Meromorphic Szego functions and asymptotic series for Verblunsky coefficients

2005/02/23 by Barry Simon, Simon, Barry
Mathematics · #30D30 #42C05 #FOS: Mathematics #Holomorphic and Operator Theory #Mathematical functions and polynomials #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math.SP #msc:30D30 #msc:42C05

paper · pdf · doi:10.48550/arxiv.math/0502489

arxiv created 2005/02/23 · openalex publication_date 2005/02/23 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that the Szegő function, D(z), of a measure on the unit circle is entire meromorphic if and only if the Verblunsky coefficients have an asymptotic expansion in exponentials. We relate the positions of the poles of D(z)^-1 to the exponential rates in the asymptotic expansion. Basically, either set is contained in the sets generated from the other by considering products of the form, z₁ ... z_ℓ z_ℓ-1... z_2ℓ-1 with z_j in the set. The proofs use nothing more than iterated Szegő recursion at z and 1/ z.

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