KOROVKIN THEOREM VIA STATISTICAL e-MODULAR CONVERGENCE OF DOUBLE SEQUENCES

KOROVKIN THEOREM VIA STATISTICAL e-MODULAR CONVERGENCE OF DOUBLE SEQUENCES

In the present paper, we obtain an abstract version of the Korovkin type theorem via the concept of statistical e-convergence in modular spaces for double sequences of positive linear operators. We give an application showing that the new result is stronger than classical ones. Also, we study an extension to non-positive operators.

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  • Bardaro, C., Boccuto, A., Demirci, K., Mantellini, I., Orhan, S., Triangular A-statistical approximation by double sequences of positive linear operators. Results Math. 2015; 68: 271-291.
  • Bardaro C., Boccuto A., Demirci K., Mantellini I., Orhan S. Korovkin-type theorems for modular -A-statistical convergence, J. Funct. Spaces. Article ID 160401 2015; 2015:1-11.
  • Bardaro C., Boccuto A., Dimitriou X. and Mantellini I., Abstract Korovkin type theorems in modular spaces and applications, Cent. Eur. J. Math. 2013; 11(10): 1774-1784.
  • Bardaro, C. and Mantellini, I., Korovkin's theorem in modular spaces, Commentationes Math. 2007; 47: 239-253.
  • Bardaro, C., Musielak,J., Vinti,G., Nonlinear integral operators and applications, de Gruyter Series in Nonlinear Analysis and Appl. Vol., 9 Walter de Gruyter Publ., Berlin, 2003.
  • Boccuto, A. and Dimitriou, X., Korovkin-type theorems for abstract modular convergence, Results in Mathematics 2016; 69: 477-495.
  • Boos, J., Leiger, T., Zeller, K. Consistency theory for SM-methods. Acta. Math. Hungar. 1997; 76: 83-116.
  • Boos, J. Classical and modern methods in summability. Oxford University Press Inc., New York, 2000.
  • Demirci K., Orhan S., Statistical relative approximation on modular spaces. Results in Mathematics 2017; 71: 1167-1184.
  • Demirci, K., Kolay, B., A-Statistical Relative Modular Convergence of Positive Linear Operators, Positivity 2017; 21: 847-863.
  • Karaku¸s, S., Demirci, K., Duman, O., Statistical approximation by positive linear operators on modular spaces. Positivity 2010; 14: 321-334.
  • Kozlowski W. M., Modular function spaces, Pure Appl. Math., Vol. 122, Marcel Dekker, Inc., New York, 1988.
  • Kuratowski K., Topology, Volls I and II, Academic Press, New York-London, 1966/1968.
  • Mantellini I., Generalized sampling operators in modular spaces, Commentationes Math. 1998; 38: 77-92.
  • Musielak, J., Orlicz spaces and modular spaces, Lecture Notes in Mathematics, Vol. 1034 Springer-Verlag, Berlin, 1983.
  • Musielak J., Nonlinear approximation in some modular function spaces I, Math. Japon. 1993; 38: 83-90.
  • Pringsheim, A. Zur theorie der zweifach unendlichen zahlenfolgen. Math. Ann. 1900; 53: 289-321.
  • Sever, Y., Talo, Ö. Statistical e-convergence of double sequences and its application to Korovkin type approximation theorem for functions of two variables. Iranian Journal of Science and Technology, Transactions A: Science. 2017; 41(3): 851-857.
  • Yilmaz, B., Demirci, K., Orhan, S., Relative Modular Convergence of Positive Linear Operators, Positivity 2016; 20: 565-577.