Rough \(\mathcal I\)-convergence. (English) Zbl 1309.40002
The authors introduce the notion of rough \(\mathcal{I}\)-convergence, using the concept of \(\mathcal{I}\)-convergence and the concept of rough convergence as follows:
Let \(r\) be a nonnegative real number and \(\mathcal{I}\) be a nontrivial ideal. A sequence \((x_i)\in \mathbb{R}^{n}\) is said to be rough \(\mathcal{I}\)-convergent to \(x\) if, for every \(\epsilon>0\), \[ \{i\in \mathbb{N} : \|x_i-x\|\geq r+\epsilon \}\in \mathcal{I}. \] They define the set of rough \(\mathcal{I}\)-limit points of a sequence and investigate some properties of this set.
Let \(r\) be a nonnegative real number and \(\mathcal{I}\) be a nontrivial ideal. A sequence \((x_i)\in \mathbb{R}^{n}\) is said to be rough \(\mathcal{I}\)-convergent to \(x\) if, for every \(\epsilon>0\), \[ \{i\in \mathbb{N} : \|x_i-x\|\geq r+\epsilon \}\in \mathcal{I}. \] They define the set of rough \(\mathcal{I}\)-limit points of a sequence and investigate some properties of this set.
Reviewer: Umit Totur (Aydin)
MSC:
| 40A35 | Ideal and statistical convergence |
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