Korovkin-type Theorems for Abstract Modular Convergence

KOROVKIN THEOREM VIA STATISTICAL e-MODULAR CONVERGENCE OF DOUBLE SEQUENCES

“…Many researchers studied some versions of Korovkin type theorem by using different type of convergence methods after Bardaro and Mantellini's work [5] on modular spaces and they get interesting results [6][7][8][9][10][11]. In this paper, we study generalized version of the Korovkin type approximation theorem for the operators , i j T , , i j   , are acting on an abstract modular function space via statistical e-modular convergence.…”

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.…”

“…Remark 19. (a) Let F be an ultrafilter of N containing F cofin , let ] be as in Remark 16(b) and let , , ∈ N, be as in (22) and (21), respectively. It is not difficult to check that the sequence ( ) is weakly (F)-]-backward exhaustive but not weakly (F)-]-forward exhaustive at 0 and that is ]-lower semicontinuous but not ]-upper semicontinuous at 0.…”

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

“…In [17] it is dealt with Korovkin-type theorems for operators acting on modular function spaces, with respect to abstract convergences satisfying suitable axioms (see e.g., [10]).…”