Wavelets, Approximation, and Statistical Applications

✍ Scribed by Wolfgang Härdle, Gerard Kerkyacharian, Dominique Picard, Alexander Tsybakov (auth.)

• Publisher: Springer-Verlag New York
• Year: 1998
• Tongue: English
• Leaves: 275
• Series: Lecture Notes in Statistics 129
• Edition: 1
• Category: Library

✦ Synopsis

The mathematical theory of ondelettes (wavelets) was developed by Yves Meyer and many collaborators about 10 years ago. It was designed for ap­ proximation of possibly irregular functions and surfaces and was successfully applied in data compression, turbulence analysis, image and signal process­ ing. Five years ago wavelet theory progressively appeared to be a power­ ful framework for nonparametric statistical problems. Efficient computa­ tional implementations are beginning to surface in this second lustrum of the nineties. This book brings together these three main streams of wavelet theory. It presents the theory, discusses approximations and gives a variety of statistical applications. It is the aim of this text to introduce the novice in this field into the various aspects of wavelets. Wavelets require a highly interactive computing interface. We present therefore all applications with software code from an interactive statistical computing environment. Readers interested in theory and construction of wavelets will find here in a condensed form results that are somewhat scattered around in the research literature. A practioner will be able to use wavelets via the available software code. We hope therefore to address both theory and practice with this book and thus help to construct bridges between the different groups of scientists. This te. xt grew out of a French-German cooperation (Seminaire Paris­ Berlin, Seminar Berlin-Paris). This seminar brings together theoretical and applied statisticians from Berlin and Paris. This work originates in the first of these seminars organized in Garchy, Burgundy in 1994.

✦ Table of Contents

Front Matter....Pages i-xviii
Wavelets....Pages 1-16
The Haar basis wavelet system....Pages 17-23
The idea of multiresolution analysis....Pages 25-29
Some facts from Fourier analysis....Pages 31-34
Basic relations of wavelet theory....Pages 35-45
Construction of wavelet bases....Pages 47-58
Compactly supported wavelets....Pages 59-69
Wavelets and Approximation....Pages 71-100
Wavelets and Besov Spaces....Pages 101-124
Statistical estimation using wavelets....Pages 125-191
Wavelet thresholding and adaptation....Pages 193-213
Computational aspects and statistical software implementations....Pages 215-235
Back Matter....Pages 237-268

✦ Subjects

Mathematics, general