Vector-Valued Nonuniform Multiresolution Associated with Linear Canonical Transform

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 vμ0 $V_0^\mu $V0μ of L2ℝℂM) has an orthonormal basis of the form ${\left\{ {\Phi (x - \lambda ){e^ - }\frac{{ - \iota \pi A}}{B}({t^2} - {\lambda ^2})} \right\}_{\lambda \in \Lambda }}${ Φ(x−λ)e−−ιπAB(t2−λ2) }λ∈Λ 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.

Keywords

• non-uniform multiresolution analysis
• linear canonical transform
• scaling function
• vector-valued wavelets