Invariant mean and some core theorems for double sequences (Q971504)

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.