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A Strong Converse Result for Approximation by Weighted Bernstein Polynomials on the Real Line

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Trends and Applications in Constructive Approximation

Part of the book series: ISNM International Series of Numerical Mathematics ((ISNM,volume 151))

Abstract

We prove that the weighted error of approximation by generalized Bernstein polynomials introduced in [1] is equivalent to the modulus of smoothness of the function. This result is analogous to a well-known theorem of Ditzian and Ivanov [2] for the classical Bernstein polynomials.

Research supported by OTKA No. T032872.

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References

  1. B. Della Vecchia, G. Mastroianni and J. Szabados: Weighted approximation of functions on the real line by Bernstein polynomials. J. Approx. Theory (accepted).

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  2. Z. Ditzian and K.G. Ivanov: Strong converse inequalities. J. d’Analyse Math. 61 (1993), 61–111.

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  3. Z. Ditzian and V. Totik: Moduli of smoothness. Springer Series in Computational Mathematics 9, Springer-Verlag, New York (1987).

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  4. A.L. Levin and D.S. Lubinsky: Christoffel functions, orthogonal polynomials and Nevai’s conjecture for Freud weights. Constr. Approx. 8 (1992), 463–535.

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  5. G.G. Lorentz: Bernstein Polynomials. Mathematical Expositions 8, University of Toronto Press, Toronto (1953).

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Author information

Authors and Affiliations

  1. Alfréd Rényi Institute of Mathematics, P.O.B. 127, H-1364, Budapest, Hungary

    Jozsef Szabados

Authors
  1. Jozsef Szabados
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Editor information

Editors and Affiliations

  1. FB3, Applied Mathematics (Constructive Approximation), University of Applied Science Bochum, Herner Str. 45, D-44787, Bochum

    Detlef H. Mache

  2. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, P.O. Box 127, H-1364, Budapest

    József Szabados

  3. Department of Applied Mathematics, Delft University of Technology, P.O. Box 5031, 2600 GA, Delft, The Netherlands

    Marcel G. de Bruin

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© 2005 Birkhäuser Verlag Basel/Switzerland

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Szabados, J. (2005). A Strong Converse Result for Approximation by Weighted Bernstein Polynomials on the Real Line. In: Mache, D.H., Szabados, J., de Bruin, M.G. (eds) Trends and Applications in Constructive Approximation. ISNM International Series of Numerical Mathematics, vol 151. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7356-3_18

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