On Properties of Nonstationary Divided Difference Vector Subdivision Schemes

  • Maria Charina
  • Costanza Conti
Conference paper
Part of the International Series of Numerical Mathematics book series (ISNM, volume 145)


This paper studies the properties of a nonstationary divided difference vector subdivision scheme S B k derived from a stationary vector subdivision scheme SA. We show that a necessary condition for SA to be C1 (i.e., its linear polynomial reproduction property) implies the existence of a difference scheme derived from S B k. This condition turns out to be a necessary condition for the subdivision scheme S B k to be convergent.


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  1. 1.
    A. S. Cavaretta, W. Dahmen, C. A. Micchelli: Stationary Subdivision, Memoirs Amer. Math. Soc. 93, No. 453 (1991).MathSciNetGoogle Scholar
  2. 2.
    M. Charina, C. Conti: Regularity of multivariate vector subdivision schemes, Manuscript (December 2001).Google Scholar
  3. 3.
    M. Charina, C. Conti: Convergence of non—stationary vector subdivision schemes, to appear in Math. Computers in Simulation.Google Scholar
  4. 4.
    D. R. Cheng, R. Q. Jia, S. D. Riemenschneider: Convergence of vector subdivision schemes in Sobolev spaces, Applied Comp. Harm. Anal. 12 (2002), 128–149.Google Scholar
  5. 5.
    C. Conti, K. Jetter: A new subdivision method for bivariate splines on the four—directional mesh, J. Comp. Appl. Math. 119 (2000), 81–96.CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    W. Dahmen, C. A. Micchelli: Biorthogonal wavelet expansions, Constr. Approx. 13 (1997), 293–328.zbMATHMathSciNetGoogle Scholar
  7. 7.
    N. Dyn: Subdivision schemes in CAGD, in: Advances in Numerical Analysis, vol. II: Wavelets, Subdivision Algorithms and Radial Basis Functions, W. A. Light (ed.), pp. 36–104, Oxford University Press, 1992.Google Scholar
  8. 8.
    N. Dyn, D. Levin: Matrix subdivision-analysis by factorization, in: Approximation Theory, B. D. Bojanov (ed.), pp. 187–211, DARBA, Sofia, 2002.Google Scholar
  9. 9.
    B. Han: Vector cascade algorithms and refinable function vectors in Sobolev spaces, preprint, 40 p. (July 2002).Google Scholar
  10. 10.
    K. Jetter, G. Plonka: A survey on L2-approximation order from shift-invariant spaces, in: Multivariate Approximation and Applications N. Dyn, D. Leviatan, D. Levin, A. Pinkus (eds.), pp. 73–111, Cambridge University Press, 2001.Google Scholar
  11. 11.
    K. Jetter, G. Zimmermann. Constructing polynomial surfaces from vector subdivision schemes, in: Constructive Functions The-ory, Varna 2002, B. D. Bojanov (ed.), pp. 327–6332, DARBA, Sofia, 2003.Google Scholar
  12. 12.
    R. Q. Jia, S. D. Riemenschneider, D. X. Zhou: Approximation by multiple refinable functions, Can. J. Math. 49 (1997), 944–962.zbMATHMathSciNetGoogle Scholar
  13. 13.
    R. Q. Jia, S. D. Riemenschneider, D. X. Zhou: Vector subdivision schemes and multiple wavelets, Math. Comp. 67 (1998), 1533–1563.Google Scholar
  14. 14.
    R. Q. Jia, S. D. Riemenschneider, D. X. Zhou: Smoothness of multiple refinable functions and multiple wavelets, SIAM J. Matrix Anal. Appl. 21 (1999), 1–28.CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Q. T. Jiang: Multivariate matrix refinable functions with arbitrary matrix, Trans. Amer. Math. Soc. 351 (1999), 2407–2438.CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    C. A. Micchelli, T. Sauer: Regularity of multiwavelets, Adv. Comp. Math. 7 (1997), 455–545.zbMATHMathSciNetGoogle Scholar
  17. 17.
    A. Ron, Z. Shen: The Sobolev regularity of refinable functions, J. Approx. Theory 106 (2000), 185–225.CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    T. Sauer: Stationary vector subdivision - quotient ideals, differences and approximation power, Rev. R. Acad. Cienc. Serie A. Mat. 229 (2002), 621–674.MathSciNetGoogle Scholar
  19. 19.
    Z. Shen: Refinable functions vectors, SIAM J. Math. Anal. 29 (1998), 235–250.zbMATHGoogle Scholar

Copyright information

© Springer Basel AG 2003

Authors and Affiliations

  • Maria Charina
    • 1
  • Costanza Conti
    • 2
  1. 1.Institut für Angewandte Mathematik Universität DortmundDortmundGermany
  2. 2.Dipartimento di Energetica “Sergio Stecco”Costanza Conti Universitá degli Studi di FirenzeFirenzeItaly

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