The sharp form of the strong Szego theorem

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 $\{c_{j-i}{\}}_{0\le i,j\le n}$ where $c_m={\widehat{f}}_m\equiv \int e^{-im\theta }f(\theta )\text{f}d\theta 2\pi$. 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 $L^1(\text{f}d\theta 2\pi )$, we have $$\underset{n\to \mathrm{\infty }}{\lim }{D}_n(e^L)e^{-(n+1){\widehat{L}}_0}=\exp \left(\sum _{k=1}^{\mathrm{\infty }}k\text{abs}{{\widehat{L}}_k}^2\right)$$ 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.