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On generalizations of certain summability methods using ideals. (English) Zbl 1223.40004

The authors intend to unify statistical convergence and lacunary statistical convergence, and use ideals to introduce the concept of \(I\)-statistical convergence and \(I\)-lacunary statistical convergence, which naturally extend the notions of statistical convergence and lacunary statistical convergence. They try to establish the relation between these two summability notions.
Their definitions are given in the following:
Definition 1. A sequence \(x=\{x_n\}_{n\in\mathbb{N}}\) is said to be \(I\)-statistical convergent to \(L\) or \(S(I)\)-convergent to \(L\) if, for each \(\epsilon>0\) and \(\delta>0\), \[ \left\{n\in \mathbb{N}:\frac{1}{n}|\{k\leq n:|x_k-L|\geq\epsilon\}|\geq\delta\right\}\in I. \] Definition 2. Let \(\theta\) be lacunary sequence. A sequence \(x=\{x_n\}_{n\in\mathbb{N}}\) is said to be \(I\)-lacunary statistical convergent to \(L\) or \(S_\theta(I)\)-convergent to \(L\) if, for any \(\epsilon>0\) and \(\delta>0\), \[ \left\{r\in \mathbb{N}:\frac{1}{h_r}|\{k\in I_r:|x_k-L|\geq\epsilon\}|\geq\delta\right\}\in I, \] where a lacunary sequence is an increasing integer sequence \(\theta=\{k_r\}_{r\in\mathbb{N}\cup\{0\}}\) such that \(k_0=0\) and \(h_r=k_r-k_{r-1}\to\infty,\) as \(r\to\infty\) and \(I_r=(k_{r-1}, k_r]\).

MSC:

40G15 Summability methods using statistical convergence
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References:

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