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A-statistical relative modular convergence of positive linear operators

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Abstract

In this paper, we investigate the problem of statistical approximation to a function f by means of positive linear operators defined on a modular space. Particularly, in order to get stronger results than the classical aspects we mainly use the concept of statistical convergence. Also, a non-trivial application is presented.

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References

  1. Altomare, F., Campiti, M.: Korovkin type approximation theory and its application. Walter de Gruyter Publ, Berlin (1994)

    Book  MATH  Google Scholar 

  2. Bardaro, C., Boccuto, A., Dimitriou, X., Mantellini, I.: Abstract Korovkin-type theorems in modular spaces and applications. Cent. Eur. J. Math. 11(10), 1774–1784 (2013)

    MathSciNet  MATH  Google Scholar 

  3. Bardaro, C., Mantellini, I.: Approximation properties in abstract modular spaces for a class of general sampling-type operators. Appl. Anal. 85, 383–413 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bardaro, C., Mantellini, I.: Korovkin’s theorem in modular spaces. Comment. Math. 47, 239–253 (2007)

    MathSciNet  MATH  Google Scholar 

  5. Bardaro, C., Mantellini, I.: A Korovkin theorem in multivariate modular function spaces. J. Funct. Spaces Appl. 7, 105–120 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bardaro, C., Musielak, J., Vinti, G.: Nonlinear integral operators and applications, de Gruyter series in nonlinear analysis and appl. Vol. 9, 201. Walter de Gruyter Publ., Berlin (2003)

  7. Chittenden, E.W.: Relatively uniform convergence of sequences of functions. Trans. AMS 20, 197–201 (1914)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chittenden, E.W.: On the limit functions of sequences of continuous functions converging relatively uniformly. Trans. AMS 20, 179–184 (1919)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chittenden, E.W.: Relatively uniform convergence and classification of functions. Trans. AMS 23, 1–15 (1922)

    Article  MathSciNet  MATH  Google Scholar 

  10. Connor, J.S., Kline, J.: On statistical limit points and the consistency of statistical convergence. J. Math. Anal. Appl. 197, 392–399 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  11. Demirci, K.: \(A\)-statistical core of a sequence. Demonstr. Math. 33, 343–353 (2006)

    MathSciNet  MATH  Google Scholar 

  12. Demirci, K., Orhan, S.: Statistically relatively uniform convergence of positive linear operators. Results. Math. 69, 359–367 (2016)

  13. Ditzian, Z., Totik, V.: Moduli of smoothness, Springer series in computational mathematics 9. Springer-Verlag, New York (1987)

    MATH  Google Scholar 

  14. Fast, H.: Sur la convergence statistique. Colloq. Math. 2, 241–244 (1951)

    MathSciNet  MATH  Google Scholar 

  15. Fridy, J.A.: On statistical convergence. Analysis 5, 301–313 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  16. Fridy, J.A.: Statistical limit points. Proc. Am. Math. Soc. 118, 1187–1192 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fridy, J.A., Orhan, C.: Statistical limit superior and limit inferior. Proc. Am. Math. Soc. 125, 3625–3631 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  18. Freedman, A.R., Sember, J.J.: Densities and summability. Pac. J. Math. 95, 293–305 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hardy, G.H.: Divergent Series. Oxford Univ. Press, London (1949)

    MATH  Google Scholar 

  20. Kolk, E.: Matrix summability of statistically convergent sequences. Analysis 13, 77–83 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  21. Karakuş, S., Demirci, K., Duman, O.: Statistical approximation by positive linear operators on modular spaces. Positivity 14, 321–334 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Korovkin, P.P.: Linear operators and approximation theory. Hindustan Publ. Corp, Delhi (1960)

    Google Scholar 

  23. Kozlowski, W.M.: Modular function spaces. Pure Appl. Math., Vol. 122. Marcel Dekker, Inc., New York (1988)

  24. Moore, E.H.: An introduction to a form of general analysis. The New Haven Mathematical Colloquium. Yale University Press, New Haven (1910)

    Google Scholar 

  25. Mantellini, I.: Generalized sampling operators in modular spaces. Comment. Math. 38, 77–92 (1998)

    MathSciNet  MATH  Google Scholar 

  26. Musielak, J., Orlicz, W.: On modular spaces. Stud. Math. 18, 49–65 (1959)

    MathSciNet  MATH  Google Scholar 

  27. Musielak, J.: Orlicz spaces and modular spaces. Lecture notes in mathematics, vol. 1034. Springer-Verlag, Berlin (1983)

  28. Musielak, J.: Nonlinear approximation in some modular function spaces I. Math. Japon. 38, 83–90 (1993)

    MathSciNet  MATH  Google Scholar 

  29. Yilmaz, B., Demirci, K., Orhan, S.: Relative modular convergence of positive linear operators. Positivity. doi:10.1007/s11117-015-0372-2

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Sciences and Arts, Sinop University, 57000, Sinop, Turkey

    Kamil Demirci & Burçak Kolay

Authors
  1. Kamil Demirci
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  2. Burçak Kolay
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Correspondence to Kamil Demirci.

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Demirci, K., Kolay, B. A-statistical relative modular convergence of positive linear operators. Positivity 21, 847–863 (2017). https://doi.org/10.1007/s11117-016-0434-0

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  • DOI: https://doi.org/10.1007/s11117-016-0434-0

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