Abstract
We consider the univariate two-scale refinement equation $\varphi(x) = {\textstyle\Sigma^N_{k=0}}c_k\varphi(2x-k)$, where $c_0, \ldots, c_N$ are complex values and ${\textstyle\Sigma} c_k = 2$.This paper analyzes the correlation between the existence of smooth compactly supported solutions of this equation and the convergence of the corresponding cascade algorithm/subdivision scheme. We introduce a criterion 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 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.
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