Approximation of Generalized Bernstein Operators (Analysis in Theory and Applications, 2014)
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$ω_{φ^{λ}}^{2}(f,t)$, $0≤λ≤1$. We obtain main results as follows$$ ω_{φ^{λ}}^{2}(f,t)=O(t^{α})\Longleftrightarrow|C_{n}(f,s_{n},x)-f(x)|=\mathcal{O}\big((n^{-\frac{1}{2}}δ_{n}^{1-λ}(x))^{α}\big),$$where $δ_{n}^{2}(x)=\max\{φ^{2}(x),{1}/{n}\}$ and $0