Subdivision Schemes and Refinement Equations with Nonnegative Masks Original Research Article

We consider the two-scale refinement equation f(x)=∑Nn=0 cnf(2x−n) with ∑n c2n=∑n c2n+1=1 where c0, cN≠0 and the corresponding subdivision scheme. We study the convergence of the subdivision scheme and the cascade algorithm when all cn⩾0. It has long been conjectured that under such an assumption the subdivision algorithm converge, and the cascade algorithm converge uniformly to a continuous function, if and only if only if 00} is 1. We prove the conjecture for a large class of refinement equations.