Skip to main content
SpringerLink
Log in

Generating functions for B-Splines with knots in geometric or affine progression

  • Published:
Calcolo Aims and scope Submit manuscript

Abstract

We derive explicit formulas for the generating functions of B-splines with knots in either geometric or affine progression. To find generating functions for B-splines whose knots have geometric or affine ratio \(q\), we construct a PDE for these generating functions in which classical derivatives are replaced by \(q\)-derivatives. We then solve this PDE for the generating functions using \(q\)-exponential functions. We apply our generating functions to derive some known and some novel identities for B-splines with knots in geometric or affine progression, including a generalization of the Schoenberg identity, formulas for sums and alternating sums, and an explicit expression for the moments of these B-splines. Special cases include both the uniform B-splines with knots at the integers and the nonuniform B-splines with knots at the \(q\)-integers.

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

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Goldman, R.: Generating functions for uniform B-splines. In: Eighth International Conference on Mathematical Methods for Curves and Surfaces. (2012)

  2. Şimşek, Y.: Generating functions for the Bernstein type polynomials: a new approach to deriving identities and applications for these polynomials. Hacettepe J. Math. Stat. (in press)

  3. Şimşek, Y.: Interpolation function of generalized \(q\)-Bernstein-type basis polynomials and applications. In: Curves and Surfaces, Lecture Notes in Computer Science, vol 6920. Springer, Heidelberg, pp. 647–662 (2012). doi:10.1007/978-3-642-27413-8-43

  4. Phillips, G.M.: Interpolation and approximation by polynomials. In: CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 14. Springer, New York (2003)

  5. Koçak, Z.F., Phillips, G.M.: \(B\)-splines with geometric knot spacings. BIT 34(3):388–399 (1994). doi:10.1007/BF01935648

  6. Budakçı, G., Oruç, H.: Bernstein–Schoenberg operator with knots at the \(q\)-integers. Math. Comput. Model. 56(3–4):56–59 (2012). doi:10.1016/j.mcm.2011.12.049

  7. Goldman, R., Warren, J.D.: An extension of chaiken’s algorithm to b-spline curves with knots in geometric progression. CVGIP Graph. Model Image Process. 55(1):58–62 (1993)

    Google Scholar 

  8. Kac, V., Cheung, P.: Quantum Calculus. Universitext, Springer, New York (2002). doi:10.1007/978-1-4613-0071-7

  9. Ismail, M.E.H.: Classical and quantum orthogonal polynomials in one variable. In: Encyclopedia of Mathematics and its Applications, vol 98. Cambridge University Press, Cambridge (2009) (with two chapters by Walter Van Assche, With a foreword by Richard A. Askey, Reprint of the 2005 original)

  10. Andrews, G.E.: The Theory of Partitions. Cambridge Mathematical Library, Cambridge University Press, Cambridge (1998) (reprint of the 1976 original)

  11. de Boor, C.: On calculating with \(B\)-splines. J. Approx. Theory 6:50–62 (1972) (collection of articles dedicated to J. L. Walsh on his 75th birthday, V) (Proceedings of Internaternational Conference on Approximation Theory, Related Topics and their Applications, University of Maryland, College Park, MD, USA, 1970)

  12. Goldman, R.: Pyramid Algorithm. Elsevier Science, Amsterdam (2003)

  13. Shoenberg, I.J.: Contributions to the problem of approximation of equidistant data by analytic functions: Part a on the problem of smoothing or graduation. a first class of analytic approximation formulae. Q. Appl. Math. 4, 45–99 (1964)

    Google Scholar 

Download references

Acknowledgments

This research began while Ron Goldman was visiting the Department of Mathematics, Dokuz Eylül University during the summer of 2012. His visit was supported by TÜBİTAK (http://www.tubitak.gov.tr), The Scientific and Technological Research Council of Turkey, Bideb 2221 program. Gülter Budakçı is also supported by a grant from TÜBİTAK.

Author information

Authors and Affiliations

  1. Department of Mathematics, Fen Fakültesi, Dokuz Eylül University, Tınaztepe Kampüsü, 35160 , Buca, Izmir, Turkey

    Çetin Dişibüyük & Halil Oruç

  2. Department of Mathematics, Fen Bilimleri Enstitüsü, Dokuz Eylül University, Tınaztepe Kampüsü, 35160 , Buca, Izmir, Turkey

    Gülter Budakçı

  3. Department of Computer Science, Rice University, 6100 South Main, Houston, TX, 77251-1892, USA

    Ron Goldman

Authors
  1. Çetin Dişibüyük
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gülter Budakçı
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ron Goldman
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Halil Oruç
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Çetin Dişibüyük.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dişibüyük, Ç., Budakçı, G., Goldman, R. et al. Generating functions for B-Splines with knots in geometric or affine progression. Calcolo 51, 599–613 (2014). https://doi.org/10.1007/s10092-013-0102-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10092-013-0102-8

Keywords

Mathematics Subject Classification (2000)

Search

Navigation