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

##### MSC:
 41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities) 41A36 Approximation by positive operators
##### Keywords:
Bernstein operators; contraction principle; Picard operator
Zbl 0177.31302
Full Text:
##### References:
 [1] Adell, J.A.; Badia, F.G.; de la Cal, J., On the iterates on some Bernstein-type operators, J. math. anal. appl., 209, 529-541, (1997) · Zbl 0872.41009 [2] Agratini, O., Aproximare prin operatori liniari, (2000), Presa Universitară Clujeană Cluj-Napoca [3] Karlin, S.; Ziegler, Z., Iteration of positive approximation operators, J. approx. theory, 3, 310-339, (1970) · Zbl 0199.44702 [4] Kelisky, R.P.; Rivlin, T.J., Iterates of Bernstein polynomials, Pacific J. math., 21, 511-520, (1967) · Zbl 0177.31302 [5] Rus, I.A., Generalized contractions and applications, (2001), Cluj Univ. Press Cluj-Napoca · Zbl 0968.54029 [6] Zeidler, E., Nonlinear functional analysis and its applications, I, (1993), Springer-Verlag Berlin
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