Asymptotic behavior and zero distribution of Carleman orthogonal polynomials Original Research Article

Let LL be an analytic Jordan curve and let View the MathML source{pn(z)}n=0∞ be the sequence of polynomials that are orthonormal with respect to the area measure over the interior of LL. A well-known result of Carleman states that equation(1) View the MathML sourcelimn→∞pn(z)(n+1)/π[ϕ(z)]n=ϕ′(z) Turn MathJax on locally uniformly on a certain open neighborhood of the closed exterior of LL, where ϕϕ is the canonical conformal map of the exterior of LL onto the exterior of the unit circle. In this paper we extend the validity of (1) to a maximal open set, every boundary point of which is an accumulation point of the zeros of the pnpn’s. Some consequences on the limiting distribution of the zeros are discussed, and the results are illustrated with two concrete examples and numerical computations.