On higher-order Szego theorems with a single critical point of arbitrary order

Abstract

We prove the following higher-order Szego theorems: if a measure on the unit circle has absolutely continuous part $w(\theta )$ and Verblunsky coefficients $\alpha$ with square-summable variation, then for any positive integer $m$, $\int (1-\cos \theta {)}^m\log w(\theta )d\theta$ is finite if and only if $\alpha \in {\ell }^{2m+2}$. This is the first known equivalence result of this kind in the regime of very slow decay, i.e. with ${\ell }^p$ conditions with arbitrarily large $p$. The usual difficulty of controlling higher-order sum rules is avoided by a new test sequence approach.

arXiv comment13 pages, to appear in Constr. Approx

References in corpus (1)

• On a conjecture for higher-order Szego theorems