 0581 Barry Simon
 Meromorphic Szego Functions and Asymptotic Series for Verblunsky Coefficients
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Feb 23, 05

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Abstract. We prove that the Szeg\H{o} 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, $z_1 \dots z_\ell \bar z_{\ell1}\dots \bar z_{2\ell1}$ with $z_j$ in the set. The proofs use nothing more than iterated Szeg\H{o} recursion at $z$ and
$1/\bar z$.
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