The sharp form of the strong Szego theorem

Details The sharp form of the strong Szego theorem The sharp form of the strong Szego theorem

2004-02-06

arxiv.org/abs/math/0402110v1

Mathematics

Spectral Theory

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

The sharp form of the strong Szego theorem does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with The sharp form of the strong Szego theorem, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The sharp form of the strong Szego theorem will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-657043