Convergence rates of cascade algorithms associated with nonhomogeneous refinement equations ✩

This paper is concerned with nonhomogeneous refinement equations of the form ϕ(x) = α∈Zs a(α)ϕ(Mx − α)+g(x), x ∈ Rs , where the vector of functions ϕ = (ϕ1, . . .,ϕr )T is unknown, g is a given vector of compactly supported functions on Rs , a is a finitely supported sequence of r × r matrices called the refinement mask, andM is an s×s integer matrix such that limn→∞M−n = 0. Our approach will be to consider the convergence rates of the cascade algorithms associated with nonhomogeneous refinement equations mentioned above. The cascade algorithms associated with mask a, nonhomogeneous term g, and dilation matrix M generates a sequence ϕn, n = 1, 2, . . . , by the iterative process ϕn(x) = α∈Zs a(α)ϕn−1(Mx − α)+ g(x), x ∈ Rs , from a starting vector of function ϕ0 in (Lp(Rs ))r (0