Abstract Korovkin theory in modular spaces in the sense of power series method

Abstract

In this paper, using the power series method we obtain an abstract Korovkin type approximation theorem for a sequence of positive linear operators defined on modular spaces.

Keywords

• Power series method,modular spaces,abstract Korovkin theory

References

1. Altomare, F., Korovkin-type theorems and approximation by positive linear operators, Surveys in Approximation Theory 5.13, 2010. Altomare, F. and Diomede, S., Contractive Korovkin subsets in weighted spaces of continuous functions, Rend. Circ. Mat. Palermo 50, 547-568, 2001.
2. Atlihan, Ö. G. and Taş, E., An abstract version of the Korovkin theorem via A-summation process, Acta. Math. Hungar. 145, 360-368, 2015.
3. Bardaro, C. Boccuto, A. Dimitriou, X. and Mantellini, I., A Korovkin theorem in multivariate modular function spaces, J. Func. Spaces Appl., 7, 105-120, 2009.
4. Bardaro, C. Boccuto, A. Dimitriou, X. and Mantellini, I., Modular filter convergence theorems for abstract sampling-type operators, Appl. Anal. 92, 2404-2423, 2013.
5. Bardaro, C. Boccuto, A. Dimitriou, X. and Mantellini, I., Abstract Korovkin-type theorems in modular spaces and applications, Cent. Eur. J. Math., 11, 1774-1784, 2013.
6. Bardaro, C. and Mantellini, I., Korovkin's theorem in modular spaces, Comment. Math. 47, 239-253, 2007.
7. Bardaro, C. and Mantellini, I., Multivariate moment type operators: approximation properties in Orlicz spaces, J. Math. Ineq. 2, 247-259, 2008.
8. Bardaro, C. Musielak, J. and Vinti, G., Nonlinear Integral Operators and Applications, De Gruyter Ser. Nonlinear Anal. Appl. 9, Walter de Gruyter, Berlin, 2003.

9. Belen, C. and Yildirim, M., Statistical approximation in multivariate modular function spaces, Comment. Math. 51, 39-53, 2011.
10. Bernstein, F., Über eine Anwendung der Mengenlehre auf ein der Theorie der sakularen Störungen herrührendes Problem, Math. Ann., 71, 417-439, 1912.
11. Boccuto, A. and Dimitriou, X., Modular filter convergence theorems for Urysohn integral operators and applications, Acta Math. Sinica, 29, 1055-1066, 2013.
12. Boos, J., Classical and Modern Methods in Summability, Oxford University Press, 2000.
13. Butzer, P. L. and Berens, H., Semi-groups of operators and approximation, Die Grundlehren der Mathematischen Wissenschaften, 145, Springer, New York, 1967.
14. Gadjiev, A. D. and Orhan, C., Some approximation theorems via statistical convergence, Rocky Mountain J. Math. 32, 129-138, 2002.
15. Karakus, S. and Demirci, K., Matrix summability and Korovkin type approximation theorem on modular spaces, Acta Math. Univ. Commenianae, 2, 281-292, 2010.
16. Karakus, S. Demirci, K. Duman, O., Statistical approximation by positive linear operators on modular spaces, Positivity, 14, 321-334, 2010.
17. Korovkin, P. P., Linear Operators and Approximation Theory, Hindustan Publ. Co., Delhi, 1960.
18. Kratz, W. and Stadtmüller, U., Tauberian theorems for $J_{p}$-summability, J. Math. Anal. Appl. 139, 362-371, 1989.
19. Kuratowski, K., Topology I-II, Academic Press/PWN, New York-London/Warsaw, 1966-1968.
20. Musielak, J., Orlicz Spaces and Modular Spaces, Lecture Notes in Math., 1034, Springer, Berlin, 1983.
21. Musielak, J. and Orlicz, W., On modular spaces, Studia Math. 18, 49-65, 1959.
22. Özgüç, I. and Taş, E., A Korovkin-type approximation theorem and power series method, Results. Math. 69, 497-504, 2016.
23. Stadtmüller, U. and Tali, A., On certain families of generalized Nörlund methods and power series methods, J. Math. Anal. Appl. 238, 44-66, 1999.