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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 598963, 6 pages
http://dx.doi.org/10.1155/2013/598963
Research Article

Korovkin Second Theorem via -Statistical -Summability

1Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India
2Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia, 43400 Serdang, Selangor, Malaysia

Received 21 September 2012; Accepted 3 January 2013

Academic Editor: Feyzi Başar

Copyright © 2013 M. Mursaleen and A. Kiliçman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Korovkin type approximation theorems are useful tools to check whether a given sequence of positive linear operators on of all continuous functions on the real interval is an approximation process. That is, these theorems exhibit a variety of test functions which assure that the approximation property holds on the whole space if it holds for them. Such a property was discovered by Korovkin in 1953 for the functions 1, , and in the space as well as for the functions 1, cos, and sin in the space of all continuous 2 -periodic functions on the real line. In this paper, we use the notion of -statistical -summability to prove the Korovkin second approximation theorem. We also study the rate of -statistical -summability of a sequence of positive linear operators defined from into .