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

DOI:
https://doi.org/10.1090/S1061-0022-05-00881-2

Published electronically:
September 21, 2005

MathSciNet review:
2106669

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Univariate piecewise-smooth refinable functions (i.e., compactly supported solutions of the equation ) 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]**Ingrid Daubechies and Jeffrey C. Lagarias,*Two-scale difference equations. I. Existence and global regularity of solutions*, SIAM J. Math. Anal.**22**(1991), no. 5, 1388–1410. MR**1112515**, https://doi.org/10.1137/0522089**[2]**Lothar Berg and Gerlind 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****[3]**Vladimir Protasov,*A complete solution characterizing smooth refinable functions*, SIAM J. Math. Anal.**31**(2000), no. 6, 1332–1350. MR**1766557**, https://doi.org/10.1137/S0036141098342969**[4]**Ingrid Daubechies and Jeffrey C. Lagarias,*Two-scale difference equations. II. Local regularity, infinite products of matrices and fractals*, SIAM J. Math. Anal.**23**(1992), no. 4, 1031–1079. MR**1166574**, https://doi.org/10.1137/0523059**[5]**Nira Dyn, John A. Gregory, and David Levin,*Analysis of uniform binary subdivision schemes for curve design*, Constr. Approx.**7**(1991), no. 2, 127–147. MR**1101059**, https://doi.org/10.1007/BF01888150**[6]**Alfred S. Cavaretta, Wolfgang Dahmen, and Charles A. Micchelli,*Stationary subdivision*, Mem. Amer. Math. Soc.**93**(1991), no. 453, vi+186. MR**1079033**, https://doi.org/10.1090/memo/0453**[7]**Lothar Berg and Gerlind Plonka,*Spectral properties of two-slanted matrices*, Results Math.**35**(1999), no. 3-4, 201–215. MR**1694902**, https://doi.org/10.1007/BF03322813**[8]**Wayne Lawton, S. L. Lee, and Zuowei Shen,*Characterization of compactly supported refinable splines*, Adv. Comput. Math.**3**(1995), no. 1-2, 137–145. MR**1314906**, https://doi.org/10.1007/BF03028364**[9]**Bruce 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****[10]**Lars F. Villemoes,*Wavelet analysis of refinement equations*, SIAM J. Math. Anal.**25**(1994), no. 5, 1433–1460. MR**1289147**, https://doi.org/10.1137/S0036141092228179**[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]**Vladimir Protasov,*The stability of subdivision operator at its fixed point*, SIAM J. Math. Anal.**33**(2001), no. 2, 448–460. MR**1857979**, https://doi.org/10.1137/S0036141099356283**[13]**Rong Qing Jia,*Subdivision schemes in 𝐿_{𝑝} spaces*, Adv. Comput. Math.**3**(1995), no. 4, 309–341. MR**1339166**, https://doi.org/10.1007/BF03028366**[14]**Ingrid Daubechies,*Ten lectures on wavelets*, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR**1162107****[15]**L. Euler,*Introductio in analysis infinitorum*, Opera Omnia Ser. Prima Opera Math., vol. 8, Teubner, Leipzig, 1922.**[16]**Kurt Mahler,*On a special functional equation*, J. London Math. Soc.**15**(1940), 115–123. MR**0002921**, https://doi.org/10.1112/jlms/s1-15.2.115**[17]**N. G. de Bruijn,*On Mahler’s partition problem*, Nederl. Akad. Wetensch., Proc.**51**(1948), 659–669 = Indagationes Math. 10, 210–220 (1948). MR**0025502****[18]**Donald E. Knuth,*An almost linear recurrence*, Fibonacci Quart.**4**(1966), 117–128. MR**0199168****[19]**R. F. Churchhouse,*Congruence properties of the binary partition function*, Proc. Cambridge Philos. Soc.**66**(1969), 371–376. MR**0248102****[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 (Russian, with Russian summary); English transl., Sb. Math.**191**(2000), no. 3-4, 381–414. MR**1773255**, https://doi.org/10.1070/SM2000v191n03ABEH000464**[22]**L. Carlitz,*Generating functions and partition problems*, Proc. Sympos. Pure Math., Vol. VIII, Amer. Math. Soc., Providence, R.I., 1965, pp. 144–169. MR**0175796****[23]**C. A. Micchelli and H. Prautzsch,*Uniform refinement of curves*, Linear Algebra Appl.**114/115**(1989), 841-870. MR**0986909 (90k:65088)**

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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:
https://doi.org/10.1090/S1061-0022-05-00881-2

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