## $$q$$-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.

### MSC:

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

### Keywords:

$$q$$-Bernstein polynomials; convergence; iterates
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### References:

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