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 $w$ be a Fourier series with Fourier coefficients $c_k$, relating to each other as

$w(\theta )=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_k{e}^{ik\theta },\theta \in [0,2\pi ],$

$c_k=\frac{1}{2\pi }\underset{0}{\overset{2\pi }{\int }}w(\theta )e^{-ik\theta }d\theta ,$

such that the $n\times n$ Toeplitz matrices $T_n(w)={\left(c_{k-l}\right)}_{0\le k,l\le n-1}$ are Hermitian, i.e., if $T_n(w)=T_n(w{)}^{\ast }$ then $c_{-k}=\overline{c_k}$. Then both $w$ and eigenvalues $({\lambda }_m^{(n)}{)}_{0\le m\le n-1}$ are real-valued and the determinant of $T_n(w)$ is given by

$\det T_n(w)=\prod _{m=1}^{n-1}{\lambda }_m^{(n)}$.

Szegő theorem

Under suitable assumptions the Szegő theorem states that

$\underset{n\to \mathrm{\infty }}{\lim }\frac{1}{n}\sum _{m=0}^{n-1}F({\lambda }_m^{(n)})=\frac{1}{2\pi }\underset{0}{\overset{2\pi }{\int }}F(w(\theta ))d\theta$

for any function $F$ that is continuous on the range of $w$. In particular

$\underset{n\to \mathrm{\infty }}{\lim }\frac{1}{n}\sum _{m=0}^{n-1}{\lambda }_m^{(n)}=\frac{1}{2\pi }\underset{0}{\overset{2\pi }{\int }}w(\theta )d\theta <\mathrm{\infty }$ 1

such that the arithmetic mean of ${\lambda }^{(n)}$ converges to the integral of $w$.

First Szegő theorem

The first Szegő theorem states that, if right-hand side of (1) holds and $w\ge 0$, then

$\underset{n\to \mathrm{\infty }}{\lim }{(\det T_n(w))}^{\frac{1}{n}}=\underset{n\to \mathrm{\infty }}{\lim }\frac{\det T_n(w)}{\det T_{n-1}(w)}=\exp (\frac{1}{2\pi }\underset{0}{\overset{2\pi }{\int }}\log w(\theta )d\theta )$ 2

holds for $w>0$ and $w\in L_1$. The RHS of (2) is the geometric mean of $w$ (well-defined by the arithmetic-geometric mean inequality).

Second Szegő theorem

Let ${\hat{c}}_k$ be the Fourier coefficient of $\log w\in L^1$, written as

${\hat{c}}_k=\frac{1}{2\pi }\underset{0}{\overset{2\pi }{\int }}\log (w(\theta ))e^{-ik\theta }d\theta$

The second (or strong) Szegő theorem states that, if $w\ge 0$, then

$\underset{n\to \mathrm{\infty }}{\lim }\frac{\det T_n(w)}{e^{(n+1){\hat{c}}_0}}=\exp \left(\sum _{k=1}^{\mathrm{\infty }}k{\left|{\hat{c}}_k\right|}^2\right).$

See also

• Trigonometric moment problem
• Verblunsky's theorem