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2015 On Filter Convergence of Series
Alexander Leonov, Cihan Orhan
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Real Anal. Exchange 40(2): 459-474 (2015).

Abstract

A series $\sum x_k$ is $\mathcal{F}$-convergent to $s$ if the sequence $(\sum_{k=1}^n x_k)$ of its partial sums is $\mathcal{F}$-convergent to $s$. We describe filters $\mathcal{F}$ for which $\mathcal{F}$-convergence of a series $\sum x_k$ implies $\mathcal{F}$-convergence to $0$ of the series terms $x_k$. If $(x_k)$ is small enough with respect to a given filter $\mathcal{F}$, then there is an $\mathcal{F}$-subseries $\sum_{k\in I} x_k$ which is absolutely convergent in the usual sense. Filters corresponding to summable ideals, Erdős-Ulam ideals, matrix summability ideals, lacunary ideals and Louveau-Veličković ideals are considered.

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Alexander Leonov. Cihan Orhan. "On Filter Convergence of Series." Real Anal. Exchange 40 (2) 459 - 474, 2015.

Information

Published: 2015
First available in Project Euclid: 4 April 2017

zbMATH: 06848849
MathSciNet: MR3499778

Subjects:
Primary: 40A05 , 54A20
Secondary: 26A05

Keywords: $\mathcal F$-convergence , $\mathcal I$-convergence , filter‎ , ideal , series , statistical convergence

Rights: Copyright © 2015 Michigan State University Press

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Vol.40 • No. 2 • 2015
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