On Korovkin-type theorems including exponential test functions on infinite intervals through power series convergence

Abstract

Approximation theory has long been concerned with the development of positive linear operators that effectively approximate classes of functions. Among the most well-known results in this area are Korovkin-type approximation theorems, which provide simple and elegant criteria for convergence by testing only on a small set of functions. Motivated by these classical results and their extensions, we focus on versions that preserve exponential functions and incorporate modern summability techniques. In this paper, we establish Korovkin-type theorems that preserve exponential functions by employing power series convergence and a special case thereof. By considering approximation through Borel-type power series convergence via integral summability, we provide an alternative framework that applies in cases where classical convergence or ordinary Borel convergence fails, and we offer a comparative analysis of the corresponding theorems. We also present illustrative examples in which the classical results fail, while the proposed approach remains applicable. In addition, the rate of convergence is analyzed through the modulus of continuity.