Advertisement

On Iterates of Linear Variation Diminishing Operators and Characterization of Bernstein-Type Polynomials

  • Ying-Sheng Hu
Part of the International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique book series (ISNM, volume 90)

Abstract

As we all know, Bernstein approximation of a smooth function on [a,b] preserves the signs of function itself and its higher order derivatives. This beautiful property, in one hand, wins important and wide applications,but on the other hand,has to pay a precious price--very slow convergence.It is interesting to notice that the similar situation appear,more or less,in some other wellknown approximations,for instance, the Modified Bernstein-Durrmeyer operator and Schoenberg variation diminishing operator (see[1]).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Y.S.Hu, A Note on the Piecewise-Comonotonic Property of the V.D Spline, written in Oslo, Spring 1982.(unpublished)Google Scholar
  2. 2.
    H.Berens & R.Devore, A Chaaracterization of Bernstein Polynomials,in Approxition Theory III, Proc.Int.Symp.Austin 1980,ed. by E.W.Cheney.Google Scholar
  3. 3.
    R.P.Kelisky & T.J.Revlin, Iterates of Bernstein-Polynomials, Pacific J.Math. 21 (3) 1967.Google Scholar
  4. 4.
    J.Nagel,Asymptotic Properties of Powers of Kantorovic Operators J.A.T.32,1982.Google Scholar
  5. 5.
    J.Nagel,Asymptotic Properties of Powers of Bernstein Operators, J.A.T.29,1980.Google Scholar
  6. 6.
    C.A.Micchelli,The Saturation Class and Iterates of the Bernstein Polynomials, J.A.T.8(1983).Google Scholar
  7. 7.
    Y.S.Hu & S.X.Xu, The Iterative Limit for a Kind of Variation Diminishing Operators I, Acta Mathematical Applacatae Sinica Vol. 1,No. 3, 1978.Google Scholar
  8. 8.
    Y.S.Hu & S.X.Xu, The Iterative Limit for a Kind of Variation Diminishing Operators II, Acta Mathematics Sinica Vol. 22,No. 3 (1979).Google Scholar
  9. 9.
    G.Z.Chang, Generalized Bernstein Polynomials,J.Computational Math.Vol. 1,No. 4. 1983.Google Scholar
  10. 10.
    Y.S.Hu, Iterates of Bernstein Polynomials on Triangle,Report.Vol.3.No.6. 1987, Inst.of Math. Academia Sinica, Beijing.Google Scholar
  11. 11.
    Y.S.Hu, On Bernstein Polynomials in Multivariables,Report,Vol.3 No.5.1987,Inst.of Math.Academia Sinica,Beijing.Google Scholar
  12. 12.
    S.Karlin,Total Positivity,Vol.l.Stanford Univ.Press.1968.Google Scholar
  13. 13.
    W.Z.Chen, On the Modified Bernstein-Durrmeyer Operator,Report on the Fifth Chinese Conference on Approx.Theory,Zhen Zhou, China,May 1987.Google Scholar

Copyright information

© Birkhäuser Verlag Basel 1989

Authors and Affiliations

  • Ying-Sheng Hu
    • 1
  1. 1.Institute of MathematicsAcademia SinicaBeijingChina

Personalised recommendations