Generalized equi-statistical convergence of positive linear operators and associated approximation theorems

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• 作者：H.M. Srivastava^a ; ^{harimsri@math.uvic.ca} ; M. Mursaleen^b ; ^{mursaleenm@gmail.com} ; Asif Khan^b ; ^{asifjnu07@gmail.com}
• 关键词：Statistical convergence ; 位-Statistical convergence ; Equi-statistical convergence ; 位-Equi-statistical convergence ; Positive linear operators ; Korovkin type approximation theorems ; Voronovskaja type theorems
• 刊名：Mathematical and Computer Modelling
• 出版年：2012
• 期刊代码：86_08957177
• 类别：cp
• 出版时间：May, 2012
• 卷：55
• 期：9-10
• 页码：2040-2051
• 文件大小：242 K

摘要

The concepts of equi-statistical convergence, statistical pointwise convergence and statistical uniform convergence for sequences of functions were introduced recently by Balcerzak et聽al. [M. Balcerzak, K. Dems, A. Komisarski, Statistical convergence and ideal convergence for sequences of functions, J. Math. Anal. Appl. 328 (2007) 715-729]. In this paper, we use the notion of -statistical convergence in order to generalize these concepts. We establish some inclusion relations between them. We apply our new notion of -equi-statistical convergence to prove a Korovkin type approximation theorem and we show that our theorem is a non-trivial extension of some well-known Korovkin type approximation theorems. Finally, we prove a Voronovskaja type approximation theorem via the concept of -equi-statistical convergence. Some interesting examples are also displayed here in support of our definitions and results.