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 w ⁿ P _n . 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. February 10, 1998. Date revised: July 23, 1998. Date accepted: August 17, 1998.