On almost regular convergence (Q1307320)

\textit{B. E. Rhoades} and the reviewer [Math. Proc. Camb. Philos. Soc. 104, No. 2, 283-294 (1988; Zbl 0675.40004)] introduced the following definition. A double sequence \(x=(x_{ij})\) of complex (or real) numbers is called almost convergent, in symbols: \(x\in ac\), if for any fixed \(m,n=0,1,\dots\) the arithmetric means \[ \sigma^{mn}_{pq}=\frac 1{pq} \sum^{m+p-1}_{i=m} \sum^{n+q-1}_{j=n} x_{ij} \] converge to the same limit as \(p,q\to\infty\), uniformly with respect to \(m\) and \(n\). In addition, the present author considers also the arithmetic means of rows and colums of \(x\) given by \(\sigma^{mn}_{p1}\) and \(\sigma^{mn}_{1q}\), respectively. A sequence \(x\in ac\) is called regularly almost convergent, in symbols: \(x\in rac\), if all rows and columns of \(x\) are almost convergent (as single sequences). Furthermore, a sequence \(x\in rac\) is called almost regularly convergent, in symbols: \(x\in arc\), if all its rows and columns are uniformly almost convergent, that is, there exist limits \(L_n'\) and \(L_m''\) for \(m,n=0,1,\dots\) such that for every \(\varepsilon>0\) there exists \(N=N(\varepsilon)\) such that for all \(m,n=0,1,\dots\) we have \(| \sigma^{mn}_{p1}-L_n'| <\varepsilon\) and \(| \sigma^{mn}_{1q}-L_m''| <\varepsilon\), provided \(p\geq N\) and \(q\geq N\), respectively. Even further subclasses of \(arc\) are introduced. The aim of the present paper is to characterize those matrices which transform sequences from one of these subclasses to bounded and convergent double sequences with limits equal to the generalized limits of the original sequences.