Volume 30, Issue 2
Approximation of Generalized Bernstein Operators

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

Published online: 2014-06

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• Abstract

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

• Keywords

Bernstein type operator Ditzian-Totik modulus direct and converse approximation theorem