An asymptotic integral representation for Carleman orthogonal polynomials

TL;DR

This paper derives an asymptotic integral representation for Carleman orthogonal polynomials on analytic Jordan curves, providing detailed insights into their zero distribution and asymptotic behavior within the domain.

Contribution

It introduces a new integral representation involving conformal maps and kernel functions, enhancing understanding of polynomial asymptotics and zero distribution.

Findings

Asymptotic behavior characterized by integral formulas

Geometric decay of error terms enables precise asymptotics

Results on zero distribution and accumulation points

Abstract

In this paper we investigate the asymptotic behavior of polynomials that are orthonormal over the interior domain of an analytic Jordan curve L with respect to area measure. We prove that, inside L, these polynomials behave asymptotically like a sequence of certain integrals involving the canonical conformal map of the exterior of L onto the exterior of the unit circle and certain meromorphic kernel function defined in terms of a conformal map of the interior of L onto the unit disk. The error term in the integral representation is proven to decay geometrically and sufficiently fast, allowing us to obtain more precise asymptotic formulas for the polynomials under certain additional geometric considerations. These formulas yield, in turn, fine results on the location, limiting distribution and accumulation points of the zeros of the polynomials.