Iterates of Bernstein operators, via contraction principle. (English) Zbl 1056.41004

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 R. P. Kelisky and T. J. Rivlin [Pac. J. Math. 21, 511–520 (1967; Zbl 0177.31302)].
Theorem. If \(n\in N^*\) is fixed, then, for all \(f\in C[0,1]\), \(\lim_{m\to\infty} B_n^m(f) (x)= f(0)+[f(1)-f(0)]x\), \(x \in[0,1]\).


41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
41A36 Approximation by positive operators


Zbl 0177.31302
Full Text: DOI


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