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Constructive Approximation

, Volume 17, Issue 1, pp 47–57 | Cite as

Asymptotics of Differentiated Bernstein Polynomials

  • C. Impens
  • H. Vernaeve
Article

Abstract.

It is shown that the m th-order derivative of the n th-order Bernstein polynomial of a function f satisfying a certain Lipschitz condition, can be written for n\rightarrow +∈fty as a singular integral of Gauss—Weierstrass type, m times differentiated (in a certain sense) under the integral sign. The theorem is applied to yield an overdifferentiation formula, involving p times differentiated Bernstein polynomials of functions that are not C p .

Key words. Bernstein polynomial, Gauss—Weierstrass singular integral. AMS Classification. Primary 41A10, 41A35; Secondary 60F05. 

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Copyright information

© Springer-Verlag New York Inc. 2000

Authors and Affiliations

  • C. Impens
    • 1
  • H. Vernaeve
    • 1
  1. 1.University of Ghent Department of Pure Mathematics and Computer Algebra Galglaan 2 B-9000 Gent Belgium ci@cage.rug.ac.be hvernaev@cage.rug.ac.beBE

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