Korovkin type approximation for double sequences via statistical A-summation process on modular spaces

Authors

  • Sevda ORHAN Department of Mathematics Sinop University Sinop, Turkey, e-mail: sevdaorhan@sinop.edu.tr
  • Burçak KOLAY Department of Mathematics Sinop University Sinop, Turkey, e-mail: burcakyilmaz@sinop.edu.tr

DOI:

https://doi.org/10.24193/subbmath.2018.1.08

Keywords:

Modular space, A-summation process, positive linear operators, Korovkin theorem.

Abstract

n this work, we introduce the Korovkin type approximation theorems on modular spaces via statistical A-summation process for double sequences of positive linear operators and we construct an example satisfying our new approximation theorem but does not satisfy the classical one.

Mathematics Subject Classification (2010): 40B05, 41A36, 47B38, 46E30.

References

Bardaro, C., Boccuto, A., Dimitriou, X., Modular convergence theorems for abstract sampling operators, Applicable Analysis, 92(2013), no. 11, 2404-2423.

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

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., Mantellini, I., Korovkin’s theorem in modular spaces, Comment. Math., 47(2007), 239-253.

Bardaro, C., Mantellini, I., A Korovkin theorem in multivariate modular function spaces, J. Funct. Spaces Appl., 7(2009), no. 2, 105-120.

Boos, J., Classical and Modern Methods in Summability, Oxford University Press, Oxford, 2000.

C¸ akan, C., Altay, B., Statistically boundedness and statistical core of double sequences, J. Math. Anal. Appl., 317(2006), 690-697.

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

Karaku¸s, S., Demirci, K., A−summation process and Korovkin-type approximation theorem for double sequences of positive linear operators, Mathematica Slovaca, textbf62(2012), 281-292.

Karaku¸s, S., Demirci, K., Matrix summability and Korovkin type approximation theorem on modular spaces, Acta Math. Univ. Comenian., 79(2010), no. 2, 281-292.

Karaku¸s, S., Demirci, K., Duman, O., Statistical approximation by positive linear operators on modular spaces, Positivity, 14(2010), 321-334.

Korovkin, P.P., Linear operators and approximation theory, Hindustan Publ. Corp., Delhi, 1960.

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

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

Moricz, F., Statistical convergence of multiple sequences, Arch. Math., 81(2004), 82-89. [16] Musielak, J., Orlicz spaces and modular spaces, Lecture Notes in Mathematics, 1034 Springer-Verlag, Berlin, 1983.

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

Orhan, C., Sakaoglu, I˙., Rate of convergence in Lp approximation, Period. Math. Hungar., 68(2014), no. 2, 176-184.

Orhan S., Demirci, K., Statistical A−summation process and Korovkin type approximation theorem on modular spaces, Positivity, 18(2014), 669-686.

Orhan S., Demirci, K., Statistical approximation by double sequences of positive linear operators on modular spaces, Positivity, 19(2015), 23-36.

Patterson, R.F., Sava¸s, E., Uniformly summable double sequences, Studia Scientiarum Mathematicarum Hungarica, 44(2007), 147-158.

Pringsheim, A., Zur theorie der zweifach unendlichen zahlenfolgen, Math. Ann., 53(1900), 289-321.

Sakaog˘lu O¨ zgu¨c¸, I˙., Orhan, C., Strong summation process in Lp spaces, Nonlinear Analysis, 86(2013), 89-94.

Sava¸s, E., Rhoades, B.E., Double summability factor theorems and applications, Math. Inequal. Appl., 10(2007), 125-149.

Steinhaus, H., Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math., 2(1951), 73-74.

Downloads

Published

2018-03-30

How to Cite

ORHAN, S., & KOLAY, B. (2018). Korovkin type approximation for double sequences via statistical A-summation process on modular spaces. Studia Universitatis Babeș-Bolyai Mathematica, 63(1), 125–140. https://doi.org/10.24193/subbmath.2018.1.08

Issue

Section

Articles

Similar Articles

<< < 20 21 22 23 24 25 26 27 28 29 > >> 

You may also start an advanced similarity search for this article.