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

Rus, Ioan A.

J. Math. Anal. Appl. 292, No. 1, 259-261 (2004).

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 \(f\in C[0,1]\), \(\lim_{m\to\infty} B_n^m(f) (x)= f(0)+[f(1)-f(0)]x\), \(x \in[0,1]\).

Reviewer: S. M. Mazhar (Kuwait)

MSC:

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

Keywords:

Bernstein operators; contraction principle; Picard operator

Citations:

Zbl 0177.31302

References:

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