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Bleimann, Butzer, and Hahn Operators Based on the-Integers

Abstract

We give a new generalization of Bleimann, Butzer, and Hahn operators, which includes-integers. We investigate uniform approximation of these new operators on some subspace of bounded and continuous functions. In Section, we show that the rates of convergence of the new operators in uniform norm are better than the classical ones. We also obtain a pointwise estimation in a general Lipschitz-type maximal function space. Finally, we define a generalization of these new operators and study the uniform convergence of them.

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References

  1. 1.

    Phillips GM: Bernstein polynomials based on the-integers. Annals of Numerical Mathematics 1997,4(1–4):511–518.

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Goodman TNT, Oruç H, Phillips GM: Convexity and generalized Bernstein polynomials. Proceedings of the Edinburgh Mathematical Society 1999,42(1):179–190. 10.1017/S0013091500020101

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Oruç H, Phillips GM: A generalization of the Bernstein polynomials. Proceedings of the Edinburgh Mathematical Society 1999,42(2):403–413. 10.1017/S0013091500020332

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Barbosu D: Some generalized bivariate Bernstein operators. Mathematical Notes 2000,1(1):3–10.

    MathSciNet  MATH  Google Scholar 

  5. 5.

    II'nskii A, Ostrovska S: Convergence of generalized Bernstein polynomials. Journal of Approximation Theory 2002,116(1):100–112. 10.1006/jath.2001.3657

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Doğru O, Duman O: Statistical approximation of Meyer-König and Zeller operators based on-integers. Publicationes Mathematicae Debrecen 2006,68(1–2):199–214.

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Trif T: Meyer-König and Zeller operators based on the-integers. Revue d'Analyse Numérique et de Théorie de l'Approximation 2000,29(2):221–229.

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Bleimann G, Butzer PL, Hahn L: A Bernšteĭn-type operator approximating continuous functions on the semi-axis. Koninklijke Nederlandse Akademie van Wetenschappen. Indagationes Mathematicae 1980,42(3):255–262.

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Altomare F, Campiti M: Korovkin-Type Approximation Theory and Its Applications, De Gruyter Studies in Mathematics. Volume 17. Walter de Gruyter, Berlin, Germany; 1994:xii+627.

    Google Scholar 

  10. 10.

    Gadjiev AD, Çakar Ö: On uniform approximation by Bleimann, Butzer and Hahn operators on all positive semiaxis. Transactions of Academy of Sciences of Azerbaijan. Series of Physical-Technical and Mathematical Sciences 1999,19(5):21–26.

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Agratini O: Approximation properties of a generalization of Bleimann, Butzer and Hahn operators. Mathematica Pannonica 1998,9(2):165–171.

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Agratini O: A class of Bleimann, Butzer and Hahn type operators. Analele Universităţii Din Timişoara 1996,34(2):173–180.

    MathSciNet  Google Scholar 

  13. 13.

    Doğru O: On Bleimann, Butzer and Hahn type generalization of Balázs operators. Studia Universitatis Babeş-Bolyai. Mathematica 2002,47(4):37–45.

    MATH  Google Scholar 

  14. 14.

    Phillips GM: Interpolation and Approximation by Polynomials, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 14. Springer, New York, NY, USA; 2003:xiv+312.

    Google Scholar 

  15. 15.

    Agratini O: Note on a class of operators on infinite interval. Demonstratio Mathematica 1999,32(4):789–794.

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Lenze B: Bernstein-Baskakov-Kantorovič operators and Lipschitz-type maximal functions. In Approximation Theory (Kecskemét, 1990), Colloquia Mathematica Societatis János Bolyai. Volume 58. North-Holland, Amsterdam, The Netherlands; 1991:469–496.

    Google Scholar 

  17. 17.

    Abel U, Ivan M: Some identities for the operator of Bleimann, Butzer and Hahn involving divided differences. Calcolo 1999,36(3):143–160. 10.1007/s100920050028

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Aral A, Gupta V: The-derivative and applications to-Szász Mirakyan operators. Calcolo 2006,43(3):151–170. 10.1007/s10092-006-0119-3

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Stancu DD: Approximation of functions by a new class of linear polynomial operators. Revue Roumaine de Mathématiques Pures et Appliquées 1968, 13: 1173–1194.

    MathSciNet  MATH  Google Scholar 

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Correspondence to Ali Aral.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Aral, A., Doğru, O. Bleimann, Butzer, and Hahn Operators Based on the-Integers. J Inequal Appl 2007, 079410 (2008). https://doi.org/10.1155/2007/79410

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Keywords

  • Continuous Function
  • Function Space
  • Uniform Convergence
  • Maximal Function
  • Uniform Approximation
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