# 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)
Matrix summability of statistically convergent sequences. (English) Zbl 0801.40005
Let $K$ be an index set and ${\varphi }^{K}:=\left({\varphi }_{j}^{K}\right)$ be defined by ${\varphi }_{j}^{K}=1$ $\left(j\in k\right)$, ${\varphi }_{j}^{K}=0$ otherwise. For a non-negative regular $\left(c\to c\right)$ matrix $A$, the $A$- density of $K$ is ${\delta }_{A}\left(K\right):={lim}_{A}{\varphi }^{K}$ whenever ${\varphi }^{K}\in {c}_{A}$. For a sequence $x={\left({x}_{k}\right)}_{k\ge 1}$ and a number ${x}_{0}$, denote ${K}_{\epsilon }:=\left\{k$: $|{x}_{k}-{x}_{0}|\ge \epsilon \right\}$; then $x$ is said to be $A$-statistically convergent to ${x}_{0}$ $\left(x\in {\text{st}}_{A}\right)$ if ${\delta }_{A}\left({K}_{\epsilon }\right)=0$ for every $\epsilon >0$. We also denote the $K$-section of $x$ as ${x}^{\left[K\right]}$, where ${x}_{k}^{\left[K\right]}={x}_{k}$ $\left(k\in K\right)$, ${x}_{k}^{\left[K\right]}=0$ otherwise, and a sequence space $X$ is called section-closed if ${x}^{\left[K\right]}\in x$ for all $x\in X$ and for every index set $K$. The $K$-column-section of a matrix $B$ is denoted by ${B}^{\left[K\right]}:=\left({b}_{nk}^{\left[K\right]}\right)$, where ${b}_{nk}^{\left[K\right]}={b}_{nk}$ $\left(k\in K\right)$, ${b}_{nk}^{\left[K\right]}=0$ otherwise. For any two sequence spaces $W$, $Y$, we denote by $\left(W,Y\right)$ the set of all matrices $H$ which map $W$ into $Y$ $\left(W\subseteq {Y}_{H}\right)$. The main Theorem 4.1 shows: Let $X$ be a section- closed sequence space containing $e:=\left(1,1,\cdots \right)$ and let $Y$ be an arbitrary sequence space; then $B\in \left(X\cap {\text{st}}_{A},Y\right)$ if and only if $B\in \left(X\cap c,Y\right)$ and ${B}^{\left[K\right]}\in \left(X,Y\right)$ $\left({\delta }_{A}\left(K\right)=0\right)$. There are some modifications and corollaries, and a final section which considers matrix maps of statistically convergent bounded sequences (where we take $X={\ell }_{\infty }$).

##### MSC:
 40C05 Matrix methods in summability 40D25 Inclusion theorems; equivalence theorems 40J05 Summability in abstract structures 46A45 Sequence spaces 40A05 Convergence and divergence of series and sequences