Multivariate refinement equation with nonnegative masks

摘要

This paper is concerned with multivariate refinement equations of the type φ = ∑_(α∈Z~s) a(α)φ(Mx-α). 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 x 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.