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The objectives of this book are to survey the theoretical results that are involved in the numerical analysis of wavelet methods, and more generally of multiscale decomposition methods, for numerical simulation problems, and to provide the most relevant examples of such mathematical tools in this particular context.
Basic examples
ntroduction
The Haar system
The Schauder hierarchical basis
Multivariate constructions
Adaptive approximation
Multilevel preconditioning
Conclusions
Historical notes
Multiresolution approximation
ntroduction
Multiresolution analysis
Refinable functions
Subdivision schemes
Computing with refinable functions
Wavelets and multiscale algorithms
Smoothness analysis
Polynomial exactness
Duality, orthonormality and interpolation
nterpolatory and orthonormal wavelets
Wavelets and splines
Bounded domains and boundary conditions
Point values, cell averages, finite elements
Conclusions
Historical notes
Approximation and smoothness
ntroduction
Function spaces
Direct estimates
nverse estimates
nterpolation and approximation spaces
Characterization of smoothness classes
LP-unstable approximation and 0 p 1
Negative smoothness and LP-spaces
Bounded domains
Boundary conditions
Multilevel preconditioning
Conclusions
Historical notes
Adaptivity
ntroduction
Nonlinear approximation in Besov spaces
Nonlinear wavelet approximation in L p
Adaptive finite element approximation
Other types of nonlinear approximations
Adaptive approximation of operators
Nonlinear approximation and PDE's
Adaptive multiscale processing
Adaptive space refinement
Conclusions
Historical notes