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SETSCI - Volume 4 (6) (2019)
ISAS WINTER-2019 (ENS) - 4th International Symposium on Innovative Approaches in Engineering and Natural Sciences, Samsun, Turkey, Nov 22, 2019

Statistical Equal Convergence On Weighted Spaces
Fadime Dirik1*, Kamil Demirci2, Sevda Yıldız3
1Sinop University, Sinop, Turkey
2Sinop University, Sinop, Turkey
3Sinop University, Sinop, Turkey
* Corresponding author: fdirik@sinop.edu.tr
Published Date: 2019-12-22   |   Page (s): 179-184   |    220     18
https://doi.org/10.36287/setsci.4.6.054

ABSTRACT The Korovkin theory has effective role in approximation theory. This theory is connected with the approximation to continuous functions by means of positive linear operators. Many mathematicians have investigated the Korovkin-type theorems by for a sequence of positive linear operators defined on different spaces by using various types of convergence. Firstly, A.D. Gadjiev has proved the weighted Korovkin type theorems, (Math. Zamet., 20 (1976) 781-786 (in Russian)). Later, these theorems are studied by many authors by means of different convergence methods. Recently, The definition of equal convergence for real functions was introduced by Császár and Laczkovich and they improved their investigations on this convergence. Later Das et. al. introduced the ideas of I and I* equal convergence with the help of ideals by extending the equal convergence (Mat. Vesnik, vol:66, 2 (2014),165-177.). In our work, we introduce a new type of statistical convergence on weighted spaces by using the notions of the equal convergence. We study its use in the Korovkin-type approximation theory. Then, we construct an example such that our new approximation result works but its classical and statistical cases do not work
KEYWORDS Statistical equal convergence, Weighted spaces, Korovkin theorem
REFERENCES [1] F. Cao, Y. Liu, "Approximation theorems by positive linear operators in weighted spaces," Positivity vol. 15, pp. 87-103, 2011.
[2] A. Császár and M. Laczkovich, Discrete and equal convergence. Studia Sci. Math. Hungar. vol:10, (3-4)(1975), 463-472.
[3] A. Császár and M. Laczkovich, Some remarks on discrete Baire classes, Acta. Math. Acad. Sci. Hungar. vol:33, (1-2)(1979), 51-70.
[4] P. Das, S. Dutta and S. K. Pal, On and -equal convergence and an Egoroff-type theorem. Mat. Vesnik, vol: 66, 2 (2014),165-177.
[5] O. Duman, C. Orhan, Statistical approximation by positive linear operators, Studia Mathematica, 161 (2) (2004), 187-197.
[6] A. D. Gadjiev, On P. P. Korovkin type theorems, Mat. Zametki 20 (1976), 781-786 (in Russian).
[7] A. D. Gadjiev, The convergence problem for a sequence of positive linear operators on unbounded sets, and theorems analogous to that of P. P. Korovkin, Soviet Math. Dokl. 15 (1974), 1433-1436.
[8] N. Gönül Bilgin and N. Özgür, "Approximation by Two Dimensional Gadjiev-Ibragimov Type Operators," Ikonion Journal of Mathematics, vol. 1(1), pp. 1-10, 2019
[9] H. Steinhaus, Sur la convergence ordinaire et la convergence asymtotique, Colloq. Math. 2 (1951), 73-74.


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