Szegő limit theorems

Let ${\displaystyle \phi$ :\mathbb {T} \to \mathbb {C} } be a complex function ("symbol") on the unit circle. Consider the ${\displaystyle n\times n}$ Toeplitz matrices ${\displaystyle T_{n}(\phi )}$, defined by

${\displaystyle T_{n}(\phi )_{k,l}={\widehat {\phi }}(k-l),\quad 0\leq k,l\leq n-1,}$

where

${\displaystyle {\widehat {\phi }}(k)={\frac {1}{2\pi }}\int _{0}^{2\pi }\phi (e^{i\theta })e^{-ik\theta }\,d\theta }$

are the Fourier coefficients of ${\displaystyle \phi }$.

The first Szegő theorem[1][4] states that, if ${\displaystyle \phi >0}$ and ${\displaystyle \phi \in L_{1}(\mathbb {T} )}$, then

${\displaystyle \lim _{n\to \infty }{\frac {\det T_{n}(\phi )}{\det T_{n-1}(\phi )}}=\exp \left\{{\frac {1}{2\pi }}\int _{0}^{2\pi }\log \phi (e^{i\theta })\,d\theta \right\}.}$

(1)

The right-hand side of (1) is the geometric mean of ${\displaystyle \phi }$ (well-defined by the arithmetic-geometric mean inequality).

Denote the right-hand side of (1) by ${\displaystyle G}$. The second (or strong) Szegő theorem[1][5] asserts that if, in addition, the derivative of ${\displaystyle \phi }$ is Hölder continuous of order ${\displaystyle \alpha >0}$, then

${\displaystyle \lim _{n\to \infty }{\frac {\det T_{n}(\phi )}{G^{n}(\phi )}}=\exp \left\{\sum _{k=1}^{\infty }k\left|{\widehat {(\log \phi )}}(k)\right|^{2}\right\}.}$