Bases of translates and multiresolution analyses

Abstract

Using the theory of basis generators we study various properties of multivariate Riesz and orthonormal sequences of translates, with emphasis on those associated with multiresolution analyses and their connection with wavelets. In particular, we show that every multiresolution analysis of multiplicity n generated by a dilation matrix preserving the lattice Z d has an orthonormal wavelet system associated with it, and give a closed form representation in Fourier space for such wavelet systems. We illustrate these results by applying them to the case of univariate wavelets associated with multiresolution analyses with binary dilations.