Abstract
We prove a Korovkin-type approximation theorem via power series methods of summability for continuous \(2\pi \)-periodic functions of two variables and verify the convergence of approximating double sequences of positive linear operators by using modulus of continuity. An example concerning double Fourier series is also constructed to illustrate the obtained results.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alotaibi, A., Mursaleen, M., Mohiuddine, S.A.: Statistical approximation for periodic functions of two variables. J. Funct. Spaces Appl. 2013, 1–5 (2013)
Baron, S., Tietz, H.: Produkts\(\ddot{a}\)tze für Verfahren zur Limitierung von Doppelfolgen. Anal. Math. 20, 81–94 (1994)
Baron, S., Stadtmüller, U.: Tauberian theorems for power series methods applied to double sequences. J. Math. Anal. Appl. 211(2), 574–589 (1997)
Belen, C., Mursaleen, M., Yıldırım, M.: Statistical \(A\)-summability of double sequences and a Korovkin type approximation theorem. Bull. Korean Math. Soc. 49(4), 851–861 (2012)
Braha, N.L., Srivastava, H.M., Mohiuddine, S.A.: A Korovkin’s type approximation theorem for periodic functions via the statistical summability of the generalized de la Valle Poussin mean. Appl. Math. Comput. 228, 162–169 (2014)
Demirci, K., Dirik, F.: Four-dimensional matrix transformation and rate of A-statistical convergence of periodic functions. Math. Comput. Model. 52, 1858–1866 (2010)
Duman, O., Erkuş, E.: Approximation of continuous periodic functions via statistical convergence. Comput. Math. Appl. 52, 967–974 (2006)
Kadak, U.: On relative weighted summability in modular function spaces and associated approximation theorems. Positivity (2017). https://doi.org/10.1007/s11117-017-0487-8
Korovkin, P.P.: Convergence of linear positive operators in the spaces of continuous functions. Dokl. Akad. Nauk. SSSR (N.S.) 90, 961–964 (1953)
Korovkin, P.P.: Linear Operators and Approximation Theory. Hindustan Publ. Co., Delhi (1960)
Mohiuddine, S.A., Alotaibi, A., Mursaleen, M.: Statistical summability \((C, 1)\) and a Korovkin type approximation theorem. J. Inequal. Appl. 172, 1–8 (2012)
Özgüç, İ., Taş, E.: A Korovkin-type approximation theorem and power series method. Results Math. 69, 497–504 (2016)
Stadtmüller, U., Tali, A.: A family of generalized Norlund methods and related power series methods applied to double sequences. Math. Nachr. 282(2), 288–306 (2009)
Taş, E., Yurdakadim, T.: Approximation by positive linear operators in modular spaces by power series method. Positivity (2017). https://doi.org/10.1007/s11117-017-0467-z
Yurdakadim, T.: Some Korovkin type results via power series method in modular spaces. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 65(2), 65–76 (2016)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Rosihan M. Ali.
Rights and permissions
About this article
Cite this article
Yavuz, E., Talo, Ö. Approximation of Continuous Periodic Functions of Two Variables via Power Series Methods of Summability. Bull. Malays. Math. Sci. Soc. 42, 1709–1717 (2019). https://doi.org/10.1007/s40840-017-0577-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40840-017-0577-6