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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.

MSC:
40G15 Summability methods using statistical convergence
41A36 Approximation by positive operators
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