Orthogonal rational functions with real poles, root asymptotics, and GMP matrices

TL;DR

This paper extends the theory of orthogonal polynomials to rational functions with real poles, establishing invariance properties, characterizing regularity of GMP matrices, and proving a conjecture related to Jacobi matrices on finite gap sets.

Contribution

It introduces a framework for rational functions with real poles, generalizes invariance under Möbius transformations, and characterizes GMP matrix regularity, addressing a conjecture of Simon.

Findings

Characterization of Stahl--Totik regularity for GMP matrices.

Proof of a conjecture of Simon regarding Cesàro--Nevai property.

Extension of asymptotic theory to rational functions with multiple poles.

Abstract

There is a vast theory of the asymptotic behavior of orthogonal polynomials with respect to a measure on $\mathbb{R}$ and its applications to Jacobi matrices. That theory has an obvious affine invariance and a very special role for $\infty$. We extend aspects of this theory in the setting of rational functions with poles on $\overline{\mathbb{R}} = \mathbb{R} \cup \{\infty\}$, obtaining a formulation which allows multiple poles and proving an invariance with respect to $\overline{\mathbb{R}}$-preserving M\"obius transformations. We obtain a characterization of Stahl--Totik regularity of a GMP matrix in terms of its matrix elements; as an application, we give a proof of a conjecture of Simon -- a Ces\`aro--Nevai property of regular Jacobi matrices on finite gap sets.