The norm estimates of the <i>q</i>-Bernstein operators for varying <i>q</i> &gt; 1

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Date

2011

Authors

Ozban, Ahmet Yasar
Özban, Ahmet Yaşar

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Pergamon-elsevier Science Ltd

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Mathematics
(2000)
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Abstract

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,B-q in the case q > 1. While for 0 < q <= 1, vertical bar vertical bar B-n,B-q vertical bar vertical bar = 1 for all n is an element of N. in the case q > 1, the norm vertical bar vertical bar B-n,B-q vertical bar vertical bar increases rather rapidly as q -> +infinity. In this study, it is proved that vertical bar vertical bar B-n,B-q vertical bar vertical bar similar to C(n)q(n(n-1)/2), q -> +infinity with C-n = 2/n (1- 1/n)(n-1). Moreover, it is shown that vertical bar vertical bar B-n,B-q vertical bar vertical bar similar to 2q(n(n-1)/2) /ne as n -> infinity, q -> +infinity. The results of the paper are illustrated by numerical examples. (C) 2011 Elsevier Ltd. All rights reserved.

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Keywords

q-integers, q-binomial coefficients, q-Bernstein polynomials, q-Bernstein operator, Operator norm, Newton's method

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6

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Volume

62

Issue

12

Start Page

4758

End Page

4771

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