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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.

40C05Matrix methods in summability
40H05Functional analytic methods in summability
40B05Multiple sequences and series
40E05Tauberian theorems, general