Applied and computational aspects of nonlinear wavelet approximation
A. M. Cohen · Cambridge University Press eBooks · 2001
Nonlinear approximation has recently found computational applications such as data compression, statistical estimation or adaptive schemes for partial differential or integral equations, especially through the development of wavelet-based methods. The goal of this paper is to provide with a short survey of nonlinear wavelet approximation in the perspective of these applications, as well as to stress some remaining open questions. 1. Introduction Numerous problems of approximation theory have in common the following general setting: we are given a family of subspaces (SN ) N0 of a normed space X, and for f 2 X, we consider the best approximation error oe N (f) := inf g2SN kf \\Gamma gkX : (1) Typically, N represents the number of parameters needed to describe an element in SN , and in most cases of interest, oe N (f) goes to zero as this number tends to infinity. For a given f , we can then study the rate of approximation, i.e. the range of r 0 for which there exists C ? 0 such th...