Advertisement

Weighted Statistical Convergence of Bögel Continuous Functions by Positive Linear Operator

  • Fadime DirikEmail author
Chapter

Abstract

In the present work, we have introduced a weighted statistical approximation theorem for sequences of positive linear operators defined on the space of all real-valued B-continuous functions on a compact subset of \( \mathbb {R} ^{2}= \mathbb {R} \times \mathbb {R} \). Furthermore, we display an application which shows that our new result is stronger than its classical version.

Keywords

Weighted uniform convergence Double sequences Statistical convergence Korovkin-type approximation theorem 

Mathematics Subject Classification

40A35 41A36 

References

  1. 1.
    F. Altomare, M. Campiti, Korovkin-Type Approximation Theory and Its Applications, vol. 17, De Gruyter Studies in Mathematics (Walter de Gruyter, Berlin, 1994)Google Scholar
  2. 2.
    I. Badea, Modulus of continuity in Bögel sense and some applications for approximation by a Bernstein-type operator. Studia Univ. Babeş-Bolyai Ser. Math. Mech. 18, 69–78 (1973) (in Romanian)Google Scholar
  3. 3.
    C. Badea, I. Badea, H.H. Gonska, A test function and approximation by pseudopolynomials. Bull. Aust. Math. Soc. 34, 53–64 (1986)Google Scholar
  4. 4.
    C. Badea, C. Cottin, Korovkin-type theorems for generalized Boolean sum operators, Approximation Theory (Kecskemét, 1990), vol. 58, Colloquia Mathematica Societatis János Bolyai (North-Holland, Amsterdam, 1991), pp. 51–68Google Scholar
  5. 5.
    C. Bardaro, A. Boccuto, K. Demirci, I. Mantellini, S. Orhan, Triangular A-statistical approximation by double sequences of positive linear operators. Results Math. 68, 271–291 (2015)MathSciNetCrossRefGoogle Scholar
  6. 6.
    C. Bardaro, A. Boccuto, K. Demirci, I. Mantellini, S. Orhan, Korovkin-type theorems for modular -A-statistical convergence. J. Funct. Spaces 11 (2015). Article ID 160401Google Scholar
  7. 7.
    K. Bögel, Mehrdimensionale differentiation von funktionen mehrerer veränderlicher. J. Reine Angew. Math. 170, 197–217 (1934)Google Scholar
  8. 8.
    K. Bögel, Über mehrdimensionale differentiation, integration und beschränkte variation. J. Reine Angew. Math. 173, 5–29 (1935)Google Scholar
  9. 9.
    K. Bögel, Über die mehrdimensionale differentiation. Jahresber. Deutsch. Mat. Verein. 65, 45–71 (1962)Google Scholar
  10. 10.
    K. Demirci, S. Orhan, Statistically relatively uniform convergence of positive linear operators. Results Math. 69(3–4), 359–367 (2016)MathSciNetCrossRefGoogle Scholar
  11. 11.
    F. Dirik, K. Demirci, Korovkin type approximation theorem for functions of two variables in statistical sense. Turk. J. Math. 34, 73–83 (2010)Google Scholar
  12. 12.
    F. Dirik, K. Demirci, Approximation in statistical sense to n-variate B-continuous functions by positive linear operators. Math. Slovaca 60, 877–886 (2010)Google Scholar
  13. 13.
    F. Dirik, O. Duman, K. Demirci, Approximation in statistical sense to B-continuous functions by positive linear operators. Studia Sci. Math. Hungarica 47(3), 289–298 (2010)MathSciNetCrossRefGoogle Scholar
  14. 14.
    H. Fast, Sur la convergence statistique. Colloq. Math. 2, 241–244 (1951)MathSciNetCrossRefGoogle Scholar
  15. 15.
    S. Ghosal, Weighted statistical convergence of order \(\alpha \) and its applications. J. Egypt. Math. Soc. 24, 60–67 (2016)Google Scholar
  16. 16.
    V. Karakaya, T.A. Chishti, Weighted statistical convergence. Iran. J. Sci. Technol. Trans. A Sci. 33, 219–223 (2009)Google Scholar
  17. 17.
    P.P. Korovkin, Linear Operators and Approximation Theory (Hindustan, Delhi, 1960)Google Scholar
  18. 18.
    M. Mursaleen, V. Karakaya, M. Erturk, F. Gursoy, Weighted statistical convergence and its application to Korovkin type approximation theorem. Appl. Math. Comput. 218, 9132–9137 (2012)MathSciNetCrossRefGoogle Scholar
  19. 19.
    F. Móricz, Statistical convergence of multiple sequences. Arch. Math. 81(1), 82–89 (2003)Google Scholar
  20. 20.
    A. Pringsheim, Zur theorie der zweifach unendlichen zahlenfolgen. Math. Ann. 53, 289–321 (1900)MathSciNetCrossRefGoogle Scholar
  21. 21.
    H. Steinhaus, Sur la convergence ordinaire et la convergence asymtotique. Colloq. Math. 2, 73–74 (1951)Google Scholar
  22. 22.
    B. Yılmaz, K. Demirci, S. Orhan, Relative modular convergence of positive linear operators. Positivity 20(3), 565–577 (2016)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Department of MathematicsSinop UniversitySinopTurkey

Personalised recommendations