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Invariant mean and some core theorems for double sequences. (English) Zbl 1209.40003

A bounded double sequence \(x=(x_{jk})\) of real numbers is said to be \(\sigma-\)convergent to a number \(L\) if \(x\in V_2^\sigma\) with
\[ V_{2}^{\sigma}= \bigg\{x: \lim_{p,q\to \infty}\frac{1}{(p+1)(q+1)}\sum_{j=0}^p\sum_{k=0}^q x_{\sigma^j(s),\sigma^k(t)}=L\text{ uniformly in }s,t;\;L=\sigma-\lim x\bigg\}, \]
where \(\sigma^p(k)\) denotes the \(p\)th iterate of the mapping \(\sigma\) at \(k\), and \(\sigma^p(k)\neq k\) for all integer \(k\geq 0\), \(p\geq1\).
In this paper the authors define and characterize the class \((V_2^{\sigma},V_2^{\sigma})\) and establish a core theorem. They determine a Tauberian condition for core inclusion and core equivalence.
Reviewer: Umit Totur (Aydin)

MSC:

40C05 Matrix methods for summability
40H05 Functional analytic methods in summability
40B05 Multiple sequences and series
40E05 Tauberian theorems
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