Abstract
In this paper, using power series method we obtain a Korovkin type theorem for double sequences of real valued functions defined on a compact subset of \(\mathbb {R}^{2}\)(the real two-dimensional space). We also present an example that satisfies our theorem. Finally, we calculate the rate of convergence.
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References
Altomare, F., Campiti, M.: Korovkin-Type Approximation Theory and Its Applications, de Gruyter Stud. Math. 17, Walter de Gruyter, Berlin (1994)
Atlihan, O.G., Taş, E.: An abstract version of the Korovkin theorem via A-summation process. Acta Math. Hungar. 145(2), 360–368 (2015). doi:10.1007/s10474-015-0476-y
Bardaro, C., Boccuto, A., Demirci, K., Mantellini, I., Orhan, S.: Triangular A-statistical approximation by double sequences of positive linear operators. Results Math. 68, 271–291 (2015)
Baron, S., Stadtmüller, U.: Tauberian theorems for power series methods applied to double sequences. J. Math. Anal. Appl. 211(2), 574–589 (1997)
Demirci, K., Dirik, F.: Statistical extension of the Korovkin-type approximation theorem. Appl. Math. E-Notes 11, 101–109 (2011)
Demirci, K., Orhan, S.: Statistical relative approximation on modular spaces. Results Math. 71, 1167–1184 (2017)
Kadak, U., Braha, N.L., Srivastava, H.M.: Statistical weighted \(\cal{B}\)-summability and its applications to approximation theorems. Appl. Math. Comput. 302, 80–96 (2017)
Orhan, S., Demirci, K.: Statistical approximation by double sequences of positive linear operators on modular spaces. Positivity 19, 23–36 (2015)
Ozguc, I., Tas, E.: A Korovkin-type approximation theorem and power series method. Results Math. 69, 497–504 (2016)
Powell, R.E., Shah, S.M.: Summability Theory and Its Applications. Van Nostrand Reinhold Company, London (1972)
Pringsheim, A.: Zur theorie der zweifach unendlichen zahlenfolgen. Math. Ann. 53, 289–321 (1900)
Volkov, V.I.: On the convergence of sequences of linear positive operators in the space of two variables. Dokl. Akad. Nauk. SSSR (N.S.) 115, 17–19 (1957)
Yurdakadim, T.: Some Korovkin type results via power series method in modular spaces. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 65, 65–76 (2016)
Tas, E., Yurdakadim, T.: Approximation to derivatives of functions by linear operators acting on weighted spaces by power series method. In: Computational Analysis, pp. 363–372. Springer International Publishing (2016)
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Şahin, P.O., Dirik, F. A Korovkin-type theorem for double sequences of positive linear operators via power series method. Positivity 22, 209–218 (2018). https://doi.org/10.1007/s11117-017-0508-7
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DOI: https://doi.org/10.1007/s11117-017-0508-7