Convergence rates of cascade algorithms

We consider solutions of a refinement equation of the form [GRAPHICS] where a is a finitely supported sequence called the refinement mask. Associated with the mask a is a linear operator Q(a) defined on L-p(R-s) by Q(a)psi := Sigma(gammais an element ofZs) a(gamma)psi(M. - gamma). This paper is concerned with the convergence of the cascade algorithm associated with a, i.e., the convergence of the sequence (Q(a)(n)psi) n = 1, 2,... in the L-p-norm. Our main result gives estimates for the convergence rate of the cascade algorithm. Let phi be the normalized solution of the above refinement equation with the dilation matrix M being isotropic. Suppose phi lies in the Lipschitz space Lip(mu, L-p(R-s)), where mu > 0 and 1 less than or equal to p less than or equal to 1. Under appropriate conditions on, the following estimate will be established: parallel toQ(a)(n)psi - phiparallel to(p) less than or equal to C(m(-1/s))(mun) For Alln is an element of N, where m := \det M\ and C is a constant. In particular, we confirm a conjecture of A. Ron on convergence of cascade algorithms.