## 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 $B_{n,q}(f;z)\rightarrow f(z)$ as $n\rightarrow \infty,$ uniformly on any compact set in $\{z:|z| < a\}.$

Full Text (PDF) [104 KB]

### Key words

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

41A10, 30E10

### Links to the cited ETNA articles

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

< Back