A survey of results on the q-Bernstein polynomials

George M. Phillips

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61 Citations (Scopus)

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 is an element of (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. Lupas, 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.

Original languageEnglish
Pages (from-to)277-288
Number of pages12
JournalIMA Journal of Numerical Analysis
Volume30
Issue number1
DOIs
Publication statusPublished - Jan 2010

Keywords

  • Bernstein polynomials
  • q-integers
  • Convexity
  • Total positivity
  • q-Bernstein basis
  • BEZIER CURVES
  • CONVERGENCE
  • APPROXIMATION
  • SATURATION
  • FORMULAS
  • OPERATOR

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