The construction of r-regular wavelets for arbitrary dilations

Given a dilation matrix A and a natural number r we construct an associated r-regular multiresolution analysis with r-regular wavelet basis. Here a dilation is an n x n expansive matrix A (all eigenvalues lambda of A satisfy \ lambda \ > 1) with integer entries. This extends a theorem of Strichartz which assumes the existence of a self-affine tiling associated with the dilation A. We also prove that regular wavelets have vanishing moments.