Vector-valued nonuniform multiresolution analysis associated with linear canonical transform domain

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. In this paper, we continue the study based on this nonstandard setting and introduce vector-valued nonuniform multiresolution analysis associated with linear canonical transform (LCT-VNUMRA) where the associated subspace V?0 of the function space L2 (R,CM) has an orthonormal basis of the form {?(x ? ?)e? ??A B (t2??2)} ??? where ? = {0, r/N} + 2Z,N ? 1 is an integer and r is an odd integer such that r and N are relatively prime. We establish a necessary and sufficient condition for the existence of associated wavelets and derive an algorithm for the construction of vector-valued nonuniform multiresolution analysis starting from a vector refinement mask with appropriate conditions