Wavelet methods in numerical analysis (Q2702613)

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. ``\dots 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:NEWLINENEWLINENEWLINEI. Basic examples (Haar basis, Schauder basis);NEWLINENEWLINENEWLINEII. Multiresolution approximation;NEWLINENEWLINENEWLINEIII. Multiscale decomposition of function spaces;NEWLINENEWLINENEWLINEIV. Nonlinear approximation and adaptivity.NEWLINENEWLINENEWLINEThe survey includes the theory of multilevel preconditioning and the nonlinear approximation of functions (data compression) and operators (sparse representation of full matrices).NEWLINENEWLINEFor the entire collection see [Zbl 0953.00016].