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Coefficients of Orthogonal Polynomials on the Unit Circle and Higher-Order Szego Theorems

Authors:

Institution:

(1) Mathematics Division, Institute for Low Temperature Physics and Engineering, 47 Lenin Avenue, Kharkov 61103, Ukraine;(2) Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, WI 53706, USA

Abstract:

Let $\mu$ be a nontrivial probability measure on the unit circle $\partial\mbox{\bf D},\ w$ the density of its absolutely continuous part, $\alpha_n$ its Verblunsky coefficients, and $\Phi_n$ its monic orthogonal polynomials. In this paper we compute the coefficients of $\Phi_n$ in terms of the $\alpha_n$ . If the function $\log w$ is in $L^1(d\theta)$ , we do the same for its Fourier coefficients. As an application we prove that if $\alpha_n\in\ell^4$ and if $Q(z) \equiv\sum_{m=0}^N q_m z^m$ is a polynomial, then with $\bar Q(z) \equiv\sum_{m=0}^N \bar q_m z^m$ and S the left-shift operator on sequences we have

$| Q(e^{i\theta}) |^2\log w(\theta) \in L^1(d\theta) \quad \Leftrightarrow \quad \{\bar Q(S)\alpha\}_n\in\ell^2.$

We also study relative ratio asymptotics of the reversed polynomials $\Phi_{n+1}^*(\mu)/\Phi_n^*(\mu)-\Phi_{n+1}^*(\nu)/\Phi_n^*(\nu)$ and provide a necessary and sufficient condition in terms of the Verblunsky coefficients of the measures $\mu$ and $\nu$ for this difference to converge to zero uniformly on compact subsets of $\mbox{\bf D}$ .

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