A note on adaptive approximation in Sobolev spaces

In this article, we consider the adaptive approximation in Sobolev spaces. After establishing some norm equivalences and inequalities in Besov spaces, we are able to prove that the best N terms approximation with wavelet‐like basis in Sobolev spaces exhibits the proper approximation order in terms of N^?1. This indicates that the computational load in adaptive approximation is proportional to the approximation accuracy. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007