\(q\)-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 \((0,1)\), 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.


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


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