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 - <p style="text-align: justify;">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 &nbsp;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 $0&lt;\alpha&lt;2$.</p>