Anal. Theory Appl., 30 (2014), pp. 205-213.
Published online: 2014-06
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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} }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 $0<\alpha<2$.