# 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)
Statistical approximation for periodic functions. (English) Zbl 1065.41041

Let $A:=\left({a}_{nk}\right)$, $n,k=1,2,\cdots$, be a nonnegative regular summability matrix. The sequence $x:=\left\{{x}_{k}\right\}$ is called $A$-statistically convergent to $L$, notation $s{t}_{A}-lim{x}_{n}=L$, if for every $ϵ>0$,

$\underset{n\to \infty }{lim}\sum _{|{x}_{k}-L|\ge ϵ}{a}_{nk}=0·$

Let ${C}^{*}$ denote the space of $2\pi$-periodic functions on the real line with the usual sup-norm, and let $\left\{{L}_{n}\right\}$ be a sequence of positive linear operators mapping ${C}^{*}$ into itself. The author proves the following Korovkin type theorem

Theorem: Let $A$ and $\left\{{L}_{n}\right\}$ be as above. Then necessary and sufficient condition in order that

$s{t}_{A}-lim\parallel {L}_{n}\left(f,·\right)-f\parallel =0,\phantom{\rule{2.em}{0ex}}\left(1\right)$

is that (1) holds for the three functions, 1, $sinx$, and $cosx$.

An application is given describing sufficient conditions on a matrix of coefficients $\left\{{\rho }_{k}^{\left(n\right)}\right\}$, $1\le k\le n=1,2\cdots$, ensuring that the operators

${T}_{n}\left(f,x\right):=\frac{{a}_{0}}{2}+\sum _{k=1}^{n}{\rho }_{k}^{\left(n\right)}\left({a}_{k}coskx+{b}_{k}sinkx\right),$

where the ${a}_{k}$’s and ${b}_{k}$’s are the Fourier coefficients of $f$, converge to $f$ in the sup-norm.

##### MSC:
 41A36 Approximation by positive operators 41A10 Approximation by polynomials
##### Keywords:
Korovkin theorem; statistical approximation