Subdivision Schemes and Refinement Equations with Nonnegative Masks

We consider the two-scale refinement equation f(x) = ∑N n=0 cnf(2x − n) with ∑ n c2n = ∑ n c2n+1 = 1 where c0, cN 6= 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, as well as the cascade algorithm converge uniformly to a continuous function, if and only if only if 0 < c0, cN < 1 and the greatest common divisor of S = {n : cn > 0} is 1. We prove the conjecture for a large class of refinement equations.