Nonuniform wavelets and wavelet sets related to one-dimensional spectral pairs

Abstract

A generalization of Mallat's classical multiresolution analysis, based on the theory of spectral pairs, was considered in two articles by Gabardo and Nashed. In this setting, the associated translation set is no longer a discrete subgroup of R but a spectrum associated with a certain one-dimensional spectral pair and the associated dilation is an even positive integer related to the given spectral pair. As a generalization of Dai, Larson, and Speegle's theory of wavelet sets, we prove in this paper the existence of nonuniform wavelet sets associated with the same translation and dilation parameters.