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\(R\)-type summability methods, Cauchy criteria, \(P\)-sets and statistical convergence. (English) Zbl 0765.40002

From the author’s abstract: “A summability method \(S\) is called an \(R\)- type summability method if \(S\) is regular and \(xy\) is strongly \(S\)- summable to 0 whenever \(x\) is strongly \(S\)-summable to 0 and \(y\) is a bounded sequence. Associated with each \(R\)-type summability method \(S\) are the following two methods: convergence in \(\mu\)-density and \(\mu\)- statistical convergence where \(\mu\) is a measure generated by \(S\). In this note we extend the notion of statistically Cauchy to \(\mu\)-Cauchy and show that a sequence is \(\mu\)-Cauchy if and only if it is \(\mu\)- statistically convergent. Let \(W(A)=\overline A^{\beta\mathbb{N}}\cap\beta\mathbb{N}\backslash\mathbb{N}\) for \(A\subset\mathbb{N}\) and \({\mathcal K}=\cap\{W(A):A\subseteq\mathbb{N}\), \(X_ A\) is strongly \(S\)-summable to 1}. Then \(\mu\)-Cauchy is equivalent to convergence in \(\mu\)-density if and only if every \(G_ \delta\) that contains \({\mathcal K}\) in \(\beta\mathbb{N}\backslash\mathbb{N}\) is a neighborhood of \({\mathcal K}\) in \(\beta\mathbb{N}\backslash\mathbb{N}\). As an application, we show that the bounded strong summability field of a nonnegative regular matrix admits a Cauchy criterion”.

MSC:

40D20 Summability and bounded fields of methods
40D25 Inclusion and equivalence theorems in summability theory
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