Some remarks on invariant means and almost convergence
It is well known that the almost convergence introduced by \textit{G. G. Lorentz} [A contribution to the theory of divergent sequences, Acta Math., Uppsala 80, 167-190 (1948; Zbl 0031.29501)] can be characterized as follows: A sequence \((x_k)\) almost converges to \(u\) if and only if the double sequence \((y_{nh})\) with \[ y_{nh}= {1\over n}\sum^{n+h-1}_{k=h} x_k\quad (n,h\in \mathbb{N}) \] converges to \(u\) uniformly in \(h\). This means that the columns of the double sequence \((y_{nh})\) converge uniformly to \(u\). The authors show that we can obtain the almost convergence also in the case we replace the previous convergence of \((y_{nh})\) by Pringsheim's concept of convergence of double sequences, and we get the usual convergence if we replace it by Hardy's notion of convergence.
scientific article; zbMATH DE number 1417935