Weighted Approximation with Varying Weights: The Case of a Power-Type Singularity

The class of functions that can be uniformly approximated by weighted polynomials of the form wnPnwith deg PnFn, depends on the behavior of the extremal measure associated with w as introduced by Mhaskar and Saff. It is shown that if in a neighborhood of a point t0 the extremal measure has a density with a power-type singularity at t0, then every uniform limit vanishes at t0. This complements results of Totik for continuous positive densities and Kuijlaars for densities that vanish at t0.