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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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