Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-07T10:25:41.210Z Has data issue: false hasContentIssue false
  • English
  • Français

A generalization of the Bernstein polynomials

Published online by Cambridge University Press:  20 January 2009

Haul Oruç ,
George M. Phillips  and
Philip J. Davis
Haul Oruç
Affiliation:
Mathematical Institute, University of St Andrews, North Haugh St Andrews, Fife KY16 9SS, Scotland
George M. Phillips
Affiliation:
Mathematical Institute, University of St Andrews, North Haugh St Andrews, Fife KY16 9SS, Scotland
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper is concerned with a generalization of the classical Bernstein polynomials where the function is evaluated at intervals which are in geometric progression. It is shown that, when the function is convex, the generalized Bernstein polynomials Bn are monotonic in n, as in the classical case.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 42 , Issue 2 , June 1999 , pp. 403 - 413
Copyright
Copyright © Edinburgh Mathematical Society 1999

References

REFERENCES

1.Andrews, G. E., The Theory of Partitions (Addison-Wesley, Reading, Mass., 1976).Google Scholar
2.Cheney, E. W., Introduction to Approximation Theory (McGraw-Hill, New York, 1966).Google Scholar
3.Davis, P. J., Interpolation and Approximation (Dover, New York, 1976).Google Scholar
4.Goodman, T. N. T., Oruç, H.; and Phillips, G. M., Convexity and generalized Bernstein polynomials, Proc. Edinburgh Math. Soc., submitted.Google Scholar
5.Hoschek, J. and Lasser, D., Fundamentals of Computer-Aided Geometric Design (A. K. Peters, 1993).Google Scholar
6.Lee, S. L. and Phillips, G. M., Polynomial interpolation at points of a geometric mesh on a triangle, Proc. Roy. Soc. Edinburgh 108A (1988), 75–87.CrossRefGoogle Scholar
7.Phillips, G. M., Bernstein polynomials based on the q-integers, in The heritage of P. L. Chebyshev: a Festschrift in honor of the 70th birthday of T. J. Rivlin, Ann. Numer. Math. 4 (1997), 511–518.Google Scholar
8.Phillips, G. M., On generalized Bernstein polynomials, in Numerical Analysis: A. R. Mitchell 75th Birthday Volume (Griffiths, D. F. and Watson, G. A. (eds.), World Scientific, Singapore, 1996), 263–269.CrossRefGoogle Scholar
9.Phillips, G. M., A de Casteljau algorithm for generalized Bernstein polynomials, BIT 36 (1996), 232–236.Google Scholar
10.Rivlin, T. J., An Introduction to the Approximation of Functions (Dover, New York, 1981).Google Scholar
11.Schoenberg, I. J., On polynomial interpolation at the points of a geometric progression, Proc. Roy. Soc. Edinburgh 90A (1981), 195–207.CrossRefGoogle Scholar