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

Authors: Heping Wang and Sofiya Ostrovska
Journal: Math. Comp. 79 (2010), 353-363
MSC (2000): Primary 46E15, 26A12, 47A30; Secondary 26D05, 41A10
Published electronically: July 2, 2009
MathSciNet review: 2552230
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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,\;\;\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.

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Additional Information

Heping Wang
Affiliation: School of Mathematical Sciences, Capital Normal University, Beijing 100048, People’s Republic of China

Sofiya Ostrovska
Affiliation: Atilim University, Department of Mathematics, Incek 06836, Ankara, Turkey

Keywords: $q$-integers, $q$-binomial coefficients, $q$-Bernstein polynomials, $q$-Bernstein operator, operator norm, strong asymptotic order
Received by editor(s): December 12, 2007
Received by editor(s) in revised form: November 7, 2008
Published electronically: July 2, 2009
Additional Notes: The first author was supported by National Natural Science Foundation of China (Project no. 10871132), Beijing Natural Science Foundation (1062004), and by a grant from the Key Programs of Beijing Municipal Education Commission (KZ200810028013).
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.