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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 (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.

41A36Approximation by positive operators
33D45Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
41A25Rate of convergence, degree of approximation