St. Petersburg Mathematical Society

Author(s): V. Yu. Protasov
Translated by: the author
Original publication: Algebra i Analiz, tom 16 (2004), vypusk 5.
Journal: St. Petersburg Math. J. 16 (2005), 821-835.
MSC (2000): Primary 41A15; Secondary 42C40
Posted: September 21, 2005
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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.

[1]: I. Daubechies and J. Lagarias, Two-scale difference equations. . Existence and global regularity of solutions, SIAM J. Math. Anal. 22 (1991), 1388-1410. MR 1112515 (92d:39001)
[2]: L. Berg and G. Plonka, Some notes on two-scale difference equations, Functional Equations and Inequalities, Math. Appl., vol. 518, Kluwer Acad. Publ., Dordrecht, 2000, pp. 7-29. MR 1792070 (2002f:39049)
[3]: V. Protasov, A complete solution characterizing smooth refinable functions, SIAM J. Math. Anal. 31 (2000), no. 6, 1332-1350 (electronic). MR 1766557 (2001j:42033)
[4]: I. Daubechies and J. Lagarias, Two-scale difference equations. . Local regularity, infinite products of matrices and fractals, SIAM J. Math. Anal. 23 (1992), 1031-1079. MR 1166574 (93g:39001)
[5]: N. Dyn, J. A. Gregory, and D. Levin, Analysis of uniform binary subdivision schemes for curve design, Constr. Approx. 7 (1991), 127-147. MR 1101059 (92d:65027)
[6]: A. S. Cavaretta, W. Dahmen, and C. A. Micchelli, Stationary subdivision, Mem. Amer. Math. Soc. 93 (1991), no. 453, 186 pp. MR 1079033 (92h:65017)
[7]: L. Berg and G. Plonka, Spectral properties of two-slanted matrices, Results Math. 35 (1999), no. 3-4, 201-215. MR 1694902 (2000c:15009)
[8]: W. Lawton, S. L. Lee, and Z. Shen, Characterization of compactly supported refinable splines, Adv. Comput. Math. 3 (1995), no. 1-2, 137-145. MR 1314906 (95m:41020)
[9]: B. Reznick, Some binary partition functions, Analytic Number Theory (Allerton Park, IL, 1989), Progr. Math., vol. 85, Birkhäuser Boston, Boston, MA, 1990, pp. 451-477. MR 1084197 (91k:11092)
[10]: L. Villemoes, Wavelet analysis of refinement equations, SIAM J. Math. Anal. 25 (1994), no. 5, 1433-1460. MR 1289147 (96f:39009)
[11]: V. Protasov, The correlation between the convergence of subdivision processes and solvability of refinement equations, Algorithms for Approximation IV (Proc. of the 2001 Internat. Sympos., Huddersfield, England, July 15-20, 2001), pp. 394-401.
[12]: -, The stability of subdivision operator at its fixed point, SIAM J. Math. Anal. 33 (2001), no. 2, 448-460 (electronic). MR 1857979 (2002h:26023)
[13]: R. Q. Jia, Subdivision schemes in $L\sb p$ spaces, Adv. Comput. Math. 3 (1995), no. 4, 309-341. MR 1339166 (96d:65028)
[14]: I. Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conf. Ser. in Appl. Math., vol. 61, SIAM, Philadelphia, PA, 1992, 357 pp. MR 1162107 (93e:42045)
[15]: L. Euler, Introductio in analysis infinitorum, Opera Omnia Ser. Prima Opera Math., vol. 8, Teubner, Leipzig, 1922.
[16]: K. Mahler, On a special functional equation, J. London Math. Soc. 15 (1940), 115-123. MR 0002921 (2:133e)
[17]: N. G. de Bruijn, On Mahler's partition problem, Indag. Math. (N.S.) 10 (1948), 210-220. MR 0025502 (10:16d)
[18]: D. E. Knuth, An almost linear recurrence, Fibonacci Quart. 4 (1966), 117-128. MR 0199168 (33:7317)
[19]: R. F. Churchhouse, Congruence properties of the binary partition function, Proc. Cambridge Philos. Soc. 66 (1969), 371-376. MR 0248102 (40:1356)
[20]: A. Tanturri, Sul numero delle partizioni d'un numero in potenze di 2, Atti. Accad. Naz. Lincei 27 (1918), 399-403.
[21]: V. Yu. Protasov, Asymptotics of the partition function, Mat. Sb. 191 (2000), no. 3, 65-98; English transl., Sb. Math. 191 (2000), no. 3-4, 381-414. MR 1773255 (2001h:11134)
[22]: L. Carlitz, Generating functions and partition problems, Proc. Sympos. Pure Math., vol. 8, Amer. Math. Soc., Providence, RI , 1965, pp. 144-169. MR 0175796 (31:72)
[23]: C. A. Micchelli and H. Prautzsch, Uniform refinement of curves, Linear Algebra Appl. 114/115 (1989), 841-870. MR 0986909 (90k:65088)

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2000): 41A15, 42C40

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

DOI: 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): 15/FEB/2004
Posted: 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)
Copyright of article: Copyright 2005, American Mathematical Society

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