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A pair of compatible variations for Bernstein triangular polynomials

  • Chang Geng zhe
  • Feng Yuyu
Article
  • 4 Downloads

Abstract

In papers [4] and [6], the Variation Diminition Property of Bernstein polynomials defined on a triangle was studied. It is well known that the sequence of Bezier nets\(\widehat{f_n }\) obtained by elevation formula convergence to the polynomial itself when n goes to infinite. But the Sequence of variation of Bezier nets does not convergence to the variation of the polynomial. This means that variations are not compatible. In this paper, we give a new definition of variation of Bezier net and prove that the inequality of variation diminition holds and the variations are compatible.

Keywords

Variational Inequality Simple Proof Bernstein Polynomial Degree Elevation Real Linear Space 
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References

  1. [1]
    Chang, G. Z., Bernstein Polynomials via the Shifting Operator, The Amer. Math. Monthly, 91 (1984), 634–638.MATHCrossRefGoogle Scholar
  2. [2]
    Chang, G. Z., & Davis, P. J., The Convexity of Bernstein Polynomials over Triangles, J. Approx. Theory, 40 (1984), 11–28.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    Chang, G. Z. & Feng, Y. Y.; A New Proof for the Convexity of the Bernstein-Bézier surfaces over Triangles, Chinese Annals of Math., 6B (1985), 141–146.MathSciNetGoogle Scholar
  4. [4]
    Chang. G. Z. & Hoschek, J., Convexity and Variation Diminishing Property of Bernstein Polynomials over Trianglis, (to appear in “Multivariate Approximation Theory III”).Google Scholar
  5. [5]
    Farin, G. E., Subsplines über Dreiecken, Diss. TU Braunschweig (1979).Google Scholar
  6. [6]
    Goodman, T. N. T., Variation Diminishing properties of Bernstein Polymonials on Triangles, (to appear in J. Approx. Theory).Google Scholar

Copyright information

© Springer 1989

Authors and Affiliations

  • Chang Geng zhe
    • 1
  • Feng Yuyu
    • 1
  1. 1.University of Science and Technology of ChinaChina

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