Wavelets associated with vector-valued nonuniform multiresolution analysis

Abstract

We consider a generalization of the notion of non-uniform multiresolution analysis (NUMRA) which is called vector-valued non-uniform multiresolution analysis (VNUMRA). The concept of NUMRA was introduced by Gabardo and Nashed based on the theory of spectral pairs. Xia and Suter introduced vector-valued multiresolution analysis and orthogonal vector-valued wavelets. We introduce VNUMRA where the associated subspace of has, an orthonormal basis, a collection of translates of a vector-valued function of the form where , is an integer and is an odd integer with such that and are relatively prime and is the set of all integers and the corresponding dilation factor is . We obtain the necessary and sufficient condition for the existence of associated wavelets and present a construction of VNUMRA.