Approximation by weighted polynomials

It is proven that if xQ′(x) is increasing on (0,+∞) and w(x) = exp(−Q) 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 wPn. This problem was raised by V. 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. Introduction In [6], V.Totik settled a basic conjecture in the theory of approximation by weighted polynomials of the form wPn, where w = exp(−Q) is a given weight function. This conjecture was the Borwein-Saff conjecture and it was stated for convex Q. In this paper the same theorem is stated and proved, but for more general Q. The criterion we use covers the case when a) Q is convex, and also the case when b) Q is defined on [0,+∞) and xQ′(x) is increasing. In logarithmic potential theory many theorems use either a) or b) as their assumption on Q, since both of them guarantee that the support Sw is an interval. The criterion to be introduced is weaker than both a) and b), but still guarantees that the support Sw is an interval. (Furthermore, we merely assume that Q is absolutely continuous and not necessarily differentiable.) Thus this criterion itself is a useful result of this paper. The reader can find the definition of the logarithmic capacity in [4], I.1. We say that a property holds quasi-everywhere, if the set where it does not hold has capacity 0.