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Volume 30, Issue 2
Approximation of Generalized Bernstein Operators

X. R. Yang, C. G. Zhang & Y. D. Ma

Anal. Theory Appl., 30 (2014), pp. 205-213.

Published online: 2014-06

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  • Abstract

This paper is devoted to studying direct and converse approximation theorems of the generalized Bernstein operators $C_{n}(f,s_{n},x)$ via so-called unified modulus$\omega_{\varphi^{\lambda}}^{2}(f,t)$, $0\leq\lambda\leq1$. We obtain  main results as follows$$ \omega_{\varphi^{\lambda}}^{2}(f,t)=O(t^{\alpha})\Longleftrightarrow|C_{n}(f,s_{n},x)-f(x)|=\mathcal{O}\big((n^{-\frac{1}{2}}\delta_{n}^{1-\lambda}(x))^{\alpha}\big),$$where $\delta_{n}^{2}(x)=\max\{\varphi^{2}(x),{1}/{n}\}$ and $0<\alpha<2$.

  • AMS Subject Headings

41A25, 41A27, 41A36

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{ATA-30-205, author = {}, title = {Approximation of Generalized Bernstein Operators}, journal = {Analysis in Theory and Applications}, year = {2014}, volume = {30}, number = {2}, pages = {205--213}, abstract = {

This paper is devoted to studying direct and converse approximation theorems of the generalized Bernstein operators $C_{n}(f,s_{n},x)$ via so-called unified modulus$\omega_{\varphi^{\lambda}}^{2}(f,t)$, $0\leq\lambda\leq1$. We obtain  main results as follows$$ \omega_{\varphi^{\lambda}}^{2}(f,t)=O(t^{\alpha})\Longleftrightarrow|C_{n}(f,s_{n},x)-f(x)|=\mathcal{O}\big((n^{-\frac{1}{2}}\delta_{n}^{1-\lambda}(x))^{\alpha}\big),$$where $\delta_{n}^{2}(x)=\max\{\varphi^{2}(x),{1}/{n}\}$ and $0<\alpha<2$.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.2014.v30.n2.6}, url = {http://global-sci.org/intro/article_detail/ata/4485.html} }
TY - JOUR T1 - Approximation of Generalized Bernstein Operators JO - Analysis in Theory and Applications VL - 2 SP - 205 EP - 213 PY - 2014 DA - 2014/06 SN - 30 DO - http://doi.org/10.4208/ata.2014.v30.n2.6 UR - https://global-sci.org/intro/article_detail/ata/4485.html KW - Bernstein type operator, Ditzian-Totik modulus, direct and converse approximation theorem. AB -

This paper is devoted to studying direct and converse approximation theorems of the generalized Bernstein operators $C_{n}(f,s_{n},x)$ via so-called unified modulus$\omega_{\varphi^{\lambda}}^{2}(f,t)$, $0\leq\lambda\leq1$. We obtain  main results as follows$$ \omega_{\varphi^{\lambda}}^{2}(f,t)=O(t^{\alpha})\Longleftrightarrow|C_{n}(f,s_{n},x)-f(x)|=\mathcal{O}\big((n^{-\frac{1}{2}}\delta_{n}^{1-\lambda}(x))^{\alpha}\big),$$where $\delta_{n}^{2}(x)=\max\{\varphi^{2}(x),{1}/{n}\}$ and $0<\alpha<2$.

X. R. Yang, C. G. Zhang & Y. D. Ma. (1970). Approximation of Generalized Bernstein Operators. Analysis in Theory and Applications. 30 (2). 205-213. doi:10.4208/ata.2014.v30.n2.6
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