Weighted Polynomial Approximation for Convex External Fields

It is proven that if Q is convex and w(x)= exp(-Q(x)) is the corresponding weight, then every continuous function that vanishes outside the support of the extremal measure associated with w can be uniformly approximated by weighted polynomials of the form wnPn . This solves a problem of P. Borwein and E. B. Saff. Actually, a similar result is true locally for any parts of the extremal support where Q is convex.