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On partial derivatives of multivariate Bernstein polynomials


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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  1. 1.

    U. Abel and M. Ivan, “Asymptotic expansion of the multivariate Bernstein polynomials on a simplex,” Approx. Theory Appl. N.S. 16 (3), 85 (2000).

    MathSciNet  MATH  Google Scholar 

  2. 2.

    S. N. Bernstein, “Démonstration du théorème deWeierstrass fondée sur le calcul des probabilités,” Communications de la Sociétémathématique de Kharkow. 2-éme série 13 (1), 1 (1912).

    Google Scholar 

  3. 3.

    S. N. Bernstein, “Complement à l’article de E. Voronovskaja “Determination de la forme asymptotique de l’approximation des fonctions par les polynômes de S. Bernstein,”” Dokl. Akad. Nauk SSSR, Ser. A 4, 86 (1932) [in Russian].

    MATH  Google Scholar 

  4. 4.

    I. N. Chlodovsky, “On some properties of S.N. Bernstein polynomials,” Proc. 1st All-Union congress of mathematics, Kharkov, 1930 (ONTI NKTP SSSR, Moscow–Leningrad, 1936), p. 22 [in Russian].

    Google Scholar 

  5. 5.

    M. S. Floater, “On the convergence of derivatives of Bernstein approximation,” J. Approx. Theory 134 (1), 130 (2005).

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    N. V. Krylov, Introduction to the Theory of Random Processes (AMS, Providence, Rhode Island, 2002).

    Book  MATH  Google Scholar 

  7. 7.

    G. G. Lorentz, “Zur Theorie der Polynome von S. Bernstein,” Mathematical Collections 2(44) (3), 543 (1937).

    MATH  Google Scholar 

  8. 8.

    G. G. Lorentz, Bernstein Polynomials, 2nd ed. (AMS Chelsea Publishing Co., New York, 1986).

    MATH  Google Scholar 

  9. 9.

    O. T. Pop, “Voronovskaja-type theorem for certain GBS operators,” Glas. Mat. Ser III, 43(63) (1), 179 (2008).

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    T. Popoviciu, “Sur l’approximation des fonctions convexes d’ordre supérieur,” Mathematica 10, 49 (1935).

    MATH  Google Scholar 

  11. 11.

    A. N. Shiryaev, Probability, 2nd ed. (Springer, New York, 1995).

    MATH  Google Scholar 

  12. 12.

    S. A. Telyakovskii, “On the rate of approximation of functions by the Bernstein polynomials,” Proc. Inst. Math. Mech. 264, suppl. 1., 177 (2009) [in Russian].

    MathSciNet  MATH  Google Scholar 

  13. 13.

    I. V. Tikhonov and V. B. Sherstyukov, “The module function approximation by Bernstein polynomials,” Bulletin of Chelyabinsk State University 26 (15), 6 (2012) [in Russian].

    MathSciNet  Google Scholar 

  14. 14.

    A. Yu. Veretennikov and E. V. Veretennnikova, “On convergence of partial derivatives of multivariate Bernstein polynomials,” Mathematics, Informatics and Physics in Science and Education, Collection of scientific papers for 140th anniversary of Moscow State Ped. Uni. (“Prometei”, Moscow, 2012), p. 39 [in Russian].

    Google Scholar 

  15. 15.

    E. V. Voronovskaya, “Determination of the asymptotic form of approximation of functions by the Bernstein polynomials,” Dokl. Akad. Nauk SSSR, Ser. A 4, 74 (1932) [in Russian].

    Google Scholar 

  16. 16.

    V. A. Zorich, Mathematical Analysis II (Springer, Berlin, 2004).

    MATH  Google Scholar 

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Corresponding authors

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

Additional information

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).

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  • multivariate Bernstein polynomial
  • partial derivative
  • convergence