Korovkin type approximation on an infinite interval via generalized matrix summability method using ideal

Abstract

Following the notion of $A^\mathcal{I}$-summability method for
real sequences \cite{espdsd2} we 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 $f\in UC_{*}[0,\infty)$.  In the last section, we extend the Korovkin type approximation theorem for positive linear operators on $UC_{*}\left([0,\infty)\times[0,\infty)\right)$. We then construct an example which shows that our new result is stronger than its classical version.