Convergence of subdivision schemes associated with nonnegative masks

This paper is concerned with refinement equations of the type f = ∑ a(α)f(M · - α), α∈ℤ^s where f is the unknown function defined on the s-dimensional Euclidean space ℝ^s-a, a is a finitely supported sequence on ℤ^s, and M is an s × s dilation matrix with m := | det M|. The solution of a refinement equation can be obtained by using the subdivision scheme associated with the mask. In this paper we give a characterization for the convergence of the subdivision scheme when the mask is nonnegative. Our method is to relate the problem of convergence to m column-stochastic matrices induced by the mask. In this way, the convergence of the subdivision scheme can be determined in a finite number of steps by checking whether each finite product of those column-stochastic matrices has a positive row. As a consequence of our characterization, we show that the convergence of the subdivision scheme with a nonnegative mask depends only on the location of its positive coefficients. Several examples are provided to demonstrate the power and applicability of our approach.