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

2020/08/27 by Eichinger, Benjamin, Lukić, Milivoje, Young, Giorgio
#42C05 #47B36 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2008.11884

Abstract

There is a vast theory of the asymptotic behavior of orthogonal polynomials with respect to a measure on ℝ 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 ℝ = ℝ ∪ \∞\, obtaining a formulation which allows multiple poles and proving an invariance with respect to 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.

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