Restricted nonlinear approximation

We introduce a new form of nonlinear approximation called restricted approx imation. It is a generalization of n-term wavelet approximation in which a weight function is used to control the terms in the wavelet expansion of th e approximant. This form of approximation occurs in statistical estimation and in the characterization of interpolation spaces for certain pairs of Lp and Besov spaces. We characterize, both in terms of their wavelet coeffici ents and also in terms of their smoothness, the functions which are approxi mated with a specified rate by restricted approximation. We also show the r elation of this form of approximation with certain types of thresholding of wavelet coefficients.