Abstract Korovkin-type theorems in the filter setting with respect to relative uniform convergence

Abstract

We prove a Korovkin-type approximation theorem using abstract relative uniform filter convergence of a net of functions with respect to another fixed filter, a particular case of which is that of all neighborhoods of a point, belonging to the domain of the involved functions. We give some examples, in which we show that our results are strict generalizations of the classical ones.

A survey on recent results in Korovkin’s approximation theory in modular spaces

“…The theory of Korovkin-type approximation in modular spaces enables one to obtain a unifying approach, which includes by a unique method, many previous results on the subject. Also, it is possible to obtain general results using a different notion of convergence, namely, filter-type convergence, including the case of the statistical convergence, relative uniform convergence with respect to scale functions, or an axiomatic abstract convergence, which includes filter convergence, triangular matrix statistical convergence and even almost convergence, which is not generated by any filter (see also [1], [4], [11], [15], [16], [17], [23], [24], [27], [28], [31], [39], [44]).…”

Statistical relative uniform convergence of a double sequence of functions at a point and applications to approximation theory

“…Another interesting type of convergence is the uniform convergence of a sequence of functions at a point [18]. More recently, Demirci et al [19] extended this type of convergence to relative uniform convergence of a sequence of functions at a point where the set of the neighborhoods of the point at which relative uniform convergence is considered (see, e.g., [20,21]).…”