Lacunary statistical convergence in topological groups
type: Item
instance of: scholarly article
title: Lacunary statistical convergence in topological groups
zbMATH Open document ID: 0835.43006
published in: Indian Journal of Pure \& Applied Mathematics
publication date: 1995-04-04T00:00:00Z
review text: A sequence \((x(k))\) in a topological group \(X\) is called statistically convergent to an element \(l\) of \(X\) if for each neighbourhood \(U\) of 0 \(\lim_{m \to \infty} m^{-1} |\{k \leq m : x(k) - l \notin U\} |= 0\) where the vertical bars indicate the number of elements in the enclosed set. This concept was first given by \textit{H. Fast} [Colloq. Math. 2, 141-144 (1951; Zbl 0044.336)] and studied by \textit{J. A. Fridy} [Analysis 5, 301-313 (1985; Zbl 0588.40001)], \textit{K. Kolk} [Tartu Ul. Toimetised 928, 41-52 (1991)] and \textit{I. J. Maddox} [Math. Proc. Camb. Philos. Soc. 83, 61-64 (1978; Zbl 0392.40001); 104, 141-145 (1988; Zbl 0674.40008)]. \textit{J. A. Fridy} and \textit{C. Orhan} [Pac. J. Math. 160, 43- 51 (1993; Zbl 0794.60012)] studied lacunary statistical convergence by giving the definition as follows: a sequence \((x(k))\) of real or complex numbers is said to be lacunary statistically convergent to a number \(L\) if for each \(\varepsilon > 0\) \[ \lim_{r \to \infty} (h_r)^{-1} \Bigl |\biggl \{k \in I_r : \bigl |x(k) - L \bigr |\geq \varepsilon \biggr\} \Bigr |= 0. \] The purpose of this paper is to introduce lacunary statistical convergence in metrizable topological groups and to prove some inclusion theorems between the set of all statistical convergent sequences and the set of all lacunary statistically convergent sequences.
Mathematics Subject Classification ID: 43A55
zbMATH DE Number: 740390
zbMATH Keywords: topological group
zbMATH Keywords: statistically convergent
zbMATH Keywords: lacunary statistical convergence
zbMATH Keywords: metrizable topological groups
zbMATH Keywords: statistically convergent sequences
author: Hüseyin Çakallı
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 740390