The stability of subdivision operator at its fixed point

We consider the univariate two-scale refinement equation phi (x) = Sigma (N )(k=0) c(k)phi (2x - k), where c(0),..., c(N) are complex values and Sigmac (k) = 2. This paper analyzes the correlation between the existence of smoot h compactly supported solutions of this equation and the convergence of the corresponding cascade algorithm/subdivision scheme. We introduce a criteri on that expresses this correlation in terms of the mask of the equation. We show that the convergence of the subdivision scheme depends on values that the mask takes at the points of its generalized cycles. This means in part icular that the stability of shifts of refinable function is not necessary for the convergence of the subdivision process. This also leads to some res ults on the degree of convergence of subdivision processes and on factoriza tions of refinable functions.