Eichinger, BenjaminLukić, MilivojeYoung, Giorgio2023-03-102023-03-102023Eichinger, Benjamin, Lukić, Milivoje and Young, Giorgio. "Orthogonal rational functions with real poles, root asymptotics, and GMP matrices." <i>Transactions of the American Mathematical Society, Series B,</i> 10, (2023) American Mathematical Society: 1-47. https://doi.org/10.1090/btran/117.https://hdl.handle.net/1911/114496There is a vast theory of the asymptotic behavior of orthogonal polynomials with respect to a measure on and its applications to Jacobi matrices. That theory has an obvious affine invariance and a very special role for . We extend aspects of this theory in the setting of rational functions with poles on, obtaining a formulation which allows multiple poles and proving an invariance with respect to 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.engCopyright 2023 by the author(s) under Creative Commons Attribution-NonCommercial 3.0 License (CC BY NC 3.0)Journal articleS2330-0000-2023-00117-0https://doi.org/10.1090/btran/117