Convergence of cascade algorithms in Sobolev spaces and integrals of wavelets

The cascade algorithm with mask a and dilation M generates a sequence φ n , n = 1, 2,. .. , by the iterative process φ n (x) = α∈Z s a(α)φ n−1 (Mx − α) x ∈ R s , from a starting function φ 0 , where M is a dilation matrix. A complete characterization is given for the strong convergence of cascade algorithms in Sobolev spaces for the case in which M is isotropic. The results on the convergence of cascade algorithms are used to deduce simple conditions for the computation of integrals of products of derivatives of refinable functions and wavelets.