The norm estimates for the $q$-Bernstein operator in the case $q>1$

The $q$-Bernstein basis with $0<q<1$ emerges as an extension of the Bernstein basis corresponding to a stochastic process generalizing Bernoulli trials forming a totally positive system on $[0,1].$ In the case $q>1,$ the behavior of the $q$-Bernstein basic polynomials on $[0,1]$ combines the fast increase in magnitude with sign oscillations. This seriously complicates the study of $q$-Bernstein polynomials in the case of $q>1.$

The aim of this paper is to present norm estimates in $C[0,1]$ for the $q$-Bernstein basic polynomials and the $q$-Bernstein operator $B_{n,q}$ in the case $q>1.$ While for $0<q\leq 1,\;\;\Vert B_{n,q}\Vert=1$ for all $n\in \mathbb{N},$ in the case $q>1,$ the norm $\Vert B_{n,q}\Vert$ increases rather rapidly as $n\rightarrow \infty .$ We prove here that $\Vert B_{n,q}\Vert\sim C_{q} q^{n(n-1)/2}/n,\;\;n \rightarrow \infty \;\;$with $\;\;C_{q}=2 (q^{-2};q^{-2})_{\infty }/e.$ Such a fast growth of norms provides an explanation for the unpredictable behavior of $q$-Bernstein polynomials $(q>1)$ with respect to convergence.