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Weighted statistical convergence and its application to Korovkin type approximation theorem. (English) Zbl 1262.40004
A sequence \(x=(x_k)\) is said to be statistically convergent to \(L\), \(L=st-\lim x\), if and only if for all \(\varepsilon>0\) the set \(K_\varepsilon=\{k\in\mathbb N: |x_k-L|\geq\varepsilon\}\) has natural density zero. If \(p=(p_k)\) is a sequence of nonnegative integers, with \(p_0>0\) and \(P_n=\sum^n_0 p_k\to +\infty\) as \(n\to+\infty\), and if \(t_n=P_n^{-1}\sum_0^n p_k x_k\), \(n=0,1,\dots\), then \(x=(x_k)\) is said to be statistically summable to \(L\) by the weighted mean method determined by the sequence \((p_k)\) if and only if \(st-\lim_n t_n=L\). Using here a weighted density function, the authors define a notion of weighted statistical convergence \((S_{\overline N}\)-convergence), a modification of a concept defined by V. Karakaya and T. A. Chishti [“Weighted statistical convergence”, Iran. J. Sci. Technol. Trans. A Sci. 33, 219–223 (2009)], and determine the relationship between this concept and the statistical summability method given by F. Móricz and C. Orhan [Stud. Sci. Math. Hung. 41, No. 4, 391–403 (2004; Zbl 1063.40007)]. Finally, the authors employ this new method so as to establish another approximation theorem of Korovkin type.

40G15 Summability methods using statistical convergence
41A36 Approximation by positive operators
Zbl 1063.40007
Full Text: DOI
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