Vector-valued nonuniform multiresolution analysis on local fields

摘要

A multiresolution analysis (MRA) on local fields of positive characteristic was defined by Shah and Abdullah 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 (VNUMRA) where the associated subspace V-0 of L-2(K, C-M) has an orthonormal basis of the form {phi(x - lambda)}(lambda is an element of Lambda) where Lambda = {0, r/N} + Z, N >= 1 is an integer and r is an odd integer such that r and N are relatively prime and Z = {u(n) : n is an element of N-0}. We establish a necessary and sufficient condition for the existence of associated wavelets and derive an algorithm for the construction of VNUMRA on local fields starting from a vector refinement mask G(xi) with appropriate conditions. Further, these results also hold for Cantor and Vilenkin groups.