Approximation by weighted polynomials Original Research Article

It is proven that if xQ′(x) is increasing on (0,+∞) and w(x)=exp(−Q(x)) is the corresponding weight on [0,+∞), 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 problem was raised by Totik, who proved a similar result (the Borwein–Saff conjecture) for convex Q. A general criterion is introduced, too, which guarantees that the support of the extremal measure is an interval. With this criterion we generalize the above approximation theorem as well as that one, where Q is supposed to be convex.