Abstract: | We consider solutions of a refinement equation of the form where is a finitely supported sequence called the refinement mask. Associated with the mask is a linear operator defined on by . This paper is concerned with the convergence of the cascade algorithm associated with , i.e., the convergence of the sequence in the -norm. Our main result gives estimates for the convergence rate of the cascade algorithm. Let be the normalized solution of the above refinement equation with the dilation matrix being isotropic. Suppose lies in the Lipschitz space , where and . Under appropriate conditions on , the following estimate will be established:
where and is a constant. In particular, we confirm a conjecture of A. Ron on convergence of cascade algorithms. |