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in Fourier Analysis, Splines and Wavelet Approximations Abdul J. Jerri Clarkson University KLUWER ACADEMIC PUBLISHERS DORDRECHT/BOSTON/LONDON 1998 |
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in Fourier Analysis, Splines and Wavelet Approximations Abdul J. Jerri Clarkson University KLUWER ACADEMIC PUBLISHERS DORDRECHT/BOSTON/LONDON 1998 |
Preface
This book represents the first attempt at a unified picture for the presence of the Gibbs (or Gibbs-Wilbraham) phenomenon in applications, its analysis and the different methods of filtering it out. The analysis and filtering cover the familiar Gibbs phenomenon in Fourier series and integral representations of functions with jump discontinuities. In addition it will include other representations, such as general orthogonal series expansions, general integral transforms, splines approximation, and continuous as well as discrete wavelet approximations. The material in this book is presented in a manner accessible to upperclassmen and graduate students in science and engineering, as well as researchers who may face the Gibbs phenomenon in the varied applications that involve the Fourier and the other approximations of functions with jump discontinuities. Those with more advanced backgrounds in analysis will find basic material, results, and motivations from which they can begin to develop deeper and more general results. We must emphasize that the aim of this book (the first on the subject): to satisfy such a diverse audience, is quite difficult. In particular, our detailed derivations and their illustrations for an introductory book may very well sound repetitive to the experts in the field who are expecting a research monograph. To answer the concern of the researchers, we can only hope that this book will prove helpful as a basic reference for their research papers. In addition, there is always the possibility of following it by a research monograph. To accommodate all those concerned with emphasis on the clarity with some intuition, we shall quote only the very basic theorems, such as those of Fourier and wavelet analysis. The basic and most likely familiar results and theorems of Fourier analysis are reviewed in Chapter 1. We will rely on a good number of the basic references that date back to Wilbraham in 1848. For completeness, we are also including most other references that deal with the Gibbs phenomenon in some way or another. They are placed separately as ``Other Related References" in an Appendix following the main bibliography of this book. To distinguish these references from the ones used in the text, we have added a prefix A to their (separate) numbers. These references also include some very recent ones, or few ``somewhat" relevant ones, that were discovered too late to be included in our general discussion. For completeness, we shall list such references at the end of their corresponding sections.
Aside from a number of additions, including theses done in the US since 1930, and other updatings since 1980, the detailed historical notes will have the spirit of the historical account of E. Hewitt and R. Hewitt, that appeared in 1980. A summary, of this historical account, for the Fourier as well as other representations, is covered in Section 2.7 of Chapter 2.
The first chapter starts with a short historical overview of the Gibbs phenomenon, then concentrates on a brief review of the very basic elements and notations of Fourier analysis. This is followed by a few typical illustrations for the appearance of the Gibbs phenomenon in the truncated Fourier series and integrals. It, then, looks back at Fejer averaging and the essentials of the summability theory, which is of importance to one of the basic methods of filtering the Gibbs phenomenon. This filtering method is complemented by the other very basic method, namely the Lanczos-local filtering. Chapter 2 covers the basic detailed analysis of the Gibbs phenomenon in Fourier series and integrals representations of signals, and the basic methods of filtering it out. It also includes other typical methods of filtering, a recent transform method, and some possible advantage of the presence of the Gibbs phenomenon for edge detection purposes, such as determining the locations of shocks, or sharper edges for the magnetic resonance imaging (MRI) of the defective parts of the heart, for example. In addition, we have included a close to complete but brief historical account of the Gibbs phenomenon in Fourier analysis and some orthogonal expansions of functions with jump discontinuities, for a time span of over a century. Chapter 3 is devoted, primarily, to general orthogonal series expansion, and the general integral representation of signals. There is also our most recent attempt at a Lanczos - like-local filtering besides the well known Fejer averaging (or Cesaro) summability methods. Such orthogonal expansions include the Fourier-Bessel series expansion, and the typical orthogonal polynomials series expansions such as the Legendre, the Tchebychev, the Hermite, and the Laguerre polynomials series expansions. The general integral transforms representation is illustrated with the Hankel transform. Chapter 4 is a short one that starts with the piecewise-linear approximation, then moves to its generalization of high order splines approximation of functions with jump discontinuities. The Fourier series approximations are looked at in light of the usual convergence in the mean (L2)-sense, as well as the Lp-sense, where the measure of the Gibbs phenomenon depends on p in Lp. The chapter concludes with a rather new topic of the approximation in interpolating the Discrete Fourier Transform (DFT). Chapter 5 deals with the newest topic, for the presence of a Gibbs-like phenomenon in the continuous as well as the discrete wavelets representations of signals. The overall result is that for most wavelets, the overshoots and undershoots are much fewer, and are smaller in magnitude than the typical ones of Fourier analysis. The emphasis here is on the clarity of an accessible presentation of this new and very important subject. This allows the reader an intuition for the reason behind expecting such a Gibbs phenomenon. We attempt to compare the new bases of wavelets to the traditional trigonometric bases of Fourier analysis. As in the case of the Fourier integral representation, with its relatively simple computations compared to the Fourier series, we will start the chapter with the continuous wavelet integral representation, followed by the discrete (orthonormal) wavelet series and concluded it by detailed analysis of the Gibbs phenomenon in (few particular) continuous wavelet representations. The chapter concludes with attempts at filtering the Gibbs phenomenon, which are, primarily, Fejer averaging (or summability) methods. However, we will have some remarks regarding possible other filters including our recent Lanczos-like-local filtering method.
Table of Contents
| Preface |
| Aim of the book |
| 1 Introduction |
| 1.1 The Gibbs-Wilbraham Phenomenon |
| 1.2 Some Basic Elements of Fourier Analysis |
| 1.3 Illustrations and Analysis |
| A. The Truncated Fourier Series Approximation |
| B. The Truncated Fourier Integral Approximation |
| 1.4 Filtering via the Fejer Averaging |
| A. The Fejer Averaging |
| B. The (C,a) Summability |
| 1.5 The Lanczos-Local-Type Filtering |
| 2 Analysis and Filtering |
| 2.1 The Truncated Fourier Integral |
| 2.2 The Fourier Trigonometric Polynomial |
| A. A Note Concerning the General Orthogonal Expansion |
| 2.3 The Two Basic Methods of Filtering |
| A. The Lanczos-Local-Type s-Averaging |
| B. The Method of Fejer Averaging and Summability |
| 2.4 Transform Methods of Filtering |
| A. The Gegenbauer Transform Method for the Truncated Fourier Series |
| B. The Truncated Fourier Integrals |
| 2.5 Examples of Other Filters |
| A. The Fourier Series in Two Dimensions |
| 2.6 Some Advantages for Edge Detection |
| 2.7 A Historical Note |
| 2.8 The Higher Dimensional Case |
| 3 The General Orthogonal Expansions |
| 3.1 A Brief Overview |
| 3.2 Orthogonal Series Expansions |
| A. The Sturm-Liousville Problem |
| B. The Fourier-Jn-Bessel Series Expansion |
| C. The Hankel Transform of Radially Symmetric Functions in n Dimensions |
| D. The Classical Orthogonal Polynomials Expansion |
| E. The Legendre Polynomials Series |
| F. The Tchebychev Polynomials Series |
| G. The Laguerre Polynomials Series |
| H. The Hermite Polynomials Series |
| 3.3 The Asymptotic Relation to Fourier Series |
| A. Rate of Convergence of the Sturm-Liouville Eigenfunctions Expansion |
| B. Singular Sturm-Liouville Problem |
| 3.4 The Global Effect on the Convergence in Rn |
| A. The Laplacian in n-Dimensional-Fourier Series of Radial Functions |
| B. The 3-Dimensional Case |
| C. The Fourier Integral Representation in n-Dimensions |
| 3.5 Filtering for Orthogonal Expansions |
| A. The Fejer Averaging |
| B. A Lanczos-Like s-Factor for General Orthogonal Expansions |
| 1. A Lanczos-Like s-Factor for Fourier-Jm-Bessel Series |
| 2. Orthogonal Polynomials Expansions |
| 3. Integral Transforms Represntations |
| 4 Splines and Other Approxmiations |
| 4.1 The Piecewise-Linear Approximation |
| 4.2 High Order Splines Approximation |
| 4.3 Approximation in Lp-Sense |
| 4.4 The Interpolation of the DFT |
| 5 The Wavelet Represtations |
| 5.1 Wavelets and Fourier Analyiss |
| A. The Possible REason Behind the Gibbs Phenomenon |
| B. Illustration of Some Basic Wavelets, their Fourier Transforms and a Glimpse at the Gibbs Phenomenon |
| 5.2 Elements of Wavelet Analysis |
| A. The Continous Wavelet (Double Integral) Representation of Functions |
| B. The Discrete Wavelet (Double) Series Expansion of Functions |
| 5.3 The Discrete Wavelet Series Approximation |
| A. Preliminaries for Having Discrete Orthonormal Wavelets |
| 5.4 The Continous Wavelet Represtation |
| A. Detailed Analysis of the Gibbs Phenomenon for Even WAvelets - The Mexican Hat Wavelet |
| B. The Mexican Hat Wavelet and its Gibbs Phenomenon |
| C. Hardy-Functions Wavelets |
| D. Recent Preliminary Results |
| References |
| Appendix A |
| Index of Notations |
| Subject Index |
| Author Index |
July 1998, 364pp.
Hardbound, ISBN 0-7923-5109-6
NLG 300.00/USD 162.00/GBP 102.00
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