 05308 Leonid Golinskii, Andrej Zlatos
 Coefficients of Orthogonal Polynomials on the Unit Circle and
Higher Order Szego Theorems
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Sep 8, 05

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Abstract. Let $\mu$ be a nontrivial probability measure on the unit circle
$\partial\bbD$, $w$ 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 $Q(z) \equiv
\sum_{m=0}^N q_m z^m$ 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 $Q(e^{i\theta})^2 \log w(\theta) \in L^1(d\theta)$ if and only if $\{\bar Q(S)\alpha\}_n \in \ell^2$. 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
$\bbD$.
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