Skip to main content
SpringerLink
Log in

Bezout-like polynomial equations associated with dual univariate interpolating subdivision schemes

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

The algebraic characterization of dual univariate interpolating subdivision schemes is investigated. Specifically, we provide a constructive approach for finding dual univariate interpolating subdivision schemes based on the solutions of certain associated polynomial equations. The proposed approach also makes it possible to identify conditions for the existence of the sought schemes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alekhnovich, M.: Linear Diophantine equations over polynomials and soft decoding of Reed-Solomon codes. IEEE Trans. Inform. Theory 51(7), 2257–2265 (2005)

    Article  MathSciNet  Google Scholar 

  2. Cavaretta, A.S., Dahmen, W., Micchelli, C.A.: Stationary subdivision. Mem. Amer. Math. Soc. 93, 453 (1991)

    MathSciNet  MATH  Google Scholar 

  3. Charina, M., Conti, C., Sauer, T.: Regularity of multivariate vector subdivision schemes. Numer Algorithms 39(1-3), 97–113 (2005)

    Article  MathSciNet  Google Scholar 

  4. Charina, M., Mejstrik, T.: Multiple multivariate subdivision schemes: matrix and operator approaches. J. Comput. Appl. Math. 349, 279–291 (2019)

    Article  MathSciNet  Google Scholar 

  5. Choi, S.W., Lee, Byung-Gook, Lee, Y.J., Yoon, J.: Stationary subdivision schemes reproducing polynomials. Comput. Aided Geom. Design 23(4), 351–360 (2006)

    Article  MathSciNet  Google Scholar 

  6. Chui, C., de Villiers, J.M.: Wavelet Subdivision Methods: GEMS for Rendering Curves and Surfaces. CRC Press, Boca Raton (2010)

    Book  Google Scholar 

  7. Conti, C., Hormann, K.: Polynomial reproduction for univariate subdivision schemes of any arity. J. Approx. Theory 163(4), 413–437 (2011)

    Article  MathSciNet  Google Scholar 

  8. Conti, C., Romani, L.: Dual univariate m-ary subdivision schemes of de Rham-type. J. Math. Anal. Appl. 407(2), 443–456 (2013)

    Article  MathSciNet  Google Scholar 

  9. de Villiers, J., Gavhi, M.R.: Local interpolation with optimal polynomial exactness in refinement spaces. Math Comp. 85(298), 759–782 (2016)

    Article  MathSciNet  Google Scholar 

  10. de Villiers, J.M., Micchelli, C.A., Sauer, T.: Building refinable functions from their values at integers. Calcolo 37(3), 139–158 (2000)

    Article  MathSciNet  Google Scholar 

  11. Deslauriers, G., Dubuc, S.: Symmetric iterative interpolation processes. Constr. Approx. 5(1), 49–68 (1989)

    Article  MathSciNet  Google Scholar 

  12. Dubuc, S.: Interpolation through an iterative scheme. J. Math. Anal. Appl. 114(1), 185–204 (1986)

    Article  MathSciNet  Google Scholar 

  13. Dubuc, S.: de Rham transforms for subdivision schemes. J. Approx. Theory 163(8), 966–987 (2011)

    Article  MathSciNet  Google Scholar 

  14. Dyn, N.: Subdivision schemes in computer-aided geometric design. In: Advances in Numerical Analysis, Vol. II (Lancaster, 1990), Oxford Sci. Publ., pp 36–104. Oxford Univ. Press, New York (1992)

  15. Dyn, N.: Interpolatory subdivision schemes. In: Floater, M.S., Iske, A., Quak E. (eds.) Tutorials on Multiresolution in Geometric Modelling, pp 25–50. Springer, Berlin (2002)

  16. Dyn, N., Hormann, K., Sabin, M.A., Shen, Z.: Polynomial reproduction by symmetric subdivision schemes. J. Approx. Theory 155(1), 28–42 (2008)

    Article  MathSciNet  Google Scholar 

  17. Dyn, N., Levin, D.: Subdivision schemes in geometric modelling. Acta Numer. 11, 73–144 (2002)

    Article  MathSciNet  Google Scholar 

  18. Gavhi-Molefe, M.R., de Villiers, J.: On interpolatory subdivision symbol formulation and parameter convergence intervals. J. Comput. Appl. Math. 349, 354–365 (2019)

    Article  MathSciNet  Google Scholar 

  19. Gentili, G., Struppa, D.C.: Minimal degree solutions of polynomial equations. Kybernetika 23(1), 44–53 (1987)

    MathSciNet  MATH  Google Scholar 

  20. Guglielmi, N., Protasov, V.: Exact computation of joint spectral characteristics of linear operators. Found Comput. Math. 13(1), 37–97 (2013)

    Article  MathSciNet  Google Scholar 

  21. Johnson, W.P.: The curious history of Faà di Bruno’s formula. Amer. Math. Monthly 109(3), 217–234 (2002)

    MathSciNet  MATH  Google Scholar 

  22. Mejstrik, T.: Algorithm 1011: improved invariant polytope algorithm and applications. ACM Trans. Math. Softw. 46(3), Art. 29, 26 (2020)

    Article  MathSciNet  Google Scholar 

  23. Micchelli, C.A: Interpolatory subdivision schemes and wavelets. JJ Approx. Theory 86, 41–71 (1996)

    Article  MathSciNet  Google Scholar 

  24. Muntingh, G.: Symbols and exact regularity of symmetric pseudo-splines of any arity. BIT 57(3), 867–900 (2017)

    Article  MathSciNet  Google Scholar 

  25. Romani, L.: Interpolating m-refinable functions with compact support: The second generation class. Appl. Math. Comput. 361, 735–746 (2019)

    MathSciNet  MATH  Google Scholar 

  26. Romani, L., Viscardi, A.: Dual univariate interpolatory subdivision of every arity: Algebraic characterization and construction. J. Math. Anal. Appl. 484(1), 123713 (2020)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the anonymous referees for their constructive comments and suggested improvements. All the authors are members of INdAM - GNCS, which partially supported this work. The first author is also partially supported by the project PRA_2020_61 of the University of Pisa. The research of the last two authors has been done within the Italian Network on Approximation (RITA) and the thematic group on “Approximation Theory and Applications” of the Italian Mathematical Union.

Author information

Authors and Affiliations

  1. Dipartimento di Informatica, Università di Pisa, Largo Bruno Pontecorvo, 3, 56127, Pisa, Italy

    Luca Gemignani

  2. Dipartimento di Matematica and AM2, Alma Mater Studiorum Università di Bologna, Piazza di Porta San Donato, 5, 40126, Bologna, Italy

    Lucia Romani & Alberto Viscardi

Authors
  1. Luca Gemignani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lucia Romani
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alberto Viscardi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alberto Viscardi.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Communicated by: Tomas Sauer

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gemignani, L., Romani, L. & Viscardi, A. Bezout-like polynomial equations associated with dual univariate interpolating subdivision schemes. Adv Comput Math 48, 4 (2022). https://doi.org/10.1007/s10444-021-09912-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10444-021-09912-4

Keywords

Mathematics Subject Classification (2010)

Search

Navigation