Approximation byq-Bernstein Polynomials in the Caseq→1+

Author:

Wu Xuezhi1

Affiliation:

1. Library, Capital Normal University, Beijing 100048, China

Abstract

LetBn,q(f;x),q(0,)be theq-Bernstein polynomials of a functionfC[0,1]. It has been known that, in general, the sequenceBn,qn(f)withqn1+is not an approximating sequence forfC[0,1], in contrast to the standard caseqn1-. In this paper, we give the sufficient and necessary condition under which the sequenceBn,qn(f)approximatesffor anyfC[0,1]in the caseqn>1. Based on this condition, we get that if1<qn<1+ln2/nfor sufficiently largen, thenBn,qn(f)approximatesffor anyfC[0,1]. On the other hand, ifBn,qn(f)can approximateffor anyfC[0,1]in the caseqn>1, then the sequence(qn)satisfieslim¯nn(qn-1)ln2.

Funder

National Natural Science Foundation of China

Publisher

Hindawi Limited

Subject

Applied Mathematics,Analysis

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