Weighted polynomial approximation for weights with logarithmic singularities in the extremal measure (Q1307400)

Let \(\Sigma\subset\mathbb{R}\) be a closed set and \(w:\Sigma \to[0,\infty)\). If \(w\in C(\Sigma)\), the set \(\{x\in\Sigma:w(x)>0\}\) has positive logarithmic capacity and \(|x|w(x)\to 0\) for \(|x|\to \infty\) (provided \(\Sigma\) is unbounded) then \(w\) is called an admissible weight. A function \(f\in C(\Sigma)\) is called \(w\)-approximable if \(f\) is the uniform limit on \(\Sigma\) of a sequence \(\{w^{n_p}_n\}\) with \(P_n \in \mathcal P_n\) (\(\mathcal P_n\) denotes the set of polynomials of degree \(\leq n\)). Let \(\mu_w\) the extremal measure associated to the admissible weight \(w\) and having the property that its density function \(v_w(t)\) has at most a finite number of singularities of logarithmic type. The author shows that every function \(f \in C(\Sigma)\) which vanishes on the exterior of the set where \(v_w\) is positive or has a logarithmic singularity is \(w\)-approximable.