×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
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 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.