Convergence of cascade algorithms associated with nonhomogeneous refinement equations

This paper is devoted to a study of multivariate nonhomogeneous refinement equations of the form [GRAPHICS] where phi = (phi (1),..., phi (r))(T) is the unknown, g = (g(1) ,..., g(r))(T) is a given vector of functions on R-s, M is an s x s dilation matrix, and a is a finitely supported refinement mask such that each a(alpha) is an r x r (complex) matrix. Let phi (0) be an initial vector in (L-2( R-s))(r). The corresponding cascade algorithm is given by [GRAPHICS] In this paper we give a complete characterization for the L-2-convergence of the cascade algorithm in terms of the refinement mask a, the nonhomogeneous term g, and the initial vector of functions phi (0).