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Extrapolation Properties of Multivariate Bernstein Polynomials

  • Michele CampitiEmail author
  • Ioan Raşa
Article
  • 20 Downloads

Abstract

We consider some connections between the classical sequence of Bernstein polynomials and the Taylor expansion at the point 0 of a \(C^\infty \) function f defined on a convex open subset \(\Omega \subset \mathbb {R}^d\) containing the d-dimensional simplex \(S^d\) of \(\mathbb {R}^d\). Under general assumptions, we obtain that the sequence of Bernstein polynomials converges to the Taylor expansion and hence to the function f together with derivatives of every order not only on \(S^d\) but also on the whole \(\Omega \). This result yields extrapolation properties of the classical Bernstein operators and their derivatives. An extension of the Voronovskaja’s formula is also stated.

Keywords

Bernstein operators Extrapolation Voronovskaja’s formula Taylor series 

AMS Classification

41A10 41A28 41A36 41A63 

Notes

References

  1. 1.
    Altomare, F.: Limit semigroups of Bernstein-Schnabl operators associated with positive projections. Annali Sc. Norm. Sup. Pisa 16(2), 259–279 (1989)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Altomare, F., Campiti, M.: Korovkin-type Approximation Theory and its Applications, De Gruyter Studies in Mathematics, vol. 17. W. De Gruyter, Berlin-New York (1994)CrossRefGoogle Scholar
  3. 3.
    Bustamante, J.: Bernstein Operators and Their Properties. Birkhäuser, Cham (2017).  https://doi.org/10.1007/978-3-319-55402-0 CrossRefzbMATHGoogle Scholar
  4. 4.
    Karlin, S., Ziegler, Z.: Iteration of positive approximation operators. J. Approx. Theory 3, 310–339 (1970)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Nasaireh, F., Raşa, I.: Another look at Voronovskaja type formulas. J. Math. Inequal 12(1), 95–105 (2018).  https://doi.org/10.7153/jmi-2018-12-07 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Natanson, I.P.: Constructive Function Theory Vol. I: Uniform Approximation. Frederick Ungar, New York (1964)zbMATHGoogle Scholar
  7. 7.
    Phillips, G.M.: Interpolation and Approximation by Polynomials, CMS Books in Mathematics. Springer-Verlag, Berlin Heidelberg (2003).  https://doi.org/10.1007/b97417 CrossRefGoogle Scholar
  8. 8.
    Turan, M.: The truncated-Bernstein polynomials in the case \(q>1\). Abstr. Appl. Anal. 2014, 126319 (2014)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics and PhysicsUniversity of SalentoLecceItaly
  2. 2.Department of MathematicsTechnical University of Cluj-NapocaCluj-NapocaRomania

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