Pointwise approximation for linear combinations of Bernstein operators
Let \(B_{n,r}\) be the combination of Bernstein operators on \(C([0,1])\) associated to the integers \(n\) and \(r\) (defined by Ditzian in 1979), the authors prove mainly that for \(f\in C([0,1])\), \(r\in\mathbb{N}\), \(0<\alpha<2r\), \(1-1/r\leq \lambda\leq 1\), \(\varphi(x)= (x(x-1))^{1/2}\), we have the equivalence of (i) and (ii): (i) \(B_{n,r}(f,x) -f(x)=O((n^{-1/2} \varphi^{1-\lambda} (x))^\alpha)\), (ii) \(\omega^{2r}_{\varphi^\lambda} (f,t)=O(t^\alpha)\). \(\omega^{2r}_{\varphi^\lambda} (f,t)\) denotes the Ditzian-Totik modulus of smoothness (see \textit{Z. Ditzian} and \textit{V. Totik}, ``Moduli of smoothness'' (1987; Zbl 0666.41001).
scientific article; zbMATH DE number 1552978