Convergence of cascade algorithms in Sobolev spaces associated with multivariate refinement equations

摘要

This paper is concerned with multivariate inhomogeneous refinement equations written in the form phi (x) = Sigma (s)(alpha is an element ofZ) a(alpha)phi (Mx - a) + g(x), x is an element of R-s, where phi is the unknown function defined on the s-dimentional Euclidean space R-s, g is a given compactly supported function on R-s, a is a finitely supported sequence on P, and M is an s x s dilation matrix with m = det M. Let phi (0) be an initial function in the Sobolev space W-2(k)(R-s). For n = 1, 2,..., define phi (n)(x) = Sigma (s)(alpha epsilonZ) a(alpha)phi (n-1)(Mx - alpha)+ g(x), x is an element of R-s. Iii this paper, we give a characterization for the strong convergence in the Sobolev space W-2(k)(R-s)(k is an element of N) of the cascade sequence (phi (n))(n is an element ofN) for the case in which M is isotropic. (C) 2001 Academic Press. [References: 16]