The sharp form of the strong Szego theorem

Abstract

Let $f$ be a function on the unit circle and $D_n(f)$ be the determinant of the $(n+1)\times (n+1)$ matrix with elements $\{c_{j-i}\}_{0\leq i,j\leq n}$ where $c_m =\hat f_m\equiv \int e^{-im\theta} f(\theta) \f{d\theta}{2\pi}$. The sharp form of the strong Szeg\H{o} theorem says that for any real-valued $L$ on the unit circle with $L,e^L$ in $L^1 (\f{d\theta}{2\pi})$, we have \[ \lim_{n\to\infty} D_n(e^L) e^{-(n+1)\hat L_0} = \exp \biggl(\sum_{k=1}^\infty k\abs{\hat L_k}^2\biggr) \] where the right side may be finite or infinite. We focus on two issues here: a new proof when $e^{i\theta}\to L(\theta)$ is analytic and known simple arguments that go from the analytic case to the general case. We add background material to make this article self-contained.