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

Abstract

Let $\mu$ be a non-trivial 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 $Q(z)={\sum }_{m=0}^N{q}_m{z}^m$ 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 $|Q(e^{i\theta }){|}^2\log w(\theta )\in L^1(d\theta )$ if and only if $\{\bar{Q}(S)\alpha {\}}_n\in {\ell }^2$. We also study relative ratio asymptotics of the reversed polynomials ${\Phi }_{n+1}^{\ast }(\mu )/{\Phi }_n^{\ast }(\mu )-{\Phi }_{n+1}^{\ast }(\nu )/{\Phi }_n^{\ast }(\nu )$ 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 $\bbD$.

21pp