We consider a univariate two-scale difference equation, which is studied in approximation theory, curve design and wavelets theory. This paper analysis 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 mask of the equation. It was shown that the convergence of subdivision scheme depends on values that the mask takes at the points of its generalized cycles. In this paper we show that the criterion is sharp in the sense that an arbitrary generalized cycle causes the divergence of a suitable subdivision scheme. To do this we construct a general method to produce divergent subdivision schemes having smooth refinable functions. The criterion therefore establishes a complete classification of divergent subdivision schemes.

Cascade algorithm, Cycles, Rate of convergence, Refinement equations, Subdivision process
hdl.handle.net/1765/594
Econometric Institute Research Papers
Report / Econometric Institute, Erasmus University Rotterdam
Erasmus School of Economics

Protassov, V. (2001). The correlation between the convergence of subdivision processes and solvability of refinement equations. (No. EI 2001-45). Report / Econometric Institute, Erasmus University Rotterdam. Retrieved from http://hdl.handle.net/1765/594