Abstract Korovkin-type theorems in modular spaces and applications

Abstract

Abstract:We prove some versions of abstract Korovkin-type theorems in modular function spaces, with respect to filter convergence for linear positive operators, by considering several kinds of test functions. We give some results with respect to an axiomatic convergence, including almost convergence. An extension to non positive operators is also studied. Finally, we give some examples and applications to moment and bivariate Kantorovich-type operators, showing that our results are proper extensions of the corresponding classical ones. MSC:40A35, 41A35, 46E30

KOROVKIN THEOREM VIA STATISTICAL e-MODULAR CONVERGENCE OF DOUBLE SEQUENCES

“…In ( [17], [19]) there are some positive answers. Following this approach, we give some positive answers also  for all , i j   .…”

“…"Sevda Yıldız Korovkin theorem via statistical e-modular convergence of double sequences…" [17]). Now, we can give our main theorem of this paper.…”

Abstract Theorems on Exchange of Limits and Preservation of (Semi)continuity of Functions and Measures in the Filter Convergence Setting

“…We begin with giving our axiomatic approach, which deals with abstract convergence with respect to filters, without using necessarily nets. For a literature about these topics, see, for instance, [16,17,19,[21][22][23][24] and their bibliographies. Definition 1.…”

Relative modular uniform approximation by means of the power series method with applications

“…Many other versions of the Korovkin theorem have been studied by several authors. Likewise, Bardaro et al [6] studied some version of abstract Korovkin type theorem on modular spaces with respect to the filter convergence on positive linear operators. Mursaleen and Mohiuddine [20] proved the Korovkin type approximation theorem with respect to almost convergence and statistical convergence by using the test functions {1, e −x , e −2x }.…”