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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
DOI: https://doi.org/10.1090/S0002-9939-05-08118-9
Published electronically: July 18, 2005
MathSciNet review: 2180904
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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.


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  • 1. M. Bröker and X. Zhou, Characterization of continuous, four-coefficient scaling functions via matrix spectral radius, SIAM J. Matrix Anal. Appl. 22 (2000), 242-257. MR 1779727 (2001h:65169)
  • 2. A. S. Cavaretta, W. Dahmen, and C.A. Micchelli, Stationary Subdivision, Mem. Amer. Math. Soc. 453 (1991). MR 1079033 (92h:65017)
  • 3. I. Daubechies and J. C. Lagarias, Two-scale difference equations I. Existence and global regularity of solutions, SIAM J. Math. Anal. 22 (1991), 1388-1410. MR 1112515 (92d:39001)
  • 4. T. N. T. Goodman, C. M. Micchelli and J. Ward, Spectral radius formulas for subdivision operators, Recent advances in Wavelet Analysis, L.L. Schumaker and G. Webb (eds.), Academic Press, Inc. 1994, 335-360. MR 1244611 (94m:47076)
  • 5. B. Han, Projectable multivariate refinable functions and biorthogonal wavelets, Appl. Comput. Harmon. Anal. 13 (2002), 89-102. MR 1930178 (2003f:42054)
  • 6. B. Han and R.-Q. Jia, Multivariate refinement equations and convergence of subdivision schemes, SIAM J. Anal. 29 (1998), 1177-1199. MR 1618691 (99f:41018)
  • 7. R.-Q. Jia and D.-X. Zhou, Convergence of subdivision schemes associated with nonnegative masks, SIAM J. Matrix Anal. Appl. 21 (1999), 418-430. MR 1718338 (2001a:42041)
  • 8. A. A. Melkman, Subdivision schemes with non-negative masks always converge - unless they obviously cannot? Ann. Numer. Math. 4 (1997), 451-460. MR 1422696 (97i:41014)
  • 9. C. A. Micchelli and H. Prautzsch, Uniform refinement of curves, Linear Algebra Appl. 114/115 (1989), 841-870. MR 0986909 (90k:65088)
  • 10. J. N. Tsitsiklis and V. D. Blondel, The Lyapunov exponent and joint spectral radius of pairs of matrices are hard, when not impossible, to compute and to approximate, Mathematics of Control, Signals and Systems 10 (1997), 31-41. MR 1462278 (99h:65238a)
  • 11. Y. Wang, Subdivision schemes and refinement equations with nonnegative masks, J. Approx. Th. 113 (2001), 207-220. MR 1876323 (2002i:42054)
  • 12. X. Zhou, Characterization of convergent subdivision schemes, J. Approx. and its Appl. 14 (1998), 11-24. MR 1668983 (99m:65041)
  • 13. X. Zhou, Subdivision schemes with nonnegative masks, Math. Comp. 74 (2005), 819-839. MR 2114650

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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: https://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.

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