# zbMATH — the first resource for mathematics

Lacunary statistical summability. (English) Zbl 0786.40004
Let $$\theta=(k_ r)$$ be an increasing sequence of integers such that $$k_ 0=0$$, $$h_ r:= k_ r- k_{r-1}\to\infty$$ as $$r\to\infty$$, and $$I_ r:=(k_{r-1}, k_ r]$$. A complex-valued sequence $$x=(x_ k)$$ is said to be $$S_ \theta$$-convergent to $$L$$ if, for each $$\varepsilon>0$$, we have $$\lim_ r h_ r^{-1}|\{k\in I_ r$$: $$| x_ k- L|\geq\varepsilon\}| =0$$, where $$|\{\cdot\}|$$ denotes the cardinality of the set; we then write $$x_ k\to L(S_ \theta)$$. Likewise, $$x$$ is an $$S_ \theta$$-Cauchy sequence if there is a subsequence $$(x_{k'(r)})$$ with $$k'(r)\in I_ r$$ for each $$r$$, $$\lim_ r x_{k'(r)}=L$$, and for each $$\varepsilon>0$$, $$\lim_ r h_ r^{-1} |\{k\in I_ r$$: $$| x_ k-x_{k'(r)}| \geq\varepsilon\}|=0$$. It is first shown (Theorem 2) that $$x$$ is $$S_ \theta$$-convergent if and only if $$x$$ is an $$S_ \theta$$-Cauchy sequence. Further (Theorem 4) if $$x$$ is a bounded sequence and $$x_ k\to L(S_ \theta)$$ then $$x_ k\to L(C_ 1)$$; that is, $$l_ \infty\cap S_ \theta\subseteq C_ 1$$. On the other hand (Theorem 6), if $$x$$ is unrestricted, then no matrix summability method can include $$S_ \theta$$. Finally, let $$T_ \theta$$ denote the class of non-negative summability matrices $$A=(a_{nk})$$ such that (a) $$\sum_{k=1}^ \infty a_{nk}=1$$ for every $$n$$, and (b) if $$K\subseteq\mathbb{N}$$ with $$\lim_ r h_ r^{-1} | K\cap I_ r|=0$$ then $$\lim_ n \sum_{k\in\mathbb{N}} a_{nk}=0$$. It is shown (Theorem 9) that $$x\in l_ \infty\cap S_ \theta$$ if and only if $$x$$ is $$A$$-summable for every $$A\in T_ \theta$$.

##### MSC:
 40G99 Special methods of summability 40A05 Convergence and divergence of series and sequences 40D20 Summability and bounded fields of methods 40C05 Matrix methods for summability
Full Text: