Regular behavior of orthogonal polynomials and its localization
Let \(P_ n(z,\mu)\), \(n=0,1,2,\dots\), be the orthonormal polynomials with respect to a positive measure \(\mu\) with compact support \(S(\mu)\subset\mathbb{R}\). These polynomials are said to possess regular asymptotic behavior if \(\lim_{n\to\infty}| P_ n(z,\mu)|^{1/n}=1\) quasi everywhere on \(S(\mu)\). \textit{H. Stahl} [J. Approximation Theory 66, No. 2, 125-161 (1991; Zbl 0728.42016)]\ (the reference [2]\ in the paper under review is wrong) showed that regular asymptotic behavior is basically a local property of the orthogonality measure. Stahl's proof is rather complicated and uses very fine techniques from logarithmic potential theory. In this paper a simple proof is presented of Stahl's localization result and actually a slight improvement of the result is proved.
scientific article; zbMATH DE number 13525