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Wavelet methods in numerical analysis. (English) Zbl 0976.65124
Ciarlet, P. G. (ed.) et al., Solution of equations in \(\mathbb{R}^n\) (Part 3). Techniques of scientific computing (Part 3). Amsterdam: North-Holland/ Elsevier (ISBN 0-444-50350-1). Handbook of Numerical Analysis 7, 417-711 (2000).
The paper gives a survey to the theoretical results that are involved in the numerical analysis of wavelet methods. Special attention is paid to the relation between wavelet decompositions and function spaces, in particular Besov spaces. “…the possibility of characterizing various smoothness classes, e.g. Sobolev, Hölder and Besov spaces, from the numerical properties of multiscale decompositions, turns out to play a key role in the application of wavelet methods.” The paper is organized as follows:
I. Basic examples (Haar basis, Schauder basis);
II. Multiresolution approximation;
III. Multiscale decomposition of function spaces;
IV. Nonlinear approximation and adaptivity.
The survey includes the theory of multilevel preconditioning and the nonlinear approximation of functions (data compression) and operators (sparse representation of full matrices).
For the entire collection see [Zbl 0953.00016].

MSC:
65T60 Numerical methods for wavelets
65Dxx Numerical approximation and computational geometry (primarily algorithms)
41A46 Approximation by arbitrary nonlinear expressions; widths and entropy
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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