The norm estimates for the -Bernstein operator in the case

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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The -Bernstein basis with emerges as an extension of the Bernstein basis corresponding to a stochastic process generalizing Bernoulli trials forming a totally positive system on In the case the behavior of the -Bernstein basic polynomials on combines the fast increase in magnitude with sign oscillations. This seriously complicates the study of -Bernstein polynomials in the case of

The aim of this paper is to present norm estimates in for the -Bernstein basic polynomials and the -Bernstein operator in the case While for for all in the case the norm increases rather rapidly as We prove here that with Such a fast growth of norms provides an explanation for the unpredictable behavior of -Bernstein polynomials 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

Email:
wanghp@yahoo.cn

**Sofiya Ostrovska**

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

Email:
ostrovskasofiya@yahoo.com

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

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.