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A note on strong matrix summability via ideals. (English) Zbl 1251.40002
Ideal convergence is a generalization of statistical convergence, and the concept of statistical convergence is a generalization of the usual notion of convergence that, for real-valued sequences, parallels the usual theory of convergence. For a subset $M$ of the set of positive integers the asymptotic density of $M,$ denoted by $\delta(M)$, is given by $$ \delta(M)=\lim_{n\rightarrow\infty}\frac{1}{n}|\{k\leq n: k\in M\}|, $$ if this limit exists, where $|\{k \leq n: k\in {M}\}|$ denotes the cardinality of the set $\{k \leq n : k \in{M}\}$. A sequence $(x_{n})$ of real numbers is statistically convergent to a real number $\ell$ if $$ \delta(\{n:|x_{n}-\ell|\geq \varepsilon \})=0 $$ for every positive real number $\varepsilon$. An ideal $I$ is a family of subsets of positive integers, which is closed under taking finite unions and subsets of its elements. A sequence $\bold{x}=({x_{n}})$ of real numbers is said to be $I$-convergent to a real number $\ell$ if $ \{n\in \mathbb{N}: |x_n-\ell|\geq \varepsilon \} \in {I}$ for every positive real number $\varepsilon$. In the present paper under review the authors apply the notion of ideals to strong matrix summability, they introduce the notion of strong $A^{I}$-summability with respect to an Orlicz function, and make certain investigations on the classes of sequences arising out of this new summability method.

40A35Ideal and statistical convergence
40G15Summability methods using statistical convergence
40F05Absolute and strong summability
Full Text: DOI
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