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Ukrainian Mathematical Journal

, Volume 62, Issue 7, pp 1061–1072 | Cite as

Quantitative convergence theorems for a class of Bernstein–Durrmeyer operators preserving linear functions

  • H. Gonska
  • R. Păltănea
Article

We supplement recent results on a class of Bernstein–Durrmeyer operators preserving linear functions. This is done by discussing two limiting cases and proving quantitative Voronovskaya-type assertions involving the first-order and second-order moduli of smoothness. The results generalize and improve earlier statements for Bernstein and genuine Bernstein–Durrmeyer operators.

Keywords

Uniform Convergence Bernstein Polynomial Positive Linear Operator Bernstein Operator Durrmeyer Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    H. Gonska and R. Păltănea, “Simultaneous approximation by a class of Bernstein–Durrmeyer operators preserving linear functions,” Czech. Math. J. (to appear).Google Scholar
  2. 2.
    R. Păltănea, “A class of Durrmeyer-type operators preserving linear functions,” Ann. Tiberiu Popoviciu Sem. Funct. Equat. Approxim. Convex. (Cluj-Napoca), 5, 109–117 (2007).zbMATHGoogle Scholar
  3. 3.
    H. Gonska, Quantitative Aussagen zur Approximation durch positive lineare Operatoren, Doctoral-Degree Thesis, University of Duisburg (1979).Google Scholar
  4. 4.
    H. Gonska, D. Kacsó, and P. Piţul, “The degree of convergence of overiterated positive linear operators,” J. Appl. Funct. Anal., 1, 403–423 (2006).zbMATHMathSciNetGoogle Scholar
  5. 5.
    H. Gonska and I. Raşa, “A Voronovskaya estimate with second-order modulus of smoothness,” Proc. Math. Inequal. (Sibiu / Romania, Sept. 2008) (to appear).Google Scholar
  6. 6.
    H. Gonska and I. Raşa, Four Notes on Voronovskaya’s Theorem, Schriftenr. Fachbereichs Math. Univ. Duisburg-Essen (2009).Google Scholar
  7. 7.
    R. Păltănea, Approximation Theory Using Positive Linear Operators, Birkhäuser, Boston (2004).zbMATHGoogle Scholar
  8. 8.
    H. Gonska, “On the degree of approximation in Voronovskaja’s theorem,” Stud. Univ. Babeş-Bolyai. Math., 52, 103–115 (2007).zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2010

Authors and Affiliations

  • H. Gonska
    • 1
  • R. Păltănea
    • 2
  1. 1.University of Duisburg-EssenEssenGermany
  2. 2.Transilvania UniversityBraşovRomania

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