The Logarithmic Asymptotic Expansions for the Norms of Evaluation Functionals

Abstract

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