Meromorphic Szego functions and asymptotic series for Verblunsky coefficients

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, $z_1...z_{\ell }{\bar{z}}_{\ell -1}...{\bar{z}}_{2\ell -1}$ with $z_j$ in the set. The proofs use nothing more than iterated Szegő recursion at $z$ and $1/\bar{z}$.

References in corpus (2)

• Fine Structure of the Zeros of Orthogonal Polynomials, I. A Tale of Two Pictures
• Fine Structure of the Zeros of Orthogonal Polynomials, II. OPUC With Competing Exponential Decay