\(q\)-Bernstein polynomials and their iterates.
\(q\)-Bernstein polynomials have been introduced by \textit{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.
scientific article; zbMATH DE number 1970539
scientific article