Multivariate refinement equation with nonnegative masks

This paper is concerned with multivariate refinement equations of the type , where ϕ is the unknown function defined on the s-dimensional Euclidean space ℝ^s, a is a finitely supported nonnegative sequence on ℤ^s, and M is an s × s dilation matrix with m:=|detM|. We characterize the existence of L ₂-solution of refinement equation in terms of spectral radius of a certain finite matrix or transition operator associated with refinement mask a and dilation matrix M. For s = 1 and M = 2, the sufficient and necessary conditions are obtained to characterize the existence of continuous solution of this refinement equation.