On regular \(n\)th root asymptotic behavior of orthonormal polynomials
Let \(P_ n(z)\), \(n=0,1,2,...\), 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 \(\lim_{n\to \infty}| P_ n(z)|^{1/n}=e^{g(z,\infty)}\) locally uniformly on \({\mathbb{C}}\setminus [a,b],\) where \([a,b]\) is the smallest interval containing S(\(\mu\)) and \(g(z,\infty)\) is the (generalized) Green's function of the domain \({\mathbb{C}}\setminus S(\mu)\) with logarithmic pole at \(\infty\). Two important theorems are proved. The first theorem is a characterization theorem in which regular asymptotic behavior on a subset of the support S(\(\mu\)) is compared with the asymptotic behavior of other polynomials. Equivalences between different types of asymptotic behavior (on and off the support) are proved. The second theorem is a localization theorem in which the asymptotic behavior of the orthonormal polynomials is characterized by the asymptotic behavior of polynomials orthonormal with respect to restrictions of the measure. Extensive use is made of logarithmic potential theory.
scientific article; zbMATH DE number 4203014
scientific article