A survey of results on the q-Bernstein polynomials
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- 23 June 2009
- journal article
- research article
- Published by Oxford University Press (OUP) in IMA Journal of Numerical Analysis
- Vol. 30 (1) , 277-288
- https://doi.org/10.1093/imanum/drn088
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 ∈ (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. Lupaş, 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.Keywords
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