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

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 Möbius 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 Cesàro–Nevai property of regular Jacobi matrices on finite gap sets.