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\(\sigma\)-regular matrices and a \(\sigma\)-core theorem for double sequences. (English) Zbl 1182.40003

Let \(A= [a_{jk}^{mn}]_{j,k=0}^{\infty}\) be a four dimensional infinite matrix of real numbers for all \(m,n= 0,1,\dots\). The sums
\[ y_{mn} = \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}a_{jk}^{mn}x_{jk} \]
are called the \(A\)-transformations of the double sequence \(x\). We say that a sequence \(x\) is \(A\)-summable to the limit \(l\) if the \(A\)-transforms of \(x\) exist for all \( m,n=0,1,\dots\) and are convergent in the sense of Pringsheim, i.e.,
\[ \lim_{p,q} \sum_{j=0}^{p}\sum_{k=0}^{q}a_{jk}^{mn}x_{jk}=y_{mn} \]
and
\[ \lim_{mn}y_{mn}= l. \]
Let us write
\[ L^{*}(x) = \limsup_{p,q} \sup_{s,t}\frac{1}{pq}\sum_{j=0}^{p}\sum_{k=0}^{q}x_{j+s,k+t} \]
and
\[ C_{\sigma}(x) = \limsup_{p,q} \sup_{s,t}\frac{1}{pq}\sum_{j=0}^{p}\sum_{k=0}^{q}x_{\sigma^{j}(s)\sigma^{k}(t)}. \]
In this paper the authors investigate necessary and sufficient conditions for the inequality \[ C_{\sigma}(Ax) \leq L(x) \tag{1} \]
for all \(x\in l_{\infty}^{2}\). The authors also indicate that in the case \(\sigma(i) = i+1,\) the inequality in (1) reduces to \(L^{*}(Ax)\leq L(x)\).

MSC:

40C05 Matrix methods for summability
40B05 Multiple sequences and series
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