Overview

The mathematical theory of wavelets was developed by Yes 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 toghether 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.

Full Product Details

Author:   Wolfgang Härdle ,  Gerard Kerkyacharian ,  Dominique Picard ,  Alexander Tsybakov
Publisher:   Springer-Verlag New York Inc.
Imprint:   Springer-Verlag New York Inc.
Volume:   129
Dimensions:   Width: 15.50cm , Height: 1.50cm , Length: 23.50cm
Weight:   0.441kg
ISBN:  

9780387984537

ISBN 10:   0387984534
Pages:   265
Publication Date:   30 April 1998
Audience:   College/higher education ,  Professional and scholarly ,  Postgraduate, Research & Scholarly ,  Professional & Vocational
Format:   Paperback
Publisher's Status:   Active
Availability:   In Print  
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Table of Contents

1 Wavelets.- 1.1 What can wavelets offer?.- 1.2 General remarks.- 1.3 Data compression.- 1.4 Local adaptivity.- 1.5 Nonlinear smoothing properties.- 1.6 Synopsis.- 2 The Haar basis wavelet system.- 3 The idea of multiresolution analysis.- 3.1 Multiresolution analysis.- 3.2 Wavelet system construction.- 3.3 An example.- 4 Some facts from Fourier analysis.- 5 Basic relations of wavelet theory.- 5.1 When do we have a wavelet expansion?.- 5.2 How to construct mothers from a father.- 5.3 Additional remarks.- 6 Construction of wavelet bases.- 6.1 Construction starting from Riesz bases.- 6.2 Construction starting from m0.- 7 Compactly supported wavelets.- 7.1 Daubechies’ construction.- 7.2 Coiflets.- 7.3 Symmlets.- 8 Wavelets and Approximation.- 8.1 Introduction.- 8.2 Sobolev Spaces.- 8.3 Approximation kernels.- 8.4 Approximation theorem in Sobolev spaces.- 8.5 Periodic kernels and projection operators.- 8.6 Moment condition for projection kernels.- 8.7 Moment condition in the wavelet case.-9 Wavelets and Besov Spaces.- 9.1 Introduction.- 9.2 Besov spaces.- 9.3 Littlewood-Paley decomposition.- 9.4 Approximation theorem in Besov spaces.- 9.5 Wavelets and approximation in Besov spaces.- 10 Statistical estimation using wavelets.- 10.1 Introduction.- 10.2 Linear wavelet density estimation.- 10.3 Soft and hard thresholding.- 10.4 Linear versus nonlinear wavelet density estimation.- 10.5 Asymptotic properties of wavelet thresholding estimates.- 10.6 Some real data examples.- 10.7 Comparison with kernel estimates.- 10.8 Regression estimation.- 10.9 Other statistical models.- 11 Wavelet thresholding and adaptation.- 11.1 Introduction.- 11.2 Different forms of wavelet thresholding.- 11.3 Adaptivity properties of wavelet estimates.- 11.4 Thresholding in sequence space.- 11.5 Adaptive thresholding and Stein’s principle.- 11.6 Oracle inequalities.- 11.7 Bibliographic remarks.- 12 Computational aspects and software.- 12.1 Introduction.- 12.2 The cascade algorithm.- 12.3 Discrete wavelet transform.- 12.4 Statistical implementation of the DWT.- 12.5 Translation invariant wavelet estimation.- 12.6 Main wavelet commands in XploRe.- A Tables.- A.1 Wavelet Coefficients.- A.2.- B Software Availability.- C Bernstein and Rosenthal inequalities.- D A Lemma on the Riesz basis.- Author Index.

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