The stability of subdivision operator at its fixed point

The paper analysis the correlation between the existence of smooth compactly supported solutions of the univariate two-scale refinement equation and the convergence of the corresponding cascade lgorithm/subdivision scheme. We introduce a criterion that expresses this correlation in terms of mask of the equation. We show that the convergence of subdivision scheme depends on values that the mask takes at the points of its generalized cycles. This means in particular that the stability of shifts of refinable function is not necessary for the convergence of the subdivision process. This also leads to some results on the degree of convergence of subdivision processes and on factorizations of refinable functions.