The correlation between the convergence of subdivision processes and solvability of refinement equations.

Abstract

We consider a univariate two-scale difference equation,\nwhich is studied in approximation theory, curve design and\nwavelets theory. This paper analysis the correlation between the existence of\nsmooth compactly supported solutions of this equation and the convergence \nof the corresponding cascade algorithm/subdivision scheme. We introduce a criterion that\nexpresses this correlation in terms of mask of the equation. \nIt was shown that the convergence of subdivision scheme\ndepends on values that the mask takes at the points of its \ngeneralized cycles. In this paper we show that the criterion is sharp in the\nsense that an arbitrary generalized cycle causes the divergence\nof a suitable subdivision scheme. To do this we construct\na general method to produce divergent subdivision schemes\nhaving smooth refinable functions. The criterion therefore\nestablishes a complete classification of divergent subdivision schemes