Wavelets: approximation and statistical applications

Wolfgang Härdle, Gerard Kerkyacharian, Dominique Picard, Alexander Tsybakov9780387984537, 0387984534

FROM THE PUBLISHERThe mathematical theory of wavelets was developed by Yves Meyer and many collaborators about ten years ago. It was designed for approximation of possibly irregular functions and surfaces and was successfully applied in data compression, turbulence analysis, and image and signal processing. Five years ago wavelet theory progressively appeared to be a powerful framework for nonparametric statistical problems. Efficient computation implementations are beginning to surface in the nineties. This book brings together these three streams of wavelet theory and introduces the novice in this field to these aspects. Readers interested in the theory and construction of wavelets will find in a condensed form results that are scattered in the research literature. A practitioner will be able to use wavelets via the available software code.

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Table of contents :
What can wavelets offer?……Page 13
General remarks……Page 19
Data compression……Page 20
Nonlinear smoothing properties……Page 25
Synopsis……Page 26
The Haar basis wavelet system……Page 29
Multiresolution analysis……Page 35
Wavelet system construction……Page 37
An example……Page 38
Some facts from Fourier analysis……Page 41
When do we have a wavelet expansion?……Page 45
How to construct mothers from a father……Page 52
Additional remarks……Page 54
Construction starting from Riesz bases……Page 57
Construction starting from m0……Page 64
Daubechies’ construction……Page 69
Coiflets……Page 73
Symmlets……Page 75
Introduction……Page 79
Sobolev Spaces……Page 80
Approximation kernels……Page 83
Approximation theorem in Sobolev spaces……Page 84
Periodic kernels and projection operators……Page 88
Moment condition for projection kernels……Page 92
Moment condition in the wavelet case……Page 97
Besov spaces……Page 109
Littlewood-Paley decomposition……Page 114
Approximation theorem in Besov spaces……Page 123
Wavelets and approximation in Besov spaces……Page 125
Introduction……Page 133
Linear wavelet density estimation……Page 134
Soft and hard thresholding……Page 146
Linear versus nonlinear wavelet density estimation……Page 155
Asymptotic properties of wavelet thresholding estimates……Page 170
Some real data examples……Page 178
Comparison with kernel estimates……Page 185
Regression estimation……Page 189
Other statistical models……Page 195
Different forms of wavelet thresholding……Page 199
Adaptivity properties of wavelet estimates……Page 203
Thresholding in sequence space……Page 207
Adaptive thresholding and Stein’s principle……Page 211
Oracle inequalities……Page 216
Bibliographic remarks……Page 218
Introduction……Page 221
The cascade algorithm……Page 222
Discrete wavelet transform……Page 226
Statistical implementation of the DWT……Page 228
Translation invariant wavelet estimation……Page 233
Main wavelet commands in XploRe……Page 236
Wavelet Coefficients……Page 241
BookmarkTitle:……Page 243
Software Availability……Page 244
Bernstein and Rosenthal inequalities……Page 245
A Lemma on the Riesz basis……Page 250
Bibliography……Page 252