Szegő limit theorems

In mathematical analysis, the Szegő limit theorems describe the asymptotic behaviour of the determinants of large Toeplitz matrices. They were first proved by Gábor Szegő.

Notation

Let φ : T→C be a complex function ("symbol") on the unit circle. Consider the n×n Toeplitz matrices T_n(φ), defined by

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

where

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

are the Fourier coefficients of φ.

First Szegő theorem

The first Szegő theorem states that, if φ > 0 and φ ∈ L₁(T), then

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

The right-hand side of (1) is the geometric mean of φ (well-defined by the arithmetic-geometric mean inequality).

Second Szegő theorem

Denote the right-hand side of (1) by G. The second (or strong) Szegő theorem asserts that if, in addition, the derivative of φ is Hölder continuous of order α > 0, then

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