The Truncated q -Bernstein Polynomials in the Case q > 1

Abstract

The truncated $q$-Bernstein polynomials $B_{n,m,q}\left(f;x\right)$, $n\in \mathbb{N}$, and $\mathrm{\hspace{0.17em}m}\in {\mathbb{N}}_0$ emerge naturally when the $q$-Bernstein polynomials of functions vanishing in some neighbourhood of 0 are considered. In this paper, the convergence of the truncated $q$-polynomials on $\left[0,1\right]$ is studied. To support the theoretical results, some numerical examples are provided.