Abstract
In this paper, we will show a new weighted equi-statistical convergence and based on this definition we will prove a kind of the Korovkin type theorems. Also we will show the rate of the convergence for this kind of weighted statistical convergence and Voronovskaya type theorem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Altomare F.: Korovkin-type theorems and approximation by positive linear operators. Survey Approx. Theory 5, 92–164 (2010)
Balcerzak M., Dems K., Komisarski A.: Statistical convergence and ideal convergence for sequences of functions. J. Math. Anal. Appl. 328, 715–729 (2007)
Braha N.L., Loku V., Srivastava H.M.: \({\Lambda^2}\)-weighted statistical convergence and Korovkin and Voronovskaya type theorems. Appl. Math. Comput. 266, 675–686 (2015)
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 Vallée Poussin mean. Appl. Math. Comput. 228, 162–169 (2014)
Edely O.H.H., Mohiuddine S.A., Noman A.K.: Korovkin type approximation theorems obtained through generalized statistical convergence. Appl. Math. Lett. 23, 1382–1387 (2010)
Fast H.: Sur la convergence statistique. Colloq. Math. 2, 241–244 (1951)
Karakus S., Demirci K., Duman O.: Equi-statistical convergence of positive linear operators. J. Math. Anal. Appl. 339(2), 1065–1072 (2008)
Korovkin, P.P.: Convergence of linear positive operators in the spaces of continuous functions (Russian). Doklady Akad. Nauk. SSSR (N.S.) 90, 961–964 (1953)
Korovkin, P.P.: Linear Operators and Approximation Theory. Hindustan Publ. Co., Delhi (1960)
Loku, V., Braha N.L.: \({\Lambda^2}\)− statistical convergence and its applications to Korovkin second type theorem. Thai J. Math. (2016, to appear)
Moricz F., Orhan C.: Tauberian conditions under which statistical convergence follows from statistical summability by weighted means. Studia Sci. Math. Hung. 41(4), 391–403 (2004)
Mursaleen M., Alotaibi A.: Statistical lacunary summability and a Korovkin type approximation theorem. Ann. Univ. Ferrara 57(2), 373–381 (2011)
Mursaleen M., Karakaya V., Erturk M., Gursoy F.: Weighted statistical convergence and its application to Korovkin type approximation theorem. Appl. Math. Comput. 218, 9132–9137 (2012)
Srivastava H.M., Mursaleen M., Khan A.: Generalized equi-statistical convergence of positive linear operators and associated approximation theorems. Math. Comput. Model. 55, 2040–2051 (2012)
Steinhaus H.: Sur la convergence ordinaire et la convergence asymptotique. Colloq. Math. 2, 73–74 (1951)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Braha, N.L. Some Weighted Equi-Statistical Convergence and Korovkin Type-Theorem. Results. Math. 70, 433–446 (2016). https://doi.org/10.1007/s00025-016-0578-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00025-016-0578-z