Convergence Rates of Multiscale and Wavelet Expansions

Abstract

Several results are proved which characterize the rate at which wavelet and multiresolution expansions converge to functions in a given Sobolev space in the supremum error norm. Some of the results are proved without assuming existence of a scaling function in the multiresolution analysis. Necessary and sufficient conditions are given for convergence at given rates in terms of behavior of Fourier transforms of the wavelet or scaling function near the origin. Such conditions turn out in special cases to be equivalent to moment conditions and other known conditions determining convergence rates.