Dietrich Braess
"Approximation by Bernstein Polynomials in Learning Theory"

Asymptotics of Bernstein polynomial are of interest when by a limiting process the Bernouilli distribution is replaced by the Poisson distribution. The approximation of the entropy function f(z)=-z*log z is known as a hard problem in this context (private communication by Berens). We import a tool used in learning theory to tackle the approximation problem at the boundary of the interval. Convexity, comparison arguments for Bernstein polynomials, and sufficiently good approximations of the entropy function are the tools for estimates in the interior. Combining the results we get the correct asymptotics for the learning process [also in the multidimensional case not discussed here].