On multivariate subdivision schemes with nonnegative finite masks

Author:
Xinlong Zhou

Journal:
Proc. Amer. Math. Soc. **134** (2006), 859-869

MSC (2000):
Primary 65D17, 26A15, 26A18

Published electronically:
July 18, 2005

MathSciNet review:
2180904

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the convergence of multivariate subdivision schemes with nonnegative finite masks. Consequently, the convergence problem for the multivariate subdivision schemes with nonnegative finite masks supported on centered zonotopes is solved. Roughly speaking, the subdivision schemes defined by these masks are always convergent, which gives an answer to a question raised by Cavaretta, Dahmen and Micchelli in 1991.

**1.**Markus Bröker and Xinlong Zhou,*Characterization of continuous, four-coefficient scaling functions via matrix spectral radius*, SIAM J. Matrix Anal. Appl.**22**(2000), no. 1, 242–257 (electronic). MR**1779727**, 10.1137/S0895479897323750**2.**Alfred S. Cavaretta, Wolfgang Dahmen, and Charles A. Micchelli,*Stationary subdivision*, Mem. Amer. Math. Soc.**93**(1991), no. 453, vi+186. MR**1079033**, 10.1090/memo/0453**3.**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**, 10.1137/0522089**4.**T. N. T. Goodman, Charles A. Micchelli, and J. D. Ward,*Spectral radius formulas for subdivision operators*, Recent advances in wavelet analysis, Wavelet Anal. Appl., vol. 3, Academic Press, Boston, MA, 1994, pp. 335–360. MR**1244611****5.**Bin Han,*Projectable multivariate refinable functions and biorthogonal wavelets*, Appl. Comput. Harmon. Anal.**13**(2002), no. 1, 89–102. MR**1930178**, 10.1016/S1063-5203(02)00007-6**6.**Bin Han and Rong-Qing Jia,*Multivariate refinement equations and convergence of subdivision schemes*, SIAM J. Math. Anal.**29**(1998), no. 5, 1177–1199. MR**1618691**, 10.1137/S0036141097294032**7.**Rong-Qing Jia and Ding-Xuan Zhou,*Convergence of subdivision schemes associated with nonnegative masks*, SIAM J. Matrix Anal. Appl.**21**(1999), no. 2, 418–430 (electronic). MR**1718338**, 10.1137/S0895479898342432**8.**Avraham A. Melkman,*Subdivision schemes with non-negative masks converge always—unless they obviously cannot?*, Ann. Numer. Math.**4**(1997), no. 1-4, 451–460. The heritage of P. L. Chebyshev: a Festschrift in honor of the 70th birthday of T. J. Rivlin. MR**1422696****9.**Charles A. Micchelli and Hartmut Prautzsch,*Uniform refinement of curves*, Linear Algebra Appl.**114/115**(1989), 841–870. MR**986909**, 10.1016/0024-3795(89)90495-3**10.**John N. Tsitsiklis and Vincent D. Blondel,*The Lyapunov exponent and joint spectral radius of pairs of matrices are hard—when not impossible—to compute and to approximate*, Math. Control Signals Systems**10**(1997), no. 1, 31–40. MR**1462278**, 10.1007/BF01219774**11.**Yang Wang,*Subdivision schemes and refinement equations with nonnegative masks*, J. Approx. Theory**113**(2001), no. 2, 207–220. MR**1876323**, 10.1006/jath.2001.3623**12.**Xinlong Zhou,*Characterization of convergent subdivision schemes*, Approx. Theory Appl. (N.S.)**14**(1998), no. 3, 11–24. MR**1668983****13.**Xinlong Zhou,*Subdivision schemes with nonnegative masks*, Math. Comp.**74**(2005), no. 250, 819–839 (electronic). MR**2114650**, 10.1090/S0025-5718-04-01712-0

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
65D17,
26A15,
26A18

Retrieve articles in all journals with MSC (2000): 65D17, 26A15, 26A18

Additional Information

**Xinlong Zhou**

Affiliation:
Department of Mathematics, China Jiliang University, 310018 Hangzhou, People’s Republic of China – and – Department of Mathematics, University of Duisburg-Essen, D-47057 Duisburg, Germany

Email:
zhou@math.uni-duisburg.de

DOI:
http://dx.doi.org/10.1090/S0002-9939-05-08118-9

Keywords:
Nonnegative mask,
subdivision scheme,
zonotope.

Received by editor(s):
March 22, 2004

Received by editor(s) in revised form:
October 8, 2004

Published electronically:
July 18, 2005

Communicated by:
David R. Larson

Article copyright:
© Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.