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Zbl 1093.41013
Ostrovska, Sofiya
$q$-Bernstein polynomials and their iterates. (English)
[J] J. Approximation Theory 123, No. 2, 232-255 (2003). ISSN 0021-9045

$q$-Bernstein polynomials have been introduced by {\it G. M. Phillips} [in Numerical Analysis: A. R. Mitchell 75th Birthday Volume, World Scientific, Singapore, 263--269 (1996)]. For $q=1$ they reduce to the classical Bernstein polynomials. When $q$ is in $(0,1)$, the corresponding linear operators are positive; several papers deal with this case. When $q>1$, the positivity fails. The author shows that in this case the approximation properties of $q$-Bernstein polynomials may be better than in the case $q<1$ or $q=1$; in particular, for entire functions the rate of convergence is exponential. The iterates of $q$-Bernstein operators are also investigated. For $q>1$, the situation is very similar to the classical case $q=1$; for $0<q<1$, it is essentially different.
[Ion Raşa (Cluj-Napoca)]

MSC 2000:: *41A36 Approximation by positive operators
33D45 Basic hypergeometric functions and integrals in several variables
41A25 Degree of approximation, etc.

Keywords: $q$-Bernstein polynomials; convergence; iterates

Cited in: Zbl pre06099685 Zbl 1098.41006

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