Proposed Running Head : ADAPTIVE APPROXIMATIONS BY WAVELETS

Given the wavelet expansion of a function v, a non-linear adaptive approximation of v is obtained by neglecting those coeecients whose size drops below a certain threshold. We propose several ways to deene the threshold: all are based on the characterization of the local regularity of v (in a Sobolev or Besov scale) in terms of summability of properly deened subsets of its coeecients. A-priori estimates of the approximation error are derived. For the Haar system, the asymptotic behavior of both the approximation error and the number of survived coeecients is thoroughly investigated for a class of functions having HH older-type singularities.