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

Wang, Heping; Ostrovska, Sofiya

doi:10.1090/S0025-5718-09-02273-X

The norm estimates for the $q$-Bernstein operator in the case $q>1$
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by Heping Wang and Sofiya Ostrovska;

Math. Comp. 79 (2010), 353-363

DOI: https://doi.org/10.1090/S0025-5718-09-02273-X

Published electronically: July 2, 2009

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Abstract:

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 $0<q\leq 1,\;\;\|B_{n,q}\|=1$ for all $n\in \mathbb {N},$ in the case $q>1,$ the norm $\|B_{n,q}\|$ increases rather rapidly as $n\rightarrow \infty .$ We prove here that $\|B_{n,q}\|\sim C_{q} q^{n(n-1)/2}/n,\;\;n \rightarrow \infty \;\;\text {with }\;\;C_{q}=2 (q^{-2};q^{-2})_{\infty }/e.$ Such a fast growth of norms provides an explanation for the unpredictable behavior of $q$-Bernstein polynomials $(q>1)$ with respect to convergence.

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