The ANZIAM Journal

Table of Contents - Volume 42 - Issue01

Research Article

A generalization of the Bernstein polynomials based on the q-integers

George M. Phillipsa1

a1 Mathematical Institute, University of St Andrews, St Andrews, Scotland.

Abstract

This paper is concerned with a generalization of the Bernstein polynomials in which the approximated function is evaluated at points spaced in geometric progression instead of the equal spacing of the original polynomials.

(Received March 16 1998)

References

  • [1] Andrews, G. E., The Theory of Partitions (Addison-Wesley, Reading, Mass., 1976). [Google Scholar]
  • [2] Bernstein, S. N., “Démonstration du théorème de Weierstrass fondée”, Comm. Kharkov Math. Soc. 13 (1912) 1–2. [OpenURL Query Data]  [Google Scholar]
  • [3] Cheney, E. W., Introduction to Approximation Theory (McGraw-Hill, New York, 1966). [Google Scholar]
  • [4] Davis, P. J., Interpolation and Approximation (Dover, New York, 1976). [Google Scholar]
  • [5] Goodman, T. N. T., Oruç, H., and Phillips, G. M., “Convexity and generalized Bernstein polynomials”, Proc. Roy. Soc. Edin. 42 (1999) 179–190. [OpenURL Query Data]  [CrossRef]  [CJO Abstract]  [Google Scholar]
  • [6] Hoschek, J. and Lasser, D., Fundamentals of Computer-Aided Geometric Design (Wellesley, Mass., 1993). [Google Scholar]
  • [7] Koçak, Z. F. and Phillips, G. M., “B-splines with geometric knot spacings”, BIT 34 (1994) 388–399. [OpenURL Query Data]  [CrossRef]  [MathSciNet]  [Google Scholar]
  • [8] Lee, S. L. and Phillips, G. M., “Polynomial interpolation at points of a geometric mesh on a triangle”, Proc. Roy. Soc. Edin. 108A (1988) 75–87. [OpenURL Query Data]  [Google Scholar]
  • [9] Lorentz, G. G., Approximation of Functions (Holt, Rinehart and Winston, New York, 1966). [Google Scholar]
  • [10] Oruç, H. and Phillips, G. M., “A generalization of the Bernstein polynomials”, Proc. Roy. Soc. Edin. 42 (1999) 403–413. [OpenURL Query Data]  [CJO Abstract]  [Google Scholar]
  • [11] Phillips, G. M., “On generalized Bernstein polynomials”, in Numerical Analysis: A. R. Mitchell 75th Birthday Volume, (World Sci. Publishing, River Edge, NJ, 1996) 263–269. [MathSciNet]  [Google Scholar]
  • [12] Phillips, G. M., “Bernstein polynomials based on the qintegers”, 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. [OpenURL Query Data]  [Google Scholar]
  • [13] Phillips, G. M., “A de Casteljau algorithm for generalized Bernstein polynomials”, BIT 37 (1997) 232–236. [OpenURL Query Data]  [CrossRef]  [MathSciNet]  [Google Scholar]
  • [14] Phillips, G. M. and Taylor, P. J., Theory and Applications of Numerical Analysis, second ed. (Academic Press, London, 1996). [Google Scholar]
  • [15] Rivlin, T. J., An Introduction to the Approximation of Functions (Dover, New York, 1981). [Google Scholar]
  • [16] Schoenberg, I. J., “On polynomial interpolation at the points of a geometric progression”, Proc. Roy. Soc. Edin. 90A (1981) 195–207. [OpenURL Query Data]  [Google Scholar]