Dinh Dung

Abstract
We give a brief survey on  non-linear $n$-approximations  using wavelet decompositions. In these approximations, we resolve a target function  by linear combinations of $n$ free terms of a wavelet series, which constitute a non-linear set. The wavelets which  form wavelet decompositions are  B-splines and dyadic scales of de la Vall\'ee Pussin kernels. The selection of $n$ terms depends on the target function. Central questions  to be focused are what, if any,  the advantages of non-linear $n$-term approximations over linear ones, and the differences between univariate and multivariate non-linear $n$-term approximations. These questions will be discussed mainly in terms of  asymptotic orders of error of the best non-linear $n$-term approximation and non-linear $n$-widths based on optimal continuous algorithms of $n$-term approximation, for smoothness classes of functions.