# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Lacunary statistical summability. (English) Zbl 0786.40004
Let $\theta =\left({k}_{r}\right)$ 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 ${I}_{r}:=\left({k}_{r-1},{k}_{r}\right]$. A complex-valued sequence $x=\left({x}_{k}\right)$ is said to be ${S}_{\theta }$-convergent to $L$ if, for each $\epsilon >0$, we have ${lim}_{r}{h}_{r}^{-1}|\left\{k\in {I}_{r}$: $|{x}_{k}-L|\ge \epsilon \right\}|=0$, where $|\left\{·\right\}|$ denotes the cardinality of the set; we then write ${x}_{k}\to L\left({S}_{\theta }\right)$. Likewise, $x$ is an ${S}_{\theta }$-Cauchy sequence if there is a subsequence $\left({x}_{{k}^{\text{'}}\left(r\right)}\right)$ with ${k}^{\text{'}}\left(r\right)\in {I}_{r}$ for each $r$, ${lim}_{r}{x}_{{k}^{\text{'}}\left(r\right)}=L$, and for each $\epsilon >0$, ${lim}_{r}{h}_{r}^{-1}|\left\{k\in {I}_{r}$: $|{x}_{k}-{x}_{{k}^{\text{'}}\left(r\right)}|\ge \epsilon \right\}|=0$. 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 ${x}_{k}\to L\left({S}_{\theta }\right)$ then ${x}_{k}\to L\left({C}_{1}\right)$; 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 $A=\left({a}_{nk}\right)$ such that (a) ${\sum }_{k=1}^{\infty }{a}_{nk}=1$ for every $n$, and (b) if $K\subseteq ℕ$ with ${lim}_{r}{h}_{r}^{-1}|K\cap {I}_{r}|=0$ then ${lim}_{n}{\sum }_{k\in ℕ}{a}_{nk}=0$. 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 in summability