The Two-directional Refinement Equation on a Locally Compact Abel Group

Abstract

In this paper, we mainly describe the situation of establishing the bidirectional multiresolution analysis on a locally compact Abel group G which is the weak direct product of a countable set of cyclic groups of p th order. We first introduce the concept of the Abel group. Then we study the existence of the solution of the two-directional refinement equation and establish the necessary conditions to ensure that the equation has a compactly supported distribution. Furthermore, the necessary conditions for the solution of the distribution of the two-directional refinement equation to generate a bi-directional multiresolution analysis are given.