Korovkin Second Theorem via B -Statistical A -Summability

Abstract

Korovkin type approximation theorems are useful tools to check whether a given sequence ${(L_n)}_{n\ge 1}$ of positive linear operators on $C[\mathrm{0,1}]$ of all continuous functions on the real interval $[\mathrm{0,1}]$ 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 $C[\mathrm{0,1}]$ 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 $C_{2\pi }(ℝ)$ into $C_{2\pi }(ℝ)$.