Iterates of Bernstein operators, via contraction principle
Let \((B_n^m)_{m\in\mathbb N}\) denote the sequence of the iterates of the classical Bernstein operators \(B_n\). The author gives a simple proof of the following theorem, proved earlier, by \textit{R. P. Kelisky} and \textit{T. J. Rivlin} [Pac. J. Math. 21, 511--520 (1967; Zbl 0177.31302)]. Theorem. If \(n\in N^*\) is fixed, then, for all \(f\in C[0,1]\), \(\lim_{m\to\infty} B_n^m(f) (x)= f(0)+[f(1)-f(0)]x\), \(x \in[0,1]\).
scientific article; zbMATH DE number 2081795