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 lim n→∞ p n (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.

Asymptotics of Carleman Polynomials for Level Curves of the Inverse of a Shifted Zhukovsky Transformation

“…Recently, the subject has experienced a new surge, with many new interesting results in a variety of topics such as the asymptotic behavior and zero distribution of the orthogonal polynomials [4,6,7,8,10,11,12,14,16], universality and Christoffel functions [9,14,20], and the existence of recurrence relations [1,18,19]. In particular, [6] and [20] consider orthogonality over several domains, while [14] considers orthogonality with respect to certain potential theoretic varying weights.…”

“…This establishes the asymptotic behavior of p n (z) on the closed exterior Ω 1 of L 1 , and on a portion of its interior G 1 , namely, on the "strip" Ω ρ ∩ G 1 . What happens at the remaining points of G 1 has been recently investigated in [4,10]. In turns out that there is a subset Σ 1 ⊂ G 1 , which is, in general, larger than the strip Ω ρ ∩ G 1 , on which an asymptotic formula just like (1) holds true.…”

On a series representation for Carleman orthogonal polynomials

“…The study of the asymptotic properties of these polynomials as their degree n → ∞ was initiated by T. Carleman in [1], who established the strong asymptotic formula (3) lim…”

“…2]. Recently in [3], we have determined the largest open set where an asymptotic formula such as (3) holds true, a set with the property that each of its boundary points is a limit point of the zeros of the p n 's. The main tool in the derivation of this result has been the asymptotic integral representation…”

Asymptotics of polynomials orthogonal over circular multiply connected domains

“…This representation can be derived by using the summation by parts formula in conjunction with (6.25); for the details see Lemma 3.3 in [6] and its proof. From the definition (6.22), we find 4 , so that, as n → ∞,…”