L p Solutions of Vector Refinement Equations with General Dilation Matrix

The purpose of this paper is to investigate the solutions of refinement equations of the form .(x)=...Zsa(.).(Mx..),x.Rs, where the vector of functions . = (.1, . . . ,.r)T is in (L p (.s))r, 0 < p . ., a(.), . . .s. is a finitely supported sequence of r. r matrices called the refinement mask, and M is an s . s integer matrix such that limn. . M .n = 0. In this article, we characterize the existence of an L p .solution of the refinement equation for 0 < p . .. Our characterizations are based on the p.norm joint spectral radius.