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$q$-Bernstein polynomials and their iterates. (English) Zbl 1093.41013
$q$-Bernstein polynomials have been introduced by 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 $\left(0,1\right)$, 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, it is essentially different.

##### MSC:
 41A36 Approximation by positive operators 33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.) 41A25 Rate of convergence, degree of approximation