Convergence of cascade algorithms associated with nonhomogeneous refinement equations

Abstract

This paper is devoted to a study of multivariate nonhomogeneous refinement equations of the form φ(x) = g(x) + ∑αεℤs α(α)φ(Mx - α), x ε ℝs, where φ = (φ1,. . . , φr)T is the unknown, g = (g1,. . . , gr)T 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 α(α) is an r × r (complex) matrix. Let φ0 be an initial vector in (L2(ℝs))r. The corresponding cascade algorithm is given by φk= φ + ∑αεℤs α(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. © 2000 American Mathematical Society.