Skip to main content
SpringerLink
Log in

Approximation by Chebyshevian Bernstein Operators versus Convergence of Dimension Elevation

  • Published:
Constructive Approximation Aims and scope

Abstract

On a closed bounded interval, consider a nested sequence of Extended Chebyshev spaces possessing Bernstein bases. This situation automatically generates an infinite dimension elevation algorithm transforming control polygons of any given level into control polygons of the next level. The convergence of these infinite sequences of polygons towards the corresponding curves is a classical issue in computer-aided geometric design. Moreover, according to recent work proving the existence of Bernstein-type operators in such Extended Chebyshev spaces, this nested sequence is automatically associated with an infinite sequence of Bernstein operators which all reproduce the same two-dimensional space. Whether or not this sequence of operators converges towards the identity on the space of all continuous functions is a natural issue in approximation theory. In the present article, we prove that the two issues are actually equivalent. Not only is this result interesting on the theoretical side, but it also has practical implications. For instance, it provides us with a Korovkin-type theorem of convergence of any infinite dimension elevation algorithm. It also enables us to tackle the question of convergence of the dimension elevation algorithm for any nested sequence obtained by repeated integration of the kernel of a given linear differential operator with constant coefficients.

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.

Similar content being viewed by others

References

  1. Ait-Haddou, R.: Dimension elevation in Müntz spaces: a new emergence of the Müntz condition. J. Approx. Theory 181, 6–17 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ait-Haddou, R., Mazure, M.-L.: Approximation by Müntz spaces on positive intervals. CRAS 351, 849–852 (2013)

    MathSciNet  MATH  Google Scholar 

  3. Ait-Haddou, R., Sakane, Y., Nomura, T.: A Müntz type theorem for a family of corner cutting schemes. Comput. Aided Geom. Des. 30, 240–253 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Aldaz, J.M., Kounchev, O., Render, H.: Bernstein operators for exponential polynomials. Constr. Approx. 29, 345–367 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Aldaz, J.M., Kounchev, O., Render, H.: Bernstein operators for extended Chebyshev systems. Appl. Math. Comput. 217, 790–800 (2010)

    MathSciNet  MATH  Google Scholar 

  7. Almira, J.M.: Müntz type theorems I. Surv. Approx. Theory 3, 152–194 (2007)

    MathSciNet  MATH  Google Scholar 

  8. Borwein, P., Erdélyi, T.: Polynomials and Polynomial Inequalities. Springer-Verlag, New York (1995)

  9. Brilleaud, M., Mazure, M.-L.: Design with L-splines. Num. Algorithms 65, 91–124 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Carnicer, J.-M., Peña, J.-M.: Total positivity and optimal bases. In: Gasca, M., Micchelli, C.A. (eds.) Total Positivity and its Applications, pp. 133–155. Kluwer Academic Publishers, Berlin (1996)

    Chapter  Google Scholar 

  11. Carnicer, J.-M., Mainar, E., Peña, J.-M.: Critical length for design purposes and extended Chebyshev spaces. Constr. Approx. 20, 55–71 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Carnicer, J.-M., Mainar, E., Peña, J.-M.: On the critical lengths of cycloidal spaces. Constr. Approx. 39, 573–583 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Goodman, T.N.T.: Shape preserving representations. In: Gasca, M., Micchelli, C.A. (eds.) Mathematical Methods in Computer Aided Geometric Design, pp. 333–351. Academic, San Diego (1989)

    Google Scholar 

  14. Goodman, T.N.T.: Total Positivity and the Shape of Curves. In: Gasca, M., Micchelli, C.A. (eds.) Total Positivity and its Applications, pp. 157–186. Kluwer Academic Publishers, Berlin (1996)

    Chapter  Google Scholar 

  15. Gurariy, V.I., Lusky, W.: Geometry of Müntz Spaces and Related Questions. Lecture Notes Math. (1870). Springer (2005)

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

    MATH  Google Scholar 

  17. Korovkin, P.P.: Linear Operators and Approximation Theory. Hindustan Publishing Corporation, New Delhi (1960)

    MATH  Google Scholar 

  18. Lyche, T.: A recurrence relation for Chebyshevian B-splines. Constr. Approx. 1, 155–178 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  19. Mazure, M.-L.: Blossoming: a geometrical approach. Constr. Approx. 15, 33–68 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  20. Mazure, M.-L.: Blossoms and optimal bases. Adv. Comput. Math. 20, 177–203 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  22. Mazure, M.-L.: Chebyshev spaces and Bernstein bases. Constr. Approx. 22, 347–363 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Mazure, M.-L.: Ready-to-blossom bases in Chebyshev spaces. In: Jetter, K., Buhmann, M., Haussmann, W., Schaback, R., Stoeckler, J. (eds.) Topics in Multivariate Approximation and Interpolation, pp. 109–148. Elsevier, New York (2006)

    Chapter  Google Scholar 

  24. Mazure, M.-L.: Extended Chebyshev piecewise spaces characterised via weight functions. J. Approx. Theory 145, 33–54 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. Mazure, M.-L.: Dimension elevation formula for Chebyshevian blossoms. Comput. Aided Geom. Des. 26, 1016–1027 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Mazure, M.-L.: Bernstein-type operators in Chebyshev spaces. Num. Algorithm 52, 93–128 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  29. Mazure, M.-L.: From Taylor interpolation to Hermite interpolation via duality. Jaén J. Approx. 4, 15–45 (2012)

    MathSciNet  MATH  Google Scholar 

  30. Mazure, M.-L.: Extended Chebyshev spaces in rationality. BIT Num. Math. 53, 1013–1045 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  31. Mazure, M.-L.: Polynomial spaces revisited via weight functions. Jaén J. Approx. 6, 167–198 (2014)

    Google Scholar 

  32. Mazure, M.-L., Pottmann, H.: Tchebycheff splines. In: Gasca, M., Micchelli, C.A. (eds.) Total Positivity and its Applications, pp. 187–218. Kluwer Academic Publishers, Berlin (1996)

    Chapter  Google Scholar 

  33. Mühlbach, G.: A recurrence formula for generalized divided differences and some applications. J. Approx. Theory 9, 165–172 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  34. Mühlbach, G.: The general Neville–Aitken-algorithm and some applications. Num. Math. 31, 97–110 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  35. Pottmann, H.: The geometry of Tchebycheffian splines. Comput. Aided Geom. Des. 10, 181–210 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  36. Ramshaw, L.: Blossoms are polar forms. Comput. Aided Geom. Des. 6, 323–358 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  37. Schumaker, L.L.: Spline Functions. Wiley Interscience, New York (1981)

    MATH  Google Scholar 

  38. Schwartz, L.: Etude des Sommes d’Exponentielles. Hermann, Paris (1959)

    MATH  Google Scholar 

Download references

Acknowledgments

We would like to warmly thank the anonymous referees whose comments, questions and suggestions helped us improve the paper.

Author information

Authors and Affiliations

  1. King Abdullah University of Science and Technology, Thuwal, Saudi Arabia

    Rachid Ait-Haddou

  2. Laboratoire Jean Kuntzmann, CNRS, UMR 5224, Université Grenoble Alpes, BP 53, 38041, Grenoble Cedex 9, France

    Marie-Laurence Mazure

Authors
  1. Rachid Ait-Haddou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marie-Laurence Mazure
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marie-Laurence Mazure.

Additional information

Communicated by : Wolfgang Dahmen.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ait-Haddou, R., Mazure, ML. Approximation by Chebyshevian Bernstein Operators versus Convergence of Dimension Elevation. Constr Approx 43, 425–461 (2016). https://doi.org/10.1007/s00365-016-9331-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00365-016-9331-9

Keywords

AMS subject classification

Search

Navigation