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Bernstein-type operators in Chebyshev spaces

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Abstract

We prove that it is possible to construct Bernstein-type operators in any given Extended Chebyshev space and we show how they are connected with blossoms. This generalises and explains a recent result by Aldas/Kounchev/Render on exponential spaces. We also indicate why such operators automatically possess interesting shape preserving properties and why similar operators exist in still more general frameworks, e.g., in Extended Chebyshev Piecewise spaces. We address the problem of convergence of infinite sequences of such operators, and we do prove convergence for special instances of Müntz spaces.

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References

  1. Aldaz, J.M., Kounchev, O., Render, H.: Bernstein Operators for Exponential Polynomials. Constr. Approx. (2008, Online)

  2. Bernstein, S.: Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités. Comm. Soc. Math. Kharkov 13, 1–2 (1912)

    Google Scholar 

  3. Bister, D.: Ein neuer Zugang für eine verallgemeinerte Klasse von Tschebyscheff-Splines. Dissertation, 15 November 1996, Shaker Verlag, Aachen (1997)

    Google Scholar 

  4. Bister, D., Prautzsch, H.: A new approach to Tchebycheffian B-splines. In: Curves and Surfaces with Applications in CAGD, pp. 35–41. Vanderbilt University Press, Nashville (1997)

    Google Scholar 

  5. 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, Boston (1996)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  7. Costantini, P.: Curve and surface construction using variable degree polynomial splines. Comput. Aided Geom. Des. 17, 419–446 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  8. Costantini, P., Lyche, T., Manni, C.: On a class of weak Tchebycheff systems. Numer. Math. 101, 333–354 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  9. 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, Bosotn (1996)

    Google Scholar 

  10. Karlin, S.: Total positivity. Stanford Univ. Press, Stanford (1968)

    MATH  Google Scholar 

  11. Karlin, S.J., Studden, W.J.: Tchebycheff Systems: with applications in analysis and statistics. Wiley Interscience, N.Y. (1966)

    MATH  Google Scholar 

  12. Korovkin, P.P.: Linear operators and approximation theory. Hindustan Publ. Corp., Delhi (1960)

    Google Scholar 

  13. Lorentz, G.G.: Bernstein Polynomials, 2nd edn. Chelsea, NY (1986)

    MATH  Google Scholar 

  14. Manni, C., Mazure, M.-L.: Shape constraints and optimal bases for C 1 Hermite interpolatory subdivision schemes (preprint)

  15. Mazure, M.-L.: Chebyshev spaces with polynomial blossoms. Adv. Comput. Math. 10, 219–238 (1999)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  17. Mazure, M.-L.: Chebyshev-Bernstein bases. Comput. Aided Geom. Des. 16, 649–669 (1999)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Chapter  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  23. Mazure, M.-L.: On dimension elevation in quasi extended Chebyshev spaces. Numer. Math. 109, 459–475 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  24. Mazure, M.-L.: Which spaces for design. Numer. Math. 110, 357–392 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  25. Mazure, M.-L.: On differentiation formulæ for Chebyshevian Bernstein and B-spline bases. Jaén Journal on Approximations (in press)

  26. Mazure, M.-L., Pottmann, H.: Tchebycheff curves. In: Total Positivity and its Applications, pp. 187–218. Kluwer Academic, Boston (1996)

    Google Scholar 

  27. Morigi, S., Neamtu, M.: Some results for a class of generalized polynomials. Adv. Comput. Math. 12, 133–149 (2000)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  30. Schumaker, L.L.: Spline functions. Wiley Interscience, N.Y. (1981)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Laboratoire Jean Kuntzmann, Université Joseph Fourier, BP53, 38041, Grenoble Cedex 9, France

    Marie-Laurence Mazure

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  1. Marie-Laurence Mazure
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Correspondence to Marie-Laurence Mazure.

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Mazure, ML. Bernstein-type operators in Chebyshev spaces. Numer Algor 52, 93–128 (2009). https://doi.org/10.1007/s11075-008-9260-1

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  • DOI: https://doi.org/10.1007/s11075-008-9260-1

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