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Matrix transformations between some new sequence spaces. (English) Zbl 0542.40003
Let $\sigma$ be a mapping of the set of positive integers into itself, let ${V}_{\sigma }$ be the space of bounded sequences all of whose $\sigma$-means are equal, and let $\sigma$-lim x be the common value of all $\sigma$-means on x. In this paper the author generalizes the idea of strong almost convergence of I. J. Maddox [Math. Proc. Camb. Philos. Soc. 83, 61-64 (1978; Zbl 0392.40001)]: a bounded sequence $x=\left({x}_{k}\right)$ is said to be strongly $\sigma$-convergent to a number L if and only if $\left(|{x}_{k}-L|\right)\in {V}_{\sigma }$ with $\sigma$-limit zero. He characterizes those real infinite matrices which map all convergent sequences (all sequences of $\sigma$-bounded variation) into sequences strongly $\sigma$-convergent to zero (strongly $\sigma$- convergent). The concept of sequences of $\sigma$-bounded variation was introduced by the author in an earlier paper [Q. J. Math., Oxf. II. Ser. 34, 77-86 (1983)].
Reviewer: J.Boos

##### MSC:
 40C05 Matrix methods in summability 40F05 Absolute and strong summability 40C99 General summability methods 40D25 Inclusion theorems; equivalence theorems