The logarithmic asymptotic expansions for the norms of evaluation functionals

Let $\mu$ be a compactly supported finite Borel measure in $\mathbb{C}$, and let $\Pi_n$ be the space of holomorphic polynomials of degree at most $n$ furnished with the norm of $L^2(\mu)$. We study the logarithmic asymptotic expansions of the norms of the evaluation functionals that relate to polynomials $p\in\Pi_n$ their values at a point $z\in\mathbb{C}$. The main results demonstrate how the asymptotic behavior depends on regularity of the complement of the support of $\mu$ and the Stahl–Totik regularity of the measure. In particular, we study the cases of pointwise and $\mu$-a.e. convergence as $n\to\infty$.