The correlation between the convergence of subdivision processes and solvability of refinement equations
AbstractWe consider a univariate two-scale difference equation,which is studied in approximation theory, curve design andwavelets theory. This paper analysis the correlation between the existence ofsmooth compactly supported solutions of this equation and the convergence of the corresponding cascade algorithm/subdivision scheme. We introduce a criterion thatexpresses this correlation in terms of mask of the equation. It was shown that the convergence of subdivision schemedepends on values that the mask takes at the points of its generalized cycles. In this paper we show that the criterion is sharp in thesense that an arbitrary generalized cycle causes the divergenceof a suitable subdivision scheme. To do this we constructa general method to produce divergent subdivision schemeshaving smooth refinable functions. The criterion thereforeestablishes a complete classification of divergent subdivision schemes.
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Bibliographic InfoPaper provided by Erasmus University Rotterdam, Econometric Institute in its series Econometric Institute Report with number EI 2001-45.
Date of creation: 14 Dec 2001
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Cycles; Cascade algorithm; Refinement equations; Subdivision process; Rate of convergence;
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