On the Sets of Convergence for Sequences of the q -Bernstein Polynomials with q > 1

Abstract

The aim of this paper is to present new results related to the convergence of the sequence of the $q$-Bernstein polynomials $\{B_{n,q}(f;x)\}$ in the case $q>1$, where $f$ is a continuous function on $[\mathrm{0,1}]$. It is shown that the polynomials converge to $f$ uniformly on the time scale ${𝕁}_q=\{q^{-j}{\}}_{j=0}^{\infty }\cup \{0\}$, and that this result is sharp in the sense that the sequence $\{B_{n,q}(f;x){\}}_{n=1}^{\infty }$ may be divergent for all $x\in R\setminus {𝕁}_q$. Further, the impossibility of the uniform approximation for the Weierstrass-type functions is established. Throughout the paper, the results are illustrated by numerical examples.