Restricting statistical convergence
type: Item
instance of: scholarly article
title: Restricting statistical convergence
zbMATH Open document ID: 1265.40016
author: S. Bhunia
author: Pratulananda Das
author: Sudip Kumar Pal
published in: Acta Mathematica Hungarica
publication date: 2012-02-13T00:00:00Z
review text: The ``ordinary'' concept of lower- and upper-asymptotic density for a (double) sequence is well known, along with the concept of a sequence being statistically convergent. The authors of the paper under review use an extension of the concept in the following direction: Definition 1. Let \(A\subset\mathbb N\) and denote for every pair \(m,n\in\mathbb N\) the cardinality of the set \(A\cap [m,n]\) by \(A(m,n)\). Let \(0<\alpha\leq 1\) be a real number. Then \[ \underline{\hbox{d}}^{\alpha}(A)= \liminf_{n\rightarrow\infty}\,{A(m,n)\over n^{\alpha}},\quad \overline{\hbox{d}}^{\alpha}(A)=\limsup_{n\rightarrow\infty}\,{A(m,n)\over n^{\alpha}}, \] are called lower resp. upper asymptotic density of order \(\alpha\) of the set \(A\). If \(\lim_{n\rightarrow\infty}\,{A(m,n)\over n^{\alpha}}\) exists, then \(d^{\alpha}(A)=\underline{d}^{\alpha}(A)= \overline{d}^{\alpha}(A)\) is said to be the asymptotic density of order \(\alpha\) of the set \(A\). Definition 2. Let \(x=\{x_k\}_{k\in\mathbb N}\in \omega\) (the space of all real sequences) and \(0<\alpha\leq 1\) be given. Then \(x\) is said to be statistically convergent of order \(\alpha\) if there is a real number \(L\) such that \[ \lim_{n\rightarrow\infty}\,{| \{k\leq n\,:\,| x_k-L| \geq\varepsilon\}| \over n^{\alpha}}=0 \] for all \(\varepsilon>0\). We write \(S^{\alpha}\hbox{--}\lim\,x_k=L\), and the set of all sequences that are statistically convergent of order \(\alpha\) will be denoted by \(m_0^{\alpha}\). The authors then prove: A. If \(\{x_k\}\) is statistically convergent of order \(\alpha\) to \(L\;(0<\alpha<1)\), then there exists a set \(K=\{k_n\,:\, k_1<k_2<\cdots<k_n< \cdots\}\subset\mathbb N\), such that \(\overline{\hbox{d}}^{\alpha}(K)=\infty\) and \(\lim_{n\rightarrow \infty}\,x_{k_n}=L\). B. Let \(0<\alpha\leq\beta\leq 1\), then \(m_0^{\alpha}\subseteq m_0^{\beta}\) and the inclusion is strict for at least those \(\alpha,\beta\) for which there exists a \(k\in\mathbb N\) with \(\alpha<{1\over k}<\beta\). …
reviewed by: Marcel G. de Bruin
Mathematics Subject Classification ID: 40A35
Mathematics Subject Classification ID: 40A05
Mathematics Subject Classification ID: 40B05
zbMATH DE Number: 6006174
zbMATH Keywords: natural density of order \(\alpha\)
zbMATH Keywords: statistical convergence of order \(\alpha\)
zbMATH Keywords: double density of order \((\alpha,\beta)\)
zbMATH Keywords: statistical convergence of order \((\alpha,\beta)\)
zbMATH Keywords: convergence
zbMATH Keywords: closed linear subspace
MaRDI profile type: Publication
full work available at URL: s10474-011-0122-2
OpenAlex ID: W2049644638
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DOI: 10.1007/S10474-011-0122-2
description: scientific article; zbMATH DE number 6006174