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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
Zbl 1063.40007
Full Text:
##### References:
 [1] Anastassiou, G.A.; Mursaleen, M.; Mohiuddine, S.A., Some approximation theorems for functions of two variables through almost convergence of double sequences, J. comput. anal. appl., 13, 1, 37-40, (2011) · Zbl 1222.41007 [2] Becker, M., Global approximation theorems for szasz – mirakjan and Baskakov operators in polynomial weight spaces, Indiana univ. math. J., 27, 1, 127-142, (1978) · Zbl 0358.41006 [3] Boyanov, B.D.; Veselinov, V.M., A note on the approximation of functions in an infinite interval by linear positive operators, Bull. math. soc. sci. math. roumanie (N.S.), 14, 62, 9-13, (1970) · Zbl 0226.41004 [4] O. Duman, K. Demirci, S. Karakuş, Statistical approximation for infinite intervals (preprint). [5] Fast, H., Sur la convergence statistique, Colloq. math., 2, 241-244, (1951) · Zbl 0044.33605 [6] Gadjiev, A.D.; Orhan, C., Some approximation theorems via statistical convergence, Rocky mountain J. math., 32, 129-138, (2002) · Zbl 1039.41018 [7] Karakaya, V.; Chishti, T.A., Weighted statistical convergence, Iran. J. sci. technol. trans. A sci., 33, 219-223, (2009) [8] Korovkin, P.P., Linear operators and approximation theory, (1960), Hindustan Publishing Corporation Delhi · Zbl 0094.10201 [9] Mohiuddine, S.A., An application of almost convergence in approximation theorems, Appl. math. lett., 24, 1856-1860, (2011) · Zbl 1252.41022 [10] Moricz, F.; Orhan, C., Tauberian conditions under which statistical convergence follows from statistical summability by weighted means, Studia sci. math. hungar., 41, 391-403, (2004) · Zbl 1063.40007 [11] Mursaleen, M.; Alotaibi, A., Statistical summability and approximation by de la vallée-poussin Mean, Appl. math. lett., 24, 320-324, (2011) · Zbl 1216.40003 [12] Mursaleen, M.; Alotaibi, A., Statistical lacunary summability and a Korovkin type approximation theorem, Ann. univ. ferrara, 57, 2, 373-381, (2011) · Zbl 1263.40002 [13] Srivastava, H.M.; Mursaleen, M.; Khan, Asif, Generalized equi-statistical convergence of positive linear operators and associated approximation theorems, Math. comput. mod., (2011) [14] Steinhaus, H., Quality control by sampling, Colloq. math., 2, 98-108, (1951) · Zbl 0042.38301 [15] Erkuş-Duman, E.; Duman, O., Statistical approximation properties of high order operators constructed with the chan – chyan – srivastava polynomials, Appl. math. comput., 218, 5, 927-1933, (2011) · Zbl 1236.41018 [16] Örkcü, M.; Doğru, O., Weighted statistical approximation by Kantorovich type q-szász – mirakjan operators, Appl. math. comput., 217, 20, 7913-7919, (2011) · Zbl 1232.41029 [17] Radu, C., On statistical approximation of a general class of positive linear operators extended in q-calculus, Appl. math. comput., 215, 6, 2317-2325, (2009) · Zbl 1179.41025
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