Approximation of Generalized Bernstein Operators

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 $0<\alpha<2$.