-9 (hbk) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. 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Cont ents Preface ................................................................................................... vii Read M e ................................................................................................... ix 1 Form ulating the pr o b le m ............................................................... 1 1.1 A central theme of analysis................................................................. 1 1.2 Fourier series........................................................................................ 4 1.3 Fourier transform ................................................................................ 8 1.4 Windowed Fouriertransform ...................................................................11 1.5 Wavelet transform ....................................................................14 1.6 The Haar w avelet.................................................................................... 20 2 Fourier an aly sis.................................................................................... 29 2.1 Fourier series............................................................................................29 2.2 Fourier transform on R .........................................................................34 2.3 The Heisenberg uncertainty p rin c ip le ..................................................49 2.4 The Shannon sampling theorem ............................................................. 53 3 The continuous wavelet transform .............................................. 61 3.1 Definitions and examples.........................................................................61 3.2 A Plancherel fo rm u la .............................................................................69 3.3 Inversion formulas ................................................................................ 74 3.4 The kernel function.................................................................................78 3.5 Decay of the wavelet transform ............................................................. 82 4 F r a m e s ....................................................................................................90 4.1 Geometrical considerations.....................................................................90 4.2 The general notion of a frame ..............................................................99 4.3 The discrete wavelet transform ............................................................104 4.4 Proof of theorem (4.10) ........................................................................114 VI Cont ents 5 M ultiresolution analysis ............................................................... 120 5.1 Axiomatic description...........................................................................121 5.2 The scaling function...............................................................................126 5.3 Constructions in the Fourier domain....................................................134 5.4 Algorithms..............................................................................................149 6 Orthonormal wavelets with compact support ...........................157 6.1 The basic i d e a ...................................................................................... 157 6.2 Algebraic constructions.......................................................................168 6.3 Binary interpolation...............................................................................176 6.4 Spline w avelets...................................................................................... 188 R eferen ces............................................................................................... 199 Index 201 Preface This book is neither the grand retrospective view of a protagonist nor an encyclopedic research monograph, but the approach of a working mathemati cian to a subject that has stimulated approximation theory and inspired users in many diverse domains of applied mathematics, unlike any other since the invention of the Fast Fourier Transform. As a matter of fact, I had only set out to draw up a one-semester course for our students at ETH Zurich that would introduce them to the world of wavelets ab ovo; indeed, such a course hadn’t been given here before. But in the end, thanks to encouraging com ments from colleagues and people in the audience, the present booklet came into existence. I had imagined that the target group for this course would be the following: students of mathematics in their senior year or first graduate year, having the usual basic knowledge of analysis, carrying around a knapsack full of convergence theorems, but without any practical experience, say, in Fourier analysis. In the back of my mind I also entertained the hope that some people from the field of engineering would attend the course. In fact, they did, and afterward I found out that exactly these students had profitted the most from my efforts. The contents of the book can be summarized as follows: The introductory Chapter 1 presents a tour d’horizon over various ways of signal representation; it is here that the Haar wavelet makes its first appearance. Chapter 2 serves primarily as a tutorial of Fourier analysis (without proofs); it is supplemented by the discussion of two theorems that define ultimate limits of signal theory: the Heisenberg uncertainty principle and Shannon’s sampling theorem. In Chapter 3 we are finally ready for a treatment of the continuous wavelet transform, and Chapter 4, entitled “Frames”, describes a general framework (pun not intended) allowing us to handle the continuous and the discrete wavelet transforms in a uniform way. All this being accomplished, we finally arrive at the main course: multiresolution analysis with its fast algorithms in Chapter 5 and the construction of orthonormal wavelets with compact support in Chapter 6. The book ends with a brief treatment of spline wavelets in Section 6.4. viii Preface Given the small size of this treatise, some things had to be left out: biorthogo- nal systems, wavelets in two dimensions, and a detailed description of applica tions, to name a few. Furthermore, I decided to leave distributions out of the picture. This means that there aren’t any Sobolev spaces, nor a discussion of pointwise convergence, etc., of wavelet approximations, and the Paley-Wiener theorem is not at our disposal either. Fortunately, there is an elementary ar gument coming to our rescue in proving that the Daubechies wavelets indeed have compact support. When putting the material together, I made generous use of the work of other authors. In the first place, of course, I borrowed from Ingrid Daubechies’ incomparable “Ten Lectures on Wavelets” [D], to some lesser extent from [L], which at the time (winter 1996-97) was the only wavelet book available in German, and from Kaiser’s “Friendly Guide to Wavelets” [K]. Concerning further

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