小波導(dǎo)論

前言

　　Fourier analysis is an established subject in the core of pure and applied mathematical analysis. Not only are the techniques in this subject of fundamental importance in all areas of science and technology， but both the integral Fourier transform and the Fourier series also have significant physical interpretations. In addition， the computational aspects of the Fourier series are especially attractive， mainly because of the orthogonality property of the seties and of its simple expression in terms of only two functions： sin z andCOS X.　　Recently， the subject of "vavelet analysis" has drawn much attention from both mathematicians and engineers Mike. Analogous to Fourier analysis， there are also two important mathematical entities in wavelet analysis： the "integral wavelet transform" and the "vavelet series". The integral wavelet transform is defined to be the convolution with respect to the dilation of the reflection of some function， called a "basic wavelet"， while the wavelet series is expressed in terms of a single function， called an "：R-wavelet" （or simply， a wavelet） by means of two very simple operations： binary dilations and integral translations. However.， unlike Fourier analysis， the integral wavelet transform with a basic wavelet and the wavelet series in terms of a wavelet are intimately related. In fact， if is chosen to be the "dual" of ， then the coefficients of the wavelet series of any square-integrable function f are precisely the values of the integral wavelet transform， evaluated at the dyadic positions in the corresponding binary dilated scale levels. Since the integral wavelet transform of f simultaneously localizes f and its Fourier transform f with the zoom-in and zoom-out capability， and since there are real-time algorithms for obtaining the coefficient sequences of the wavelet series， and for recovering f from these sequences， the list of applications of wavelet analysis seems to be endless. On the other hand， polynomial spline functions are among the simplest functions for both computational and implementational purposes. Hence， they are most attractive for analyzing and constructing wavelets.

內(nèi)容概要

本書(shū)是一本小波分析的入門(mén)書(shū)，著重于樣條小波和時(shí)頻分析。書(shū)中基本內(nèi)容有Fourier分析、小波變換、尺度函數(shù)、基數(shù)樣條分析、基數(shù)樣條小波、小波級(jí)數(shù)、正交小波和小波包。本書(shū)內(nèi)容安排由淺入深，算法推導(dǎo)詳細(xì)，既有理論，又有應(yīng)用背景?！　”緯?shū)自成體系，只要求讀者具有函數(shù)論和實(shí)分析的一些基礎(chǔ)知識(shí)，適合作為高等院校理工科小波分析的入門(mén)教材，也適合科技工作者用作學(xué)習(xí)小波的指導(dǎo)讀物。

作者簡(jiǎn)介

崔錦泰（Charles K.Chui），國(guó)際著名的小波分析專(zhuān)家，IEEE會(huì)士，密蘇里大學(xué)路易分校數(shù)學(xué)與計(jì)算機(jī)科學(xué)系講座教授，該校計(jì)算調(diào)和分析研究所所長(zhǎng)，斯坦福大學(xué)顧部教授。曾擔(dān)任數(shù)個(gè)國(guó)際著名期刊和叢書(shū)的主編或編委。他在調(diào)和分析應(yīng)用、逼近及其應(yīng)用等領(lǐng)域也做出了杰出的貢獻(xiàn)，首創(chuàng)將樣條應(yīng)用于小波中。

書(shū)籍目錄

1. An　Overview 　1.1　From Fourier analysm to wavelet analysm　　1.2　The integral wavelet transform and time-frequency analysis　　1.3　Inversion formulas and duals　　1.4　Classification of wavelets　　1.5　Multiresolution analysis, splines, and wavelets　　1.6　Wavelet decompositions and reconstructions　2. Fourier Analysis　　2.1　Fourier and inverse Fourier transforms　　2.2　Continuous-time convolution and the delta function　　2.3　Fourier transform of square-integrable functions　　2.4　Fourier series　　2.5　Basic convergence theory and Poisson's summation formula　3. Wavelet Transforms and Time-Frequency Analysis　　3.1　The Gabor transform　　3.2　Short-time Fourier transforms and the Uncertainty Principle　　3.3　The integral wavelet transform　　3.4　Dyadic wavelets and inversions　3.5　Frames　　3.6　Waveletcseries　4.cCardinalcSplinecAnalysis　　4.1　Cardinalcsplinecspaces　　4.2　B-splinescandctheircbasiccpropertiesc　4.3　Thectwo-scalecrelationcandcancinterpolatorycgraphicalcdisplaycalgorithm　　4.4　B-netcrepresentationscandccomputationcofccardinalcsplines　　4.5　Constructioncofcsplinecapproximationcformulas　　4.6　Constructioncofcsplinecinterpolationcformulas5.cScalingcFunctionscandcWavelets　5.1　Multiresolutioncanalysis　　5.2　Scalingcfunctionscwithcfinitectwo-scalecrelations　　5.3　Direct-sumcdecompositionscofcL2(R)　　5.4　Waveletscandctheircduals　　5.5　Linear-phasecfiltering　　5.6　Compactlycsupportedcwavelets　6.cCardinalcSpline-Wavelets　　6.1　Interpolatorycspline-wavelets　　6.2　Compactlycsupportedcspline-wavelets　　6.3　Computationcofccardinalcspline-wavelets　　6.4　EulerFrobeniuspolynomials　　6.5　Errorcanalysiscincsplinecwaveletcdecomposition　　6.6　Totalcpositivity,ccompletecoscillation,czero-crossings　7.cOrthogonalcWaveletscandcWaveletcPackets　　7.1　Examplescofcorthogonalcwavelets　　7.2　Identificationcofcorthogonalctwo-scalecsymbols　　7.3　Constructioncofccompactlycsupportedcorthogonalcwavelets　　7.4　Orthogonalcwaveletcpackets　　7.5　Orthogonalcdecompositioncofcwaveletcseries　NotesReferences　Subject Index　Appendixc

章節(jié)摘錄

　　An Overview　　"Wavelets" has been a very popular topic of conversations in many scientific and engineering gatherings these days. Some view wavelets as a new basis for representing functions, some consider it as a technique for time-frequency analysis, and others think of it as a new mathematical subject. Of course, all of them are right, since "wavelets" is a versatile tool with very rich mathematical content and great potential for applications. However, as this subject is still in the midst of rapid development, it is definitely too early to give a unified presentation. The objective of this book is very modest: it is intended to be used as a textbook for an introductory one-semester course on "wavelet analysis" for upper-division undergraduate or beginning graduate mathematics and engineering students, and is also written for both mathematicians and engineers who wish to learn about the subject. For the specialists, this volume is suitable as complementary reading to the more advanced monographs, such as the two volumes of Ondelettes et Operateurs by Yves Meyer, the edited volume of Wavelets-A Tutorial in Theory and Applications in this series, and the forthcoming CBMS volume by Ingrid Danbechies.　　Since wavelet analysis is a relatively new subject and the approach and organization in this book are somewhat different from that in the others, the goal of this chapter is to convey a general idea of what wavelet analysis is about and to describe what this book aims to cover.　　1.1. From Fourier analysis to wavelet analysis

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　　《小波導(dǎo)論(英文版)》是小波分析方面奠基性的經(jīng)典著作，已被翻譯為多種語(yǔ)言，產(chǎn)生了深遠(yuǎn)影響?！　　缎〔▽?dǎo)論(英文版)》著重講述樣條小波和時(shí)頻分析，內(nèi)容安排由淺入深，算法推導(dǎo)詳細(xì)，既有理論，又有應(yīng)用背景。書(shū)中內(nèi)容只要求讀者具有函數(shù)論和實(shí)分析的一些基礎(chǔ)知識(shí)，既適合初學(xué)者以及工程、技術(shù)方面人員學(xué)習(xí)，也是研究人員不可或缺的參考書(shū)。

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