Weighted Polynomial Approximation

In this chapter, we establish the existence of weighted polynomial approximations that are a prerequisite to the estimates and asymptotics in subsequent chapters. We search for polynomials P _n of degree n such that P _n W approximates 1 in some sense on [a_−n, a _n ]. There are several approaches to this problem ([48], [86], [115], [136], [66], [178]), most based on discretisation of the potential. The most successful one has been given by Totik [178], and this is the method that we employ. Because of our need to approximate almost up to a _±n , our results do not follow from those in [178], [180], [182].