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The norm estimates of the $q$-Bernstein operators for varying $q>1$. (English) Zbl 1236.41026
Summary: The aim of this paper is to present norm estimates in $C\left[0,1\right]$ for the $q$-Bernstein basic polynomials and the $q$-Bernstein operators ${B}_{n,q}$ in the case $q>1$. While for for all $n\in ℕ$, in the case $q>1$, the norm $\parallel {B}_{n,q}\parallel$ increases rather rapidly as $q\to +\infty$. In this study, it is proved that $\parallel {B}_{n,q}\parallel \sim {C}_{n}{q}^{n\left(n-1\right)/2}$, $q\to +\infty$ with ${C}_{n}=\frac{2}{n}{\left(1-\frac{1}{n}\right)}^{n-1}$. Moreover, it is shown that $\parallel {B}_{n,q}\parallel \sim \frac{2{q}^{n\left(n-1\right)/2}}{ne}$ as $n\to \infty$, $q\to \infty$. The results of the paper are illustrated by numerical examples.
MSC:
 41A35 Approximation by operators (in particular, by integral operators) 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 05A30 $q$-calculus and related topics
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