Approximation Properties of Non-Separable Wavelet Bases With Isotropic Scaling Matrices And Their Relation to Besov Spaces

Abstract

In this thesis we investigate the connection between non-separable wavelet bases and Besov spaces. The well known results about the characterization of Besov spaces via dyadic wavelet expansions are extended for those cases where the dilation is given by a general expanding isotropic integer matrix. Beside the Quincunx matrix or the Box-spline matrix we present other scaling matrices for non-separable wavelets. Non-separable wavelets are capable to detect sufficiently precise structures that are not only horizontal, vertical or diagonal but arbitrarily orientated. So far it is not known how the proofs of the approximation theory can be adopted from the dyadic separable case to the more general non-separable case with arbitrary scaling matrices. In this thesis we close the gap between separable and non-separable wavelet approximation. We show which Besov spaces can be characterized by non-separable wavelet expansions.This thesis consists of two main parts: linear and nonlinear wavelet approximation. First we investigate approximation andsmoothness properties of shift-invariant spaces generated bynon-separable scaling functions. In particular we prove suitable Jackson and Bernstein estimates for these spaces. They are the most important ingredients for the general theory for the characterization of Besov spaces via wavelet expansions. At the end of the first part we present a norm equivalence between a discrete version of a Besov norm and a weighted sequence norm of wavelet coefficients. In the second part we investigate approximation spaces for the N-term approximation with non-separable wavelet bases. Here we use the norm equivalences from the first part to present an adaptive choice of N wavelet coefficients and to determine the rate of approximation. It will be shown that spaces with the same approximation rate are again Besov spaces.