Numerical Analysis of Wavelet Methods

Albert Cohen (Eds.)9780444511249, 0444511245

Since their introduction in the 1980’s, wavelets have become a powerful tool in mathematical analysis, with applications such as image compression, statistical estimation and numerical simulation of partial differential equations. One of their main attractive features is the ability to accurately represent fairly general functions with a small number of adaptively chosen wavelet coefficients, as well as to characterize the smoothness of such functions from the numerical behaviour of these coefficients. The theoretical pillar that underlies such properties involves approximation theory and function spaces, and plays a pivotal role in the analysis of wavelet-based numerical methods. This book offers a self-contained treatment of wavelets, which includes this theoretical pillar and it applications to the numerical treatment of partial differential equations. Its key features are:
1. Self-contained introduction to wavelet bases and related numerical algorithms, from the simplest examples to the most numerically useful general constructions.
2. Full treatment of the theoretical foundations that are crucial for the analysis of wavelets and other related multiscale methods : function spaces, linear and nonlinear approximation, interpolation theory.
3. Applications of these concepts to the numerical treatment of partial differential equations : multilevel preconditioning, sparse approximations of differential and integral operators, adaptive discretization strategies.

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Table of contents :
Content:
Foreword
Page vii

Introduction
Pages xi-xvi

Notations
Pages xvii-xviii

Chapter 1 Basic examples Original Research Article
Pages 1-42

Chapter 2 Multiresolution approximation Original Research Article
Pages 43-153

Chapter 3 Approximation and smoothness Original Research Article
Pages 155-241

Chapter 4 Adaptivity Original Research Article
Pages 243-319

References
Pages 321-333

Index
Pages 335-336