Cover; Contents; Introduction; Notations; Chapter 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; Chapter 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 wavelets2.11 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; Chapter 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 < p < 1; 3.8 Negative smoothness and LP-spaces; 3.9 Bounded domains; 3.10 Boundary conditions; 3.11 Multilevel preconditioning ; 3.12 Conclusions3.13 Historical notes; Chapter 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