The Correlation Between the Convergence of Subdivision Processes and Solvability of Refinement Equations
MOSCOW STATE UNIV (RUSSIA) FACULTY OF MECHANICS AND MATHEMATICS
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We consider the univariate two-scale refinement equation phichi. This paper analyses the correlation between the existence of smooth compactly supported solutions of this equation and the convergence of the corresponding cascade algorithmsubdivision scheme. In the work 11 we have introduced a criterion that expresses this correlation in terms of the mask of the equation. It is 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.
- Numerical Mathematics
- Theoretical Mathematics