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Zbl 1065.41041
Duman, Oktay
Statistical approximation for periodic functions. (English)
[J] Demonstr. Math. 36, No. 4, 873-878 (2003). ISSN 0420-1213

Let $A:=(a_{nk})$, $n,k=1,2,\dots$, be a nonnegative regular summability matrix. The sequence $x:=\{x_k\}$ is called $A$-statistically convergent to $L$, notation $st_A-\lim x_n=L$, if for every $\epsilon>0$, $$ \lim_{n\to\infty}\sum_{\vert x_k-L\vert \ge\epsilon}a_{nk}=0. $$ Let $C^*$ denote the space of $2\pi$-periodic functions on the real line with the usual sup-norm, and let $\{L_n\}$ be a sequence of positive linear operators mapping $C^*$ into itself. The author proves the following Korovkin type theorem \par Theorem: Let $A$ and $\{L_n\}$ be as above. Then necessary and sufficient condition in order that $$ st_A-\lim\Vert L_n(f,\cdot)-f\Vert =0,\tag1 $$ is that (1) holds for the three functions, 1, $\sin x$, and $\cos x$. \par An application is given describing sufficient conditions on a matrix of coefficients $\{\rho^{(n)}_k\}$, $1\le k\le n=1,2\dots$, ensuring that the operators $$ T_n(f,x):=\frac{a_0}2+\sum_{k=1}^n\rho^{(n)}_k(a_k\cos{kx}+b_k\sin{kx}), $$ where the $a_k$'s and $b_k$'s are the Fourier coefficients of $f$, converge to $f$ in the sup-norm.
[Dany Leviatan (Tel Aviv)]

MSC 2000:: *41A36 Approximation by positive operators
41A10 Approximation by polynomials

Keywords: Korovkin theorem; statistical approximation

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Zentralblatt MATH Berlin [Germany]