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.
1. Basic examples.
1.2 The Haar system.
1.3 The Schauder hierarchical basis.
1.4 Multivariate constructions.
1.5 Adaptive approximation.
1.6 Multilevel preconditioning.
1.8 Historical notes.
2. Multiresolution approximation.
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.
2.15 Historical notes.
3. Approximation and smoothness.
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<pLp-spaces.
3.9 Bounded domains.
3.10 Boundary conditions.
3.11 Multilevel preconditioning.
3.13 Historical notes.
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.
4.11 Historical notes.