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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
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