On the q -Bernstein Polynomials of Unbounded Functions with q > 1

Abstract

The aim of this paper is to present new results related to the $q$-Bernstein polynomials $B_{n,q}(f;x)$ of unbounded functions in the case $q>1$ and to illustrate those results using numerical examples. As a model, the behavior of polynomials $B_{n,q}(f;x)$ is examined both theoretically and numerically in detail for functions on $[0,1]$ satisfying $f(x)~K{x}^{-\alpha }$ as $x\to 0^{+}$, where $\alpha >0$ and $K\ne 0$ are real numbers.