Extrapolation Properties of Multivariate Bernstein Polynomials

  • Michele CampitiEmail author
  • Ioan Raşa


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.


Bernstein operators Extrapolation Voronovskaja’s formula Taylor series 

AMS Classification

41A10 41A28 41A36 41A63 



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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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