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

Given a finitely supported sequence a on Z(S) and an s x s dilation matrix M, the transition operator T-a is the linear operator defined by T(a)nu(alpha) := Sigma(B epsilon Zs) a(M alpha - beta)nu(beta), where alpha epsilon Z(s) and nu 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 phi = Sigma(alpha epsilon Zs) a(alpha)phi(M . -x). (C) 1999 Elsevier Science Inc. All rights reserved.