Asymptotics of Differentiated Bernstein Polynomials

Abstract.

It is shown that the m th-order derivative of the n th-order Bernstein polynomial of a function f satisfying a certain Lipschitz condition, can be written for n\rightarrow +∈fty as a singular integral of Gauss—Weierstrass type, m times differentiated (in a certain sense) under the integral sign. The theorem is applied to yield an overdifferentiation formula, involving p times differentiated Bernstein polynomials of functions that are not C ^p .