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Zbl 1236.41026
Ostrovska, Sofiya; Özban, Ahmet Yaşar
The norm estimates of the $q$-Bernstein operators for varying $q>1$. (English)
[J] Comput. Math. Appl. 62, No. 12, 4758-4771 (2011). ISSN 0898-1221

Summary: The aim of this paper is to present norm estimates in $C[0,1]$ for the $q$-Bernstein basic polynomials and the $q$-Bernstein operators $B_{n,q}$ in the case $q>1$. While for for all $n\in \Bbb N$, in the case $q>1$, the norm $\Vert B_{n,q}\Vert $ increases rather rapidly as $q\to +\infty $. In this study, it is proved that $\Vert B_{n,q}\Vert \sim C_nq^{n(n-1)/2}$, $q\to +\infty $ with $C_n=\frac{2}{n}(1-\frac{1}{n})^{n-1}$. Moreover, it is shown that $\Vert B_{n,q}\Vert \sim \frac{2q^{n(n-1)/2}}{ne}$ as $n\to\infty$, $q\to\infty$. The results of the paper are illustrated by numerical examples.

MSC 2000:: *41A35 Approximation by operators
41A60 Asymptotic problems in approximation
05A30 q-calculus and related topics

Keywords: $q$-integers; $q$-binomial coefficients; $q$-Bernstein polynomials; $q$-Bernstein operator; operator norm; Newton's method

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