A new class of modified Bernstein operators (Q1960905)

The Bernstein-Stancu operator \(B_{m,p}^{\langle\alpha ,\beta ,\gamma \rangle}f(x)\) [\textit{H. H. Gonska} and \textit{J. Meier}, Calcolo 21, 317-335 (1984; Zbl 0568.41021)] is used (for \(\beta = \gamma =p=0)\) to construct a new modified Bernstein operator \({}_{\alpha }B_n\) as the Maclaurin series truncated at degree \(\alpha \) of \(B_{m,0}^{\langle s,0,0\rangle}f(x)\) regarded as a function of \(s\) at the point \(s=-1/n.\) The main result is as follows. 1) For all \(\alpha ,p,q,r \in {\mathbb N}_0,\) there exists a constant \(M\) such that for all \(n\in {\mathbb N}\) and for all \(f\in C^r[0,1]\) \[ |(x-x^2)^p\bigl( {}_{\alpha }B_nf\bigr)^{(q+r)}(x)|\leq Mn^{q-\min \{p,[q/2]\}}\|f^{(r)}\|_{C[0,1]}; \] 2) For all \(\alpha ,\beta ,\gamma \in {\mathbb N}_0\) \((\beta \leq \alpha)\) and for all \(f\in C^{2\beta +\gamma }[0,1]\) \[ \|\bigl({}_{\alpha }B_n f\bigr)^{(\gamma)}-f{(\gamma)}\|=o(n^{-\beta }),n\to \infty; \] Moreover the explicit representations for the operator \({}_{\alpha }B_n\) and for the remainder \[ \lim_{n\to \infty }n^{\alpha +1}\bigl(({}_{\alpha }B_nf)^{(\gamma)}-f^{(\gamma)}\bigr) \] are given, and applications to numerical quadrature are discussed.