Open Access

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

Journal of Inequalities and Applications20082007:079410

https://doi.org/10.1155/2007/79410

Received: 29 May 2007

Accepted: 9 October 2007

Published: 17 January 2008

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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Authors’ Affiliations

(1)
Department of Mathematics, Kirikkale University
(2)
Department of Mathematics, Faculty of Sciences and Arts, Gazi University

References

  1. Phillips GM: Bernstein polynomials based on the-integers. Annals of Numerical Mathematics 1997,4(1–4):511–518.MathSciNetMATHGoogle Scholar
  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/S0013091500020101MathSciNetView ArticleMATHGoogle Scholar
  3. Oruç H, Phillips GM: A generalization of the Bernstein polynomials. Proceedings of the Edinburgh Mathematical Society 1999,42(2):403–413. 10.1017/S0013091500020332MathSciNetView ArticleMATHGoogle Scholar
  4. Barbosu D: Some generalized bivariate Bernstein operators. Mathematical Notes 2000,1(1):3–10.MathSciNetMATHGoogle Scholar
  5. II'nskii A, Ostrovska S: Convergence of generalized Bernstein polynomials. Journal of Approximation Theory 2002,116(1):100–112. 10.1006/jath.2001.3657MathSciNetView ArticleMATHGoogle Scholar
  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.MathSciNetMATHGoogle Scholar
  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.MathSciNetMATHGoogle Scholar
  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.MathSciNetMATHGoogle Scholar
  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.View ArticleMATHGoogle Scholar
  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.MathSciNetMATHGoogle Scholar
  11. Agratini O: Approximation properties of a generalization of Bleimann, Butzer and Hahn operators. Mathematica Pannonica 1998,9(2):165–171.MathSciNetMATHGoogle Scholar
  12. Agratini O: A class of Bleimann, Butzer and Hahn type operators. Analele Universităţii Din Timişoara 1996,34(2):173–180.MathSciNetGoogle Scholar
  13. Doğru O: On Bleimann, Butzer and Hahn type generalization of Balázs operators. Studia Universitatis Babeş-Bolyai. Mathematica 2002,47(4):37–45.MATHGoogle Scholar
  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.View ArticleGoogle Scholar
  15. Agratini O: Note on a class of operators on infinite interval. Demonstratio Mathematica 1999,32(4):789–794.MathSciNetMATHGoogle Scholar
  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. 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/s100920050028MathSciNetView ArticleMATHGoogle Scholar
  18. Aral A, Gupta V: The-derivative and applications to-Szász Mirakyan operators. Calcolo 2006,43(3):151–170. 10.1007/s10092-006-0119-3MathSciNetView ArticleMATHGoogle Scholar
  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.MathSciNetMATHGoogle Scholar

Copyright

© A. Aral and O. Doğru 2007

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.