## Abstract and Applied Analysis

### Korovkin Second Theorem via $B$-Statistical $A$-Summability

#### Abstract

Korovkin type approximation theorems are useful tools to check whether a given sequence ${\left({L}_{n}\right)}_{n\ge 1}$ of positive linear operators on $C\left[\mathrm{0,1}\right]$ of all continuous functions on the real interval $\left[\mathrm{0,1}\right]$ 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, $x$, and ${x}^{2}$ in the space $C\left[\mathrm{0,1}\right]$ as well as for the functions 1, cos, and sin in the space of all continuous 2$\pi$-periodic functions on the real line. In this paper, we use the notion of $B$-statistical $A$-summability to prove the Korovkin second approximation theorem. We also study the rate of $B$-statistical $A$-summability of a sequence of positive linear operators defined from ${C}_{2\pi }\left(ℝ\right)$ into ${C}_{2\pi }\left(ℝ\right)$.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 598963, 6 pages.

Dates
First available in Project Euclid: 18 April 2013

https://projecteuclid.org/euclid.aaa/1366306746

Digital Object Identifier
doi:10.1155/2013/598963

Mathematical Reviews number (MathSciNet)
MR3034945

Zentralblatt MATH identifier
06161356

#### Citation

Mursaleen, M.; Kiliçman, A. Korovkin Second Theorem via $B$ -Statistical $A$ -Summability. Abstr. Appl. Anal. 2013 (2013), Article ID 598963, 6 pages. doi:10.1155/2013/598963. https://projecteuclid.org/euclid.aaa/1366306746

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