Saturation for Bernstein type rational functions
C. Balázs introduced the Bernstein type rational functions \[ R_ n(f,x)=(1/(1+a_ nx)^ n)\sum^{n}_{k=0}f(k/b_ n)\left( \begin{matrix} n\\ k\end{matrix} \right)(a_ nx)^ k\quad(x\geq 0) \] that can be used for the approximation of \(f\in C[0,\infty)\). For \(a_ n=n^{\beta -1}\), \(b_ n=n^{\beta}\), \(0<\beta<1\) we solve the saturation problem of \(R_ n\) away from the point 0. The saturation classes are quite unlike the saturation classes of other approximation methods for which one usually gets the Lipschitz classes. For general \(a_ n\), \(b_ n\) we characterize the approximation property of \(R_ n\), e.g. \(R_ n(f,x)\to f(x)\) for every \(f\in C[0,\infty)\) uniformly on compact subsets of \([0,\infty)\) if and only if \(\lim_{n\to \infty}b_{n,[na_ nx/(1+a_ nx)]}=x\) for every \(x\geq 0\).
scientific article; zbMATH DE number 3855676