Convergence of Cascade Algorithms in Sobolev Spaces Associated with Inhomogeneous Refinement Equations Original Research Article

In this paper, we consider multivariate inhomogeneous refinement equations of the form ϕ(x)=∑α∈Zs a(α) ϕ(2x−α)+g(x), x∈Rs, where ϕ=(ϕ1, …, ϕr)T is the unknown, g=(g1, …, gr)T is a given vector of functions on Rs, and a is a finitely supported refinement mask such that each a(α) is an r×r (complex) matrix. Let ϕ0 be an initial vector of functions in the Sobolev space Wk2(Rs). The corresponding cascade algorithm is given by ϕn(x)=∑α∈Zs a(α) ϕn−1(2x−α)+g(x), x∈Rs, n=1, 2, …. A characterization is given for the strong convergence of the cascade algorithm in the Sobolev space Wk2(Rs) (k∈N) in terms of the refinement mask a, the inhomogeneous term g, and the initial vector of functions ϕ0.