Skip to main content

Quadratic convergence of approximations by CCC-Schoenberg operators

Abstract

We generalize to the Canonical Complete Chebyshev splines some properties already known for Extended Chebyshev and piecewise Extended Chebyshev splines, like Marsden identity and Greville points. Also, we represent an interesting algorithm which leads to numerically stable expressions for the Greville points for CCC-splines. We show that any CCC-spline space provides us with infinite number of operators of the Schoenberg-type, and we give a simple characterization of them. After proving few important properties, we establish a sufficient condition for quadratic convergence of approximations by CCC-Schoenberg operators to a given function.

This is a preview of subscription content, access via your institution.

References

  1. Aldaz, J.M., Kounchev, O., Render, H.: Shape preserving properties of generalized Bernstein operators on extended Chebyshev spaces. Numer. Math. 114, 1–25 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  2. Bister, D., Prautzsch, H.: A new approach to Tchebycheffian B-splines. In: Méhauteé, A.L., Rabut, C., Schumaker, L.L. (eds.) Curve and Surfaces in Geometric Design, pp. 35–43. Vanderbilt University Press, Nashville (1997)

    Google Scholar 

  3. Bosner, T.: Knot insertion algorithms for weighted splines. In: Drmač, Z., Marušić, M., Tutek, Z. (eds.) Proceedings of the Conference on Applied Mathematics and Scientific Computing, pp. 151–160. Springer, Berlin (2005)

  4. Bosner, T.: Knot insertion algorithms for Chebyshev splines. Ph.D. thesis, Dept. of Mathematics, University of Zagreb (2006). http://web.math.hr/~tinab/TinaBosnerPhD.pdf

  5. Bosner, T.: Basis of splines associated with singularly perturbed advection-diffusion problems. Math. Commun. 15(1), 1–12 (2010)

    MathSciNet  MATH  Google Scholar 

  6. Bosner, T., Rogina, M.: Non-uniform exponential tension splines. Numer. Algor. 46, 265–294 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  7. Bosner, T., Rogina, M.: Collocation by singular splines. Annali dell’Università di Ferrara 54(2), 217–227 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  8. Bosner, T., Rogina, M.: Variable degree polynomial splines are Chebyshev splines. Adv. Comput. Math. 38, 383–400 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  9. Burrill, C.W.: Measure, Integration, and Probability. McGraw-Hill Book Company, New York (1972)

    MATH  Google Scholar 

  10. de Boor, C.: A Practical Guide to Splines, revised edition. Springer, Berlin (2001)

    MATH  Google Scholar 

  11. Johnson, R.W.: Higher order B-spline collocation at the Greville abscissae. Appl. Numer. Math. 52, 63–75 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  12. Karlin, S.: Total Positivity. Stanford Univ. Press, California (1968)

    MATH  Google Scholar 

  13. Karlin, S., Studden, W.: Tchebycheff Systems: With Applications in Analysis and Statistics. Wiley Interscience, New York (1966)

    MATH  Google Scholar 

  14. Kavčič, I., Rogina, M., Bosner, T.: Singularly perturbed advection-diffusion-reaction problems: comparison of operator-fitted methods. Math. Comput. Simul. 81(10), 2215–2224 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  15. Koch, O.: Asymptotically correct error estimation for collocation methods applied to singular boundary value problems. Numer. Math. 101, 143–164 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  16. Lyche, T., Mazure, M.L.: Total positivity and the existence of piecewise exponential B-splines. Adv. Comput. Math. 25(1–3), 105–133 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  17. Marsden, M.J.: An identity for spline functions with applications to variation-diminishing spline approximation. J. Approx. Theory 3, 7–49 (1970)

    MathSciNet  MATH  Article  Google Scholar 

  18. Marušić, M.: A fourth/second order accurate collocation method for singularly perturbed two-point boundary value problems using tension splines. Numer. Math. 88, 135–158 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  19. Marušić, M., Rogina, M.: A collocation method for singularly perturbed two-point boundary value problems with splines in tension. Adv. Comput. Math. 6(1), 65–76 (1996)

    MathSciNet  MATH  Article  Google Scholar 

  20. Mazure, M.L.: Blossoms of generalized derivatives in Chebyshev spaces. J. Approx. Theory 131, 47–58 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  21. Mazure, M.L.: On Chebyshevian spline subdivision. J. Approx. Theory 143, 74–110 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  22. Mazure, M.L.: Bernstein-type operators in Chebyshev spaces. Numer. Algor. 52, 93–128 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  23. Mazure, M.L.: On differentiation formulae for Chebyshevian Bernstein and B-spline bases. Jaén J. Approx. 1, 111–143 (2009)

    MathSciNet  MATH  Google Scholar 

  24. Mazure, M.L.: Chebyshev–Schoenberg operators. Constr. Approx. 34, 181–208 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  25. Mazure, M.L.: A duality formula for chebyshevian divided differences and blossoms. Jaén J. Approx. 3, 67–86 (2011)

    MathSciNet  MATH  Google Scholar 

  26. Mazure, M.L.: Finding all systems of weight functions associated with a given extended chebyshev space. J. Approx. Theory. 163, 363–376 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  27. Mazure, M.L.: How to build all Chebyshevian spline spaces good for geometric design? Numer. Math. 119, 517–556 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  28. Mazure, M.L.: Quasi extended Chebyshev spaces and weight functions. Numer. Math. 118 (2011)

  29. Mazure, M.L.: Piecewise Chebyshev–Schoenberg operators: shape preservation, approximation and space embedding. J. Approx. Theory. 166, 106–135 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  30. Mazure, M.L., Laurent, P.J.: Piecewise smooth spaces in duality: application to blossoming. J. Approx. Theory 98, 316–353 (1999)

    MathSciNet  MATH  Article  Google Scholar 

  31. Rogina, M.: Basis of splines associated with some singular differential operators. BIT 32, 496–505 (1992)

    MathSciNet  MATH  Article  Google Scholar 

  32. Rogina, M.: On construction of fourth order Chebyshev splines. Math. Commun. 4, 83–92 (1999)

    MathSciNet  MATH  Google Scholar 

  33. Rogina, M.: Algebraic proof of the B-spline derivative formula. In: Drmač, Z., Marušić, M., Tutek, Z. (eds.) Proceedings of the Conference on Applied Mathematics and Scientific Computing, pp. 273–282. Springer, Berlin (2005)

    Chapter  Google Scholar 

  34. Rogina, M., Bosner, T.: On calculating with lower order Chebyshev splines. In: Laurent, P.J., Sabloniere, P., Schumaker, L.L. (eds.) Curves and Surfaces Design, pp. 343–353. Vanderbilt Univ. Press, Nashville (2000)

    Google Scholar 

  35. Royden, H.L., Fitzpatrick, P.M.: Real Analysis, 4th edn. China Machine Press, China (2010)

    MATH  Google Scholar 

  36. Schoenberg, I.J.: Contributions to the problem of approximation of equidistant data by analytic functions. Q. Appl. Math. 4(1), 45–99, 112–141 (1946)

  37. Schumaker, L.L.: On Tchebycheffian spline functions. J. Approx. Theory 18, 278–303 (1976)

    MathSciNet  MATH  Article  Google Scholar 

  38. Schumaker, L.L.: Spline Functions: Basic Theory. Wiley, New York (1981)

    MATH  Google Scholar 

  39. Schumaker, L.L.: On recursions for generalized splines. J. Approx. Theory 36, 16–31 (1982)

    MathSciNet  MATH  Article  Google Scholar 

Download references

Acknowledgments

This research was supported by Ministry of science, education and sports of the Republic of Croatia under Grant 037-1193086-2771.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Tina Bosner.

Additional information

This article is in memory and honor of Mladen Rogina who passed away on 24 January 2013, as our last joint work.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bosner, T., Rogina, M. Quadratic convergence of approximations by CCC-Schoenberg operators. Numer. Math. 135, 1253–1287 (2017). https://doi.org/10.1007/s00211-016-0831-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-016-0831-0

Mathematics Subject Classification

  • 41A15
  • 41A25
  • 41A35
  • 41A50
  • 65D07
  • 65D17