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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

Citations:

Zbl 0177.31302
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Full Text: DOI

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: 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 Univ. Press Cluj-Napoca · Zbl 0968.54029
[6] Zeidler, E., Nonlinear Functional Analysis and Its Applications, I (1993), Springer-Verlag: Springer-Verlag Berlin
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