Asymptotics for Christoffel functions for general measures on the real line (Q1591327)

The Christoffel function \(\lambda_n(x,\mu)\) for orthogonal polynomials for a positive measure \(\mu\) can be defined as \(1/\sum_{k=0}^n |p_k(x)|^2\) or as the infimum of the square of the \(L_2(\mu)\) norm of polynomials \(P_n\) of degree at most \(n\) for which \(P_n(x)=1\). The asymptotic behaviour of \(n\lambda_n(x,\mu)\) gives the Radon-Nikodým derivative of the measure \(\mu\), as was shown by \textit{A. Máté, B. Nevai}, and \textit{V. Totik} [Ann. Math. (2) 134, No.~2, 433-453 (1991; Zbl 0752.42015)] when supp\((\mu) = [-1,1]\) and \(\mu'\) satisfies a local Szegő condition. In the present paper the author settles the more general case when the support of \(\mu\) is a compact set \(E\) on the real line for which \({\mathbb{C}} \setminus E\) is a regular domain with respect to the Dirichlet problem, and \(\mu\) is a regular measure in the sense of \textit{H. Stahl} and \textit{V. Totik} [``General orthogonal polynomials'' (1992; Zbl 0791.33009)]. If \(\log \mu'\) is integrable on an interval \(I \subset E\), then \(n\lambda_n(x,\mu)\) converges to \(d\mu(x)/d\omega_E(x)\) for almost every \(x \in I\), where \(\omega_E\) is the equilibrium measure (from logarithmic potential theory) for the set \(E\).