Numerical Analysis of Wavelet Methods

Abstract

Introduction. Notations. 1. Basic examples. 1.1 Introduction. 1.2 The Haar system. 1.3 The Schauder hierarchical basis. 1.4 Multivariate constructions. 1.5 Adaptive approximation. 1.6 Multilevel preconditioning. 1.7 Conclusions. 1.8 Historical notes. 2. Multiresolution approximation. 2.1 Introduction. 2.2 Multiresolution analysis. 2.3 Refinable functions. 2.4 Subdivision schemes. 2.5 Computing with refinable functions. 2.6 Wavelets and multiscale algorithms. 2.7 Smoothness analysis. 2.8 Polynomial exactness. 2.9 Duality, orthonormality and interpolation. 2.10 Interpolatory and orthonormal wavelets. 2.11 Wavelets and splines. 2.12 Bounded domains and boundary conditions. 2.13 Point values, cell averages, finite elements. 2.14 Conclusions. 2.15 Historical notes. 3. Approximation and smoothness. 3.1 Introduction. 3.2 Function spaces. 3.3 Direct estimates. 3.4 Inverse estimates. 3.5 Interpolation and approximation spaces. 3.6 Characterization of smoothness classes. 3.7 Lp-unstable approximation and 0 1. 3.8 Negative smoothness and Lp-spaces. 3.9 Bounded domains. 3.10 Boundary conditions. 3.11 Multilevel preconditioning. 3.12 Conclusions. 3.13 Historical notes. 4. Adaptivity. 4.1 Introduction. 4.2 Nonlinear approximation in Besov spaces. 4.3 Nonlinear wavelet approximation in Lp. 4.4 Adaptive finite element approximation. 4.5 Other types of nonlinear approximations. 4.6 Adaptive approximation of operators. 4.7 Nonlinear approximation and PDE's. 4.8 Adaptive multiscale processing. 4.9 Adaptive space refinement. 4.10 Conclusions. 4.11 Historical notes. References. Index.