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$ R$-type summability methods, Cauchy criteria, $ P$-sets and statistical convergence


Author: Jeff Connor
Journal: Proc. Amer. Math. Soc. 115 (1992), 319-327
MSC: Primary 40D20; Secondary 40D25
DOI: https://doi.org/10.1090/S0002-9939-1992-1095221-7
MathSciNet review: 1095221
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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\left( A \right) = {\overline A ^{\beta \mathbb{N}}} \cap \beta \mathbb{N}\backslash\mathbb{N}$ for $ A \subset \mathbb{N}$ and $ \mathcal{K}{\text{ = }} \cap \left\{ {W\left( A \right):A \subseteq \mathbb{N}... ...{\text{is}}\;{\text{strongly}}\;S - {\text{summable}}\;{\text{to}}\;1} \right\}$. 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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1992-1095221-7
Keywords: Strong summability, statistical convergence, measures, additive property for null sets
Article copyright: © Copyright 1992 American Mathematical Society

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