Convergence of Cascade Algorithms in Sobolev Spaces Associated with Multivariate Refinement Equations1

This paper is concerned with multivariate inhomogeneous refinement equations written in the form ϕ x = α∈ s a α ϕ Mx − α + g x x ∈ s , where ϕ is the unknown function defined on the s-dimentional Euclidean space s g is a given compactly supported function on s , a is a finitely supported sequence on s , and M is an s × s dilation matrix with m = detM .Le t ϕ0 be an initial function in the Sobolev space W k 2 s .Fo r n = 1 2 define ϕn x = α∈ s a α ϕn−1 Mx−α + g x x ∈ s.In this paper, we give a characterization for the strong convergence in the Sobolev space W k 2 s k ∈ of the cascade sequence ϕn n∈ for the case in which M is isotropic.