## 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

### MSC:

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

### Keywords:

strongly almost convergent sequences; Banach limit