Szegő limit theorems

Template:Short description 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

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

$c_k=\frac{1}{2\pi }{\int }_0^{2\pi }w(\theta )e^{-ik\theta }d\theta ,$

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

$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n}\sum _{m=0}^{n-1}F({\lambda }_m^{(n)})=\frac{1}{2\pi }{\int }_0^{2\pi }F(w(\theta ))d\theta$

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

i.e. the arithmetic mean of ${\lambda }^{(n)}$ converges to the integral of $w$, i.e. $c_0$.

First Szegő theorem

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

holds for $w>0$ and $w\in L^1$. The RHS of (Template:EquationNote) 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

${\hat{c}}_k=\frac{1}{2\pi }{\int }_0^{2\pi }\log (w(\theta ))e^{-ik\theta }d\theta$

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