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The r-monotonicity of generalized Bernstein polynomials

Part of: Approximations and expansions

Published online by Cambridge University Press:  26 July 2012

Laiyi Zhu  and
Zhiyong Huang
Laiyi Zhu
Affiliation:
School of Information, Renmin University of China, Beijing 100872, People's Republic of China (huangzhiy@ruc.edu.cn)
Zhiyong Huang
Affiliation:
School of Information, Renmin University of China, Beijing 100872, People's Republic of China (huangzhiy@ruc.edu.cn)
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Abstract

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Let f ∊ C[0, 1] and let the Bn(f, q; x) be generalized Bernstein polynomials based on the q-integers that were introduced by Phillips. We prove that if f is r-monotone, then Bn(f, q; x) is r-monotone, generalizing well-known results when q = 1 and the results when r = 1 and r = 2 by Goodman et al. We also prove a sufficient condition for a continuous function to be r-monotone.

Keywords

generalized Bernstein polynomialr-monotonicitynumber of sign changes

MSC classification

Secondary: 41A10: Approximation by polynomials
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 55 , Issue 3 , October 2012 , pp. 797 - 807
Copyright
Copyright © Edinburgh Mathematical Society 2012

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