Spectral properties of the transition operator associated to a multivariate refinement equation

Abstract

Given a finitely supported sequence a on Z s and an s×s dilation matrix M, the transition operator T a is the linear operator defined by T av(α):=∑ β∈ Z s a(Mα−β)v(β) , where α∈ Z s and v lies in ℓ 0( Z s) , the linear space of all finitely supported sequences on Z s . In this paper we investigate the spectral properties of the transition operator T a and apply these properties to the study of the approximation and smoothness properties of the normalized solution of the refinement equation φ=∑ α∈ Z s a(α)φ(M·−α) .