\(\mathcal {I}_\lambda \) -double Statistically Convergent Sequences in Topological Groups
In this paper, we introduce new notion, namely, \(\mathcal {I}_{\lambda }-\) double statistical convergence in topological groups. We mainly investigate some inclusion relations between \(\mathcal {I}-\) double statistical and \( \mathcal {I}_{\lambda }-\
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ract. In this paper, we introduce new notion, namely, Iλ − double statistical convergence in topological groups. We mainly investigate some inclusion relations between I−double statistical and Iλ − double statistical convergence. Keywords: Ideal convergence · Ideal double statistical convergence · Double statistical convergence · λ-double statistical convergence · Topological groups
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Introduction
Looking through historically to statistical convergence of sequences, we recall that the concept of statistically convergence of sequences was first introduced by Fast [10] as an extension of the usual concept of sequential limits and also independently by Buck [2]. Schoenberg [33] gave some basic properties of the statistical convergence and also studied the concept as a summability method. Over the years and under different names statistical convergence has been discussed in the theory of Fourier analysis, ergodic theory and number theory. Later on it was further investigated from the sequence space point of view and linked with ˇ at [23] and many others. In recent years, summability theory by Fridy [11], Sal´ generalization of statistical convergence have appeared in the study of strong integral summability. Moreover statistical convergence is closely related to the concept of convergence in probability. Most of the existing works on statistical convergence have been restricted to real or complex sequences except the works of Kolk [13], Maddox [17] and Cakalli [3]. Mursaleen [19]introduced λ-statistical convergence as a generalization of statistical convergence. In [13], Kolk extended the statistical convergence to normed spaces and also Maddox [17] extended it to locally convex Hausdorff topological linear spaces giving a representation of the statistical convergence in terms of strongly summability by using a modulus function and Cakalli [3] extended this notion to topological Hausdorff groups. Di Maio and Koˇcinac [18] introduced the concept of statistical convergence in topological spaces and statistical Cauchy condition in uniform spaces and established the topological nature of this convergence. Later on Hazarika and Sava¸s [12] introduced λ−statistical convergence c Springer Nature Singapore Pte Ltd. 2017 D. Giri et al. (Eds.): ICMC 2017, CCIS 655, pp. 349–357, 2017. DOI: 10.1007/978-981-10-4642-1 30
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of double sequences in n-normed spaces and also Sava¸s and Mohiuddine [32] introduced and studied the concepts of double λ-statistically convergent and double λ−statistically Cauchy sequences in probabilistic normed space. Cakalli and Sava¸s [4] studied the statistical convergence of double sequences to topological groups. Quite recently Savas [25] studied Iλ −statistical convergence for sequences in topological groups where more references on this important summability method can be found. In many branches of science and engineering we often come across double sequences, i.e. sequences of matrices and certainly there are situations where either the idea of ordinary convergence does not work or the
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