Strongly almost convergent sequences. (English) Zbl 0588.40009

Let \(\{p_ k\}\) be a sequence of strictly positive real numbers. We write \(\ell_{\infty}(p)=\{x:\) \(\sup_{k}| x_ k|^{p_ k}<\infty \}\) and \([\hat c]=\{x:\) \(\lim_{m\to \infty}\frac{1}{m}\sum^{n+m}_{k=n+1}| x_ k-\ell | =0\) uniformly in \(n\}\). The main aim of the present note is to extend the set \([\hat c]\) to \([\hat c,p],\) where the set \([\hat c,p]\) is defined as \[ [\hat c,p]=\{x:\quad \lim_{m\to \infty}\frac{1}{m}\sum^{n+m}_{k=n+1}| x_ k-\ell |^{p_ k}=0\text{ uniformly in }n\}. \] The author obtains a number of interesting results concerning \([\hat c,p]\). Thus he shows that \([\hat c,p]_{\infty}=\ell_{\infty}(p)\), where \([\hat c,p]_{\infty}=\{x:\) \(\sup_{m,n}\frac{1}{m} \sum^{n+m}_{k=n+1}| x_ k|^{p_ k}<\infty \}\), and if \(0<p_ k\leq q_ k\) such that \(q_ k/p_ k\) is bounded, then \([\hat c,q]\subset [\hat c,p]\).
Reviewer: S.Mazhar


40F05 Absolute and strong summability
40D25 Inclusion and equivalence theorems in summability theory
40G05 Cesàro, Euler, Nörlund and Hausdorff methods