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Zbl 1209.40003
Mursaleen, M.; Mohiuddine, S.A.
Invariant mean and some core theorems for double sequences. (English)
[J] Taiwanese J. Math. 14, No. 1, 21-33 (2010). ISSN 1027-5487

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.
[Umit Totur (Aydin)]

MSC 2000:: *40C05 Matrix methods in summability
40H05 Functional analytic methods in summability
40B05 Multiple sequences and series
40E05 Tauberian theorems, general

Keywords: double sequence; $p$-convergence; invariant mean; $\sigma$-convergence; $\sigma$-core; core theorems

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