Orthogonal rational functions with real poles, root asymptotics, and GMP matrices
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There 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.
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Eichinger, Benjamin, Lukić, Milivoje and Young, Giorgio. "Orthogonal rational functions with real poles, root asymptotics, and GMP matrices." Transactions of the American Mathematical Society, Series B, 10, (2023) American Mathematical Society: 1-47. https://doi.org/10.1090/btran/117.