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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

Piecewise-smooth refinable functions


Author: V. Yu. Protasov
Translated by: the author
Original publication: Algebra i Analiz, tom 16 (2004), nomer 5.
Journal: St. Petersburg Math. J. 16 (2005), 821-835
MSC (2000): Primary 41A15; Secondary 42C40
Published electronically: September 21, 2005
MathSciNet review: 2106669
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Abstract: Univariate piecewise-smooth refinable functions (i.e., compactly supported solutions of the equation $\varphi (\frac{x}{2})=\sum_{k = 0}^N c_k \varphi (x{-}k)$) are classified completely. Characterization of the structure of refinable splines leads to a simple convergence criterion for the subdivision schemes corresponding to such splines, and to explicit computation of the rate of convergence. This makes it possible to prove a factorization theorem about decomposition of any smooth refinable function (not necessarily stable or corresponding to a convergent subdivision scheme) into a convolution of a continuous refinable function and a refinable spline of the corresponding order. These results are applied to a problem of combinatorial number theory (the asymptotics of Euler's partition function). The results of the paper generalize several previously known statements about refinement equations and help to solve two open problems.


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Additional Information

V. Yu. Protasov
Affiliation: Department of Mechanics and Mathematics, Moscow State University, Moscow 119992, Russia
Email: vladimir_protassov@yahoo.com

DOI: http://dx.doi.org/10.1090/S1061-0022-05-00881-2
PII: S 1061-0022(05)00881-2
Keywords: Refinable functions, splines, regularity, subdivision algorithms, convergence
Received by editor(s): February 15, 2004
Published electronically: September 21, 2005
Additional Notes: This work was supported by RFBR (grant nos. 02–01–00248, 03–01–06300) and by the SS Program (grant no. 304.2003.1)
Article copyright: © Copyright 2005 American Mathematical Society