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On partial derivatives of multivariate Bernstein polynomials

Abstract

It is shown that Bernstein polynomials for a multivariate function converge to this function along with partial derivatives provided that the latter derivatives exist and are continuous. This result may be useful in some issues of stochastic calculus.

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Correspondence to A. Yu. Veretennikov or E. V. Veretennikova.

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Original Russian Text ©A. Yu. Veretennikov and E. V. Veretennikova, 2015, published in Matematicheskie Trudy, 2015, Vol. 18, No. 2, pp. 22–38.

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Veretennikov, A.Y., Veretennikova, E.V. On partial derivatives of multivariate Bernstein polynomials. Sib. Adv. Math. 26, 294–305 (2016). https://doi.org/10.3103/S1055134416040039

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Keywords

  • multivariate Bernstein polynomial
  • partial derivative
  • convergence