Volume 37, pp. 105-112, 2010.

On the approximation of analytic functions by the $q$-Bernstein polynomials in the case $q>1$

Sofiya Ostrovska

Abstract

Since for $q>1,$ the $q$-Bernstein polynomials $B_{n,q}$ are not positive linear operators on $C[0,1],$ the investigation of their convergence properties turns out to be much more difficult than that in the case $0<q<1.$ In this paper, new results on the approximation of continuous functions by the $q$-Bernstein polynomials in the case $q>1$ are presented. It is shown that if $f\in C[0,1]$ and admits an analytic continuation $f(z)$ into $\{z:|z|<a\},$ then $B_{n,q}(f;z)\to f(z)$ as $n\to \mathrm{\infty },$ uniformly on any compact set in $\{z:|z|<a\}.$

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Key words

$q$-integers, $q$-binomial coefficients, $q$-Bernstein polynomials, uniform convergence

AMS subject classifications

41A10, 30E10

Links to the cited ETNA articles

[7]	Vol. 25 (2006), pp. 431-438 Uri Itai: On the eigenstructure of the Bernstein kernel