On Korovkin-type theorems including exponential test functions on infinite intervals through power series convergence
Dilek Söylemez, Mehmet Ünver
TL;DR
This work extends Korovkin-type approximation to unbounded domains by focusing on exponential test functions and leveraging power-series convergence, with Abel and Borel as special cases. It first establishes a Korovkin-type theorem via power-series convergence for positive linear operators preserving exponential behavior on [0,∞) and then strengthens the framework by introducing an integral summability approach that yields convergence where classical or pure Borel methods fail. The authors derive rate-of-convergence estimates using a modulus of continuity adapted to the exponential test set, providing quantitative insights under both power-series and integral-summability regimes. The results broaden the applicability of Korovkin-type theorems to infinite intervals and robust summability contexts, with potential extensions to other summability methods and weighted function spaces.
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.