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Zbl 1136.65018
Cárdenas-Morales, D.; Garrancho, P.; Muñoz-Delgado, F.J.
Shape preserving approximation by Bernstein-type operators which fix polynomials. (English)
[J] Appl. Math. Comput. 182, No. 2, 1615-1622 (2006). ISSN 0096-3003

A family of sequences of linear Bernstein type operators $B_{n,a}: C[0,1]\to C[0,1]$, $a\in [0,+\infty)$, $n = 1,2,\dots,$ is studied. In the case $a = 0$, the sequence was recently introduced by {\it J. P. King} [Acta Math. Hung. 99, No.~3, 203--208 (2003; Zbl 1027.41028)]. If $a\to +\infty$, every $B_{n,a} f$ becomes $B_n f=\sum^n_{v=0} f(\frac vn){n\choose v}x^v(1-x)^{n-v}$, related to the classical Bernstein polynomials. The authors show some shape preserving and convergence properties they possess and make a comparison between the approximations of a function $f\in C[0,1]$ given by $B_{n,a}f$ and $B_nf$.
[Delfina Roux (Milano)]

MSC 2000:: *65D15 Algorithms for functional approximation
41A10 Approximation by polynomials
41A29 Approximation with constraints

Keywords: Bernstein polynomials; shape preserving properties; convexity with respect to a function; convergence

Citations: Zbl 1027.41028

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