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IMA Journal of Numerical Analysis Advance Access published online on June 23, 2009
IMA Journal of Numerical Analysis, doi:10.1093/imanum/drn088
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© The author 2009. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

A survey of results on the q-Bernstein polynomials

George M. Phillips{dagger}

Mathematical Institute, University of St Andrews, St Andrews, UK

{dagger} Email: gmp{at}st-andrews.ac.uk

Received on 1 May 2008. Revised on 21 December 2008.


  Abstract

It is now nearly a century since S. N. Bernstein introduced his well-known polynomials. This paper is concerned with generalizations of the Bernstein polynomials, mainly with the so called q-Bernstein polynomials. These are due to the author of this paper and are based on the q integers. They reduce to the Bernstein polynomials when we put q = 1 and share the shape-preserving properties of the Bernstein polynomials when q isin (0, 1). This paper also describes another earlier generalization of the Bernstein polynomials, a sequence of rational functions that are also based on the q-integers, proposed by A. Lupa, and two even earlier generalizations due to D. D. Stancu. The present author summarizes various results, due to a number of authors, that are concerned with the q-Bernstein polynomials and with Stancu's two generalizations.

Key Words: Bernstein polynomials; q-integers; Convexity; Total positivity; q-Bernstein basis

Written with gratitude in memory of Ron Mitchell, mentor and friend


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