Convergence of Cascade Algorithms Associated with Nonhomogeneous Refinement Equations

This paper is devoted to a study of multivariate nonhomogeneous refinement equations of the form φ(x) = g(x) + ∑ α∈Zs a(α)φ(Mx − α), x ∈ R, where φ = (φ1, . . . , φr) is the unknown, g = (g1, . . . , gr) is a given vector of functions on Rs, M is an s× s dilation matrix, and a is a finitely supported refinement mask such that each a(α) is an r × r (complex) matrix. Let φ0 be an initial vector in (L2(R)) . The corresponding cascade algorithm is given by φk := g + ∑ α∈Zs a(α)φk−1(M · − α), k = 1, 2, . . . . In this paper we give a complete characterization for the L2-convergence of the cascade algorithm in terms of the refinement mask a, the nonhomogeneous term g, and the initial vector of functions φ0.