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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

Citations:

Zbl 0588.40009
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