Asymptotic behavior and zero distribution of Carleman orthogonal polynomials

Abstract

Let L be an analytic Jordan curve and let { p n ( z ) } n = 0 ∞ be the sequence of polynomials that are orthonormal with respect to the area measure over the interior of L . A well-known result of Carleman states that (1) lim n → ∞ p n ( z ) ( n + 1 ) / π [ ϕ ( z ) ] n = ϕ ′ ( z ) locally uniformly on a certain open neighborhood of the closed exterior of L , where ϕ is the canonical conformal map of the exterior of L 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 p n ’s. Some consequences on the limiting distribution of the zeros are discussed, and the results are illustrated with two concrete examples and numerical computations.