Weighted Polynomial Approximation for Weights with Logarithmic Singularities in the Extremal Measure

Let w : Σ → [0, ∞) be a weight function on a set Σ ⊂ R. We assume that the associated extremal measure μω has density function vω(t) with finitely many singularities of logarithmic type. We show that any continuous function f on Σ which vanishes outside the set where vω is positive or has a logarithmic singularity, is the uniform limit on Σ of a sequence of weighted polynomials of the form wn Pn, where Pn is of degree ≦ n. This extends previous results for continuous densities to densities having logarithmic singularities.