Weighted αβ-equistatistical Convergence for Double Sequences of Functions of Two Variables

  • Aktuğlu H. Department of Mathematics, EMU University, Gazimağusa, North Cyprus Mersin 10 Turkey.
  • Zaheriani S.Yashar Department of Mathematics, EMU University, Gazimağusa, North Cyprus Mersin 10 Turkey.
Keywords: Double sequences, Statistical convergence, Equistatistical convergence, Rate of convergence, Korovkin type approximation, weighted statistical convergence, Positive Linear Operator

Abstract

Karakaya and T.A. Chiristi extended the definition of statistical convergence to weighted statistical convergence in [37], using the sequence of real numbers , satisfying some conditions. The modification of this topic was fulfilled in some papers such as [21] and [30]. It is well known that if , for all k, the weighted statistical convergence reduces to statistical convergence. Karakaya and Karasia [38] defined weighted -statistical convergence of order , which does not have this property. In this extension for the case , for all k, weighted -statistical convergence of order does not reduce to -statistical convergence. Later Aktuğlu and Halil introduced a modification in [12] to remove this extension problem. In this paper we introduce weighted -equistatistical convergence of order  for  double sequences, by using two real sequences   and , considering the modified extension with improved method, also we use this definition to prove Korovkin type approximation theorem via weighted  -equistatistical convergence of order  and weighted -statistical uniform convergence of order  for bivariate functions on . Some examples of positive linear operators are constructed to show that, our approximation results work, but its uniform case does not work. Furthermore rate of weighted -equistatistical convergence of order  are studied.

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Published
2021-01-05
Section
Articles