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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.

MSC:
40A35Ideal and statistical convergence
40G15Summability methods using statistical convergence
40F05Absolute and strong summability
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References:
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