Convergence of cascade algorithms in Sobolev spaces and integrals of wavelets

Abstract

The cascade algorithm with mask a and dilation M generates a sequence φn, n = 1, 2,..., by the iterative process φn(x) = ∑αεℤs a(α)φn-1(Mx - α) x ε ℝs, from a starting function φ0, 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.