As it is well known, Bernstein polynomials were introduced by S. Bernstein in 1912 to provide a constructive proof of the Weierstrass approximation theorem, establishing that every continuous function defined on [0, 1] can be uniformly approximated by Bernstein polynomials.

In this work we study Bernstein-type operators that preserves the derivatives in the sense that the operator applied to the derivative of a function can be expressed as the derivative of the operator applied to the original function as was studied in [2], related to the operators defined in [1].

Joint work with: David Lara Velasco, Universidad de Granada (Spain).

References:

[1] M. M. Derriennic. Sur l'approximation de fonctions intégrables sur [0,1] par des polynômes de Bernstein modifiés. J. Approx. Theory 31(4):325--343, 1981.

[2] V. Gupta, A. J. López-Moreno, J. M. Latorre-Palacios. On simultaneous approximation of the Bernstein Durrmeyer operators. Appl. Math. Comput. 213(1):112--120, 2009.

[3] G. G. Lorentz. Bernstein Polynomials, second edition. Chelsea Publishing Co., New York, 1986.