## Strongly $$\sigma$$-convergent sequences.(English)Zbl 0716.40006

Let $$\sigma$$ be a mapping of the set of positive integers into itself. Let $$p=(p_ k)$$ be a sequence of positive numbers. Then $$[V_{\sigma}]_ p$$ denotes the set of sequences $$x=(x(n))$$ such that, for some constant L, $$(1/m)\sum^{m}_{k=1}| x(\sigma^ k(n))- L|^{p_ k}\to 0\quad as\quad m\to \infty,$$ uniformly in n. This is a generalisation of strong almost convergence, to which it reduces when $$\sigma$$ is given by $$\sigma (n)=n+1$$. Strong almost convergence has been considered by S. Nanda [ibid. 76, No.4, 236-240 (1984; Zbl 0588.40009)]. The results for the more general methods considered here are similar and can be proved in a similar way.

### MSC:

 40F05 Absolute and strong summability 46B99 Normed linear spaces and Banach spaces; Banach lattices

### Keywords:

strong almost convergence

Zbl 0588.40009