A General Korovkin Result Under Generalized Convergence

Abstract

In this paper, the classic result of Korovkin about the convergence of sequences of functions definedfrom sequences of linear operators is reformulated in terms of generalized convergence. This convergence extendssome others given in the literature. The operator of the sequence fulfill a shape preserving property more general thanthe positivity. This property is related with certain extension of the notion of derivative. This extended derivative isprecisely the object of the approximation process. The study is completed by analysing the conditions for the existenceof an asymptotic formula, from which some interesting consequences are derived as a local version of the shape preserving property. Finally, as applications of the previous results, the author use the following notion of generalizedconvergence, an extension of Nörlund-Cesáro summability given by V. Loku and N. L. Braha in 2017. A way to transfera notion of generalized convergence to approximation theory by means of linear operators is showed.