Multidimensional matrix characterization of equivalent double sequences (Q2915443)

The authors treat absolutely convergent double sequences. Using the notion ``Pringsheim convergence'' (A double sequence \(x=\{x_{k,\ell}\}_{k,\ell\in\mathbb N}\) has Pringsheim limit \(L\) if given \(\varepsilon>0\) there exists \(N\in\mathbb N\) such that \(| x_{k,\ell}-L| <\varepsilon \) for \(k,\ell>N\)), they introduce two notions: (1) ``asymptotically statistical equivalent'' for double sequences and (2) ``asymptotically statistical regular'' for a \(4\)-dimensional summability matrix \(A=\{a_{p,q,k,\ell}\}_{p,q,k,\ell\in\mathbb N}\). One of their results is: Let \(A\) be a nonnegative \(4\)-dimensional matrix mapping bounded Pringsheim limit \(0\) double sequences to bounded Pringsheim limit \(0\) double sequences. If the nonnegative convergent double sequences \(x,\,y\) are asymptotically statistical equivalent, and \(x\) has at most a finite number of columns and/or rows with zero entries, \(y_{k,\ell}\geq\delta\) for some \(\delta>0\), then \(\mu(Ax)=\{(\sup_{k,\ell>m,n}(Ax)_{k,\ell})_{m,n\in\mathbb N}\}\) and \(\mu(Ay)\) are asymptotically statistical equivalent.NEWLINENEWLINEThey discuss other implications and variations, and give a characterization for asymptotically statistical regular matrices.