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

Abstract

Let $μ$ be a non-trivial probability measure on the unit circle $\partial\bbD$, $w$ the density of its absolutely continuous part, $α_n$ its Verblunsky coefficients, and $Φ_n$ its monic orthogonal polynomials. In this paper we compute the coefficients of $Φ_n$ in terms of the $α_n$. If the function $\log w$ is in $L^1(dθ)$, we do the same for its Fourier coefficients. As an application we prove that if $α_n \in \ell^4$ and $Q(z) = \sum_{m=0}^N q_m z^m$ is a polynomial, then with $\bar Q(z) = \sum_{m=0}^N \bar q_m z^m$ and $S$ the left shift operator on sequences we have $|Q(e^{iθ})|^2 \log w(θ) \in L^1(dθ)$ if and only if $\{\bar Q(S)α\}_n \in \ell^2$. We also study relative ratio asymptotics of the reversed polynomials $Φ_{n+1}^*(μ)/Φ_n^*(μ)-Φ_{n+1}^*(ν)/Φ_n^*(ν)$ and provide a necessary and sufficient condition in terms of the Verblunsky coefficients of the measures $μ$ and $ν$ for this difference to converge to zero uniformly on compact subsets of $\bbD$.