Weighted Polynomial Approximation for Weights with Slowly Varying Extremal Density

Polynomial approximation by weighted polynomials of the form wⁿ(x) P_n(x) is investigated on closed subsets of the real line. It is known that the possibility of approximation is closely related to the density of an extremal measure associated with w via a weighted energy problem. It is also known that if in a neighborhood of a point x₀ this density is continuous and positive, then, in that neighborhood, any continuous function can be approximated. The aim of the present paper is twofold. On the one hand it is shown that the same approximation theorem is true if in a neighborhood of x₀ the density is slowly varying and is bounded away from 0. This allows singularities of logarithmic types. On the other hand, we also show that under some mild conditions, if the density at x₀ is slowly varying, then approximation is still possible even if the density vanishes at x₀ . This is the first positive result for approximation with a vanishing density.