Korovkin Theorem via Statistical e-Modular Convergence of Double Sequences

The main purpose of the present paper is to 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. After proving this theorem, 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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Pringsheim, "A. Zur theorie der zweifach unendlichen zahlenfolgen," Math. Ann., vol. 53, pp. 289-321, 1900.

Boos, J., Leiger, T., Zeller, "K. Consistency theory for SM-methods," Acta. Math. Hungar., vol. 76, pp. 83-116, 1997.

Boos, J. "Classical and modern methods in summability," Oxford University Press Inc., New York, 2000.

Sever, Y., Talo, Ö. "Statistical econvergence 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, vol. 41, no. 3, pp. 851-857, 2017.

Bardaro, C. and Mantellini, I., "Korovkin's theorem in modular spaces," Commentationes Math., vol. 47, pp. 239- 253, 2007.

Bardaro, C., Boccuto, A., Demirci, K., Mantellini, I., Orhan, S., "Triangular Astatistical approximation by double sequences of positive linear operators," Results Math., vol. 68, pp. 271-291, 2015.

Boccuto, A. and Dimitriou, X., "Korovkintype theorems for abstract modular convergence," Results in Mathematics, vol. 69, pp. 477-495, 2016.

Demirci K., Orhan S., "Statistical relative approximation on modular spaces," Results in Mathematics, vol. 71, pp. 1167-1184, 2017.

Demirci, K., Kolay, B., "A-Statistical Relative Modular Convergence of Positive Linear Operators," Positivity, vol. 21, pp. 847-863, 2017.

Karakuş, S., Demirci, K., Duman, O., "Statistical approximation by positive linear operators on modular spaces," Positivity, vol. 14, pp. 321-334, 2010.

Yilmaz, B., Demirci, K., Orhan, S., "Relative Modular Convergence of Positive Linear Operators," Positivity, vol. 20, pp. 565-577, 2016.

Kuratowski K., "Topology," Volls I and II, Academic Press, New York-London, 1966/1968.

Kozlowski W. M., "Modular function spaces," Pure Appl. Math., vol. 122, Marcel Dekker, Inc., New York, 1988.

Mantellini I., "Generalized sampling operators in modular spaces," Commentationes Math., vol. 38, pp. 77-92, 1998.

Musielak J., "Nonlinear approximation in some modular function spaces" I, Math. Japon. vol. 38, pp. 83-90, 1993.

Musielak, J., "Orlicz spaces and modular spaces," Lecture Notes in Mathematics, vol. 1034 Springer-Verlag, Berlin, 1983.

Bardaro C., Boccuto A., Dimitriou X. and Mantellini I., "Abstract Korovkin type theorems in modular spaces and applications," Cent. Eur. J. Math., vol. 11, no. 10, pp. 1774-1784, 2013.

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.

Bardaro C., Boccuto A., Demirci K., Mantellini I., Orhan S. "Korovkin-type theorems for modular Ψ-A-statistical convergence," J. Funct. Spaces. Article ID 160401 vol. 2015, pp. 1-11, 2015.