Piecewise-smooth refinable functions

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Abstract:
In this paper we give a complete classification of univariate piecewise smooth refinable functions(i.e. compactly supported solutions of the equation $\varphi \left( \frac{x}{2}\right) = \sum\limits_{k = 0}^N c_k \varphi (x -k)$). By characterizing the structure of refinable splineswe elaborate a simple criterion of convergence of the subdivision schemes, corresponding to such splines, and compute explicitly therate of convergence. These results make it possible to prove a factorization theorem about a decomposition of any smooth refinable function (not necessarily stable or corresponding to a convergent subdivision scheme) into a convolution of a continuous refinable function with the corresponding order refinable spline. This gives an application to one problem of the combinatorial number theory (the asymptotics of Euler's partition function). The results of the paper generalize several previously known results on this direction and help solving two open problems.

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Date: 2003-05-01