An adaptive compression algorithm in Besov spaces

Given a function f on [0, 1] and a wavelet-type expansion of f, we introduce a new algorithm providing an approximation (f) over tilde of f with a prescribed number D of nonzero coefficients in its expansion. This algorithm depends only on the number of coefficients to be kept and not on any smoothness assumption on f. Nevertheless it provides the optimal rate D-alpha of approximation with respect to the L-q-norm when f belongs to some Besov space B-p, infinity(alpha) whenever alpha > (1/p - 1/q)(+). These results extend tb more general expansions including splines and piecewise polynomials and to multivariate functions. Moreover, this construction allows us to compute easily the metric entropy of Besov balls.