An expansion for polynomials orthogonal over an analytic Jordan curve

TL;DR

This paper develops a new integral expansion for polynomials orthogonal over an analytic Jordan curve, providing uniform convergence, asymptotic formulas, and representations that generalize classical results to more complex curves.

Contribution

It introduces a novel integral transform expansion for orthogonal polynomials on analytic Jordan curves, extending classical asymptotic results beyond the unit circle case.

Findings

Uniform convergence of the expansion in the complex plane

Derivation of Szego's asymptotic formula for these polynomials

New integral representation inside the curve

Abstract

We consider polynomials that are orthogonal over an analytic Jordan curve L with respect to a positive analytic weight, and show that each such polynomial of sufficiently large degree can be expanded in a series of certain integral transforms that converges uniformly in the whole complex plane. This expansion yields, in particular and simultaneously, Szego's classical strong asymptotic formula and a new integral representation for the polynomials inside L. We further exploit such a representation to derive finer asymptotic results for weights having finitely many singularities (all of algebraic type) on a thin neighborhood of the orthogonality curve. Our results are a generalization of those previously obtained in [7] for the case of L being the unit circle.