On weighted statistical convergence based on (p,q)-integers and related approximation theorems for functions of two variables

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• 作者：Uğur Kadak ^{ugurkadak@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：The difference operator Δp ; q[m] ; Statistical convergence and statistical summability ; (p ; q)(p ; q)-analogue of Bernstein operator ; Korovkin and Voronovskaja type approximations for functions of two variables
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：15 November 2016
• 年：2016
• 卷：443
• 期：2
• 页码：752-764
• 全文大小：353 K

文摘

In this work, we first define a difference operator ${\mathrm{\Delta }}_{p,q}^{[m]}$ of natural order m   with respect to (p,q)$(p,q)$-integers. We then introduce the concepts of ${\mathrm{\Lambda }}_{p,q}^{[m]}$-statistical convergence, statistical ${\mathrm{\Lambda }}_{p,q}^{[m]}$-summability and strong ${\mathrm{\Lambda }}_{p,q}^{[m]}$-summability of order γ   by the weighted method. Furthermore, based on the definition of statistical ${\mathrm{\Lambda }}_{p,q}^{[m]}$-summability, we prove a Korovkin type approximation theorem for functions of two variables. By using (p,q)$(p,q)$-analogue of Bernstein operator of two variables we give an example which shows that proposed method successfully works. Finally, some Voronovskaja type approximation results are obtained.