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

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

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

such that the $n\times n$ Toeplitz matrices $T_{n}(w)=\left(c_{k-l}\right)_{0\leq k,l\leq n-1}$ are Hermitian, i.e., if $T_{n}(w)=T_{n}(w)^{\ast }$ then $c_{-k}={\overline {c_{k}}}$. Then both $w$ and eigenvalues $(\lambda _{m}^{(n)})_{0\leq m\leq n-1}$ 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

$\lim _{n\rightarrow \infty }{\frac {1}{n}}\sum _{m=0}^{n-1}F(\lambda _{m}^{(n)})={\frac {1}{2\pi }}\int _{0}^{2\pi }F(w(\theta ))\,d\theta$

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

$\lim _{n\rightarrow \infty }{\frac {1}{n}}\sum _{m=0}^{n-1}\lambda _{m}^{(n)}={\frac {1}{2\pi }}\int _{0}^{2\pi }w(\theta )\,d\theta <\infty$ (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 $w\geq 0$, then

$\lim _{n\to \infty }\left(\det T_{n}(w)\right)^{\frac {1}{n}}=\lim _{n\to \infty }{\frac {\det T_{n}(w)}{\det T_{n-1}(w)}}=\exp \left({\frac {1}{2\pi }}\int _{0}^{2\pi }\log w(\theta )\,d\theta \right)$ (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 ${\widehat {c}}_{k}$ be the Fourier coefficient of $\log w\in L^{1}$, written as

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

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

$\lim _{n\to \infty }{\frac {\det T_{n}(w)}{e^{(n+1){\widehat {c}}_{0}}}}=\exp \left(\sum _{k=1}^{\infty }k\left|{\widehat {c}}_{k}\right|^{2}\right).$

See also

• Trigonometric moment problem
• Verblunsky's theorem