Korovkin type approximation theorem on an infinite interval in $A^{\mathcal{I}}$-statistical sense

Abstract

In this paper we consider the notion of AI-statistical convergence forreal sequences and establish a Korovkin type approximation theorem for positive linear operators on $UC*[0,\infty)$, the Banach space of all real valued uniform continuous functions on  [0,\infty)$ with the property that $\displaystyle{\lim_{x\rightarrow \infty}f(x)}$ exists finitely for any $f\in UC_{*}[0,\infty)$. We then construct an example which shows that our new result is stronger than its classical version. In the last section, we extend the Korovkin type approximation theorem for positive linear operators on $UC_{*}\left([0,\infty)\times[0,\infty)\right)$.