Vector-Valued Nonuniform Multiresolution Associated with Linear Canonical Transform

Department of Mathematical Sciences, Islamic University of Science and Technology Awantipora, Pulwama, Jammu and Kashmir 192122, India

E-mail: gyounusg@gmail.com

Abstract

A multiresolution analysis associated with linear canonical transform was defined by Shah and Waseem for which the translation set is a discrete set which is not a group. 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^_μ0 of L²ℝℂ^M) has an orthonormal basis of the form ${\left\{\Phi (x-\lambda )e^{-}\frac{-\iota \pi A}{B}(t^2-{\lambda }^2)\right\}}_{\lambda \in \Lambda }$ where Λ = {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 on local fields starting from a vector refinement mask with appropriate conditions.