Korovkin-Type Theorems for ModularΨ-A-Statistical Convergence

Abstract

We deal with a new type of statistical convergence for double sequences, calledΨ-A-statistical convergence, and we prove a Korovkin-type approximation theorem with respect to this type of convergence in modular spaces. Finally, we give some application to moment-type operators in Orlicz spaces.

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   .…”

Abstract Korovkin Theorems via Relative Modular Convergence for Double Sequences of Linear Operators

“…In this section, we relax the positivity condition of linear operators in the Korovkin theorems. In [1,3,4] there are some positive answers. Following this approach, we give some positive answers for statistical relative modular convergence and prove a Korovkin type approximation theorem.…”

“…Many researchers studied some versions of this theorem in different spaces and Bardaro and Mantellini studied this theorem on modular spaces which is the natural generalization of L p (p > 0), Orlicz, Lorentz, and Köthe spaces ( [5]) and so on( [7,8]). In addition, general versions of the Korovkin theorem were studied in which a various kind of convergence methods is used, particularly statistical convergence methods( [2,3,14,22]). More recently, Demirci and Orhan ([9]) have introduced statistical relative uniform convergence of single sequences by using the notions of the natural density and the relative uniform convergence.…”

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

“…There have been also several studies on Korovkin-type theorems related to convergence associated with summability methods, statistical and filter convergence (see e.g., [5,8,9,29,[32][33][34]49]).…”