Szegő limit theorems

In mathematical analysis, the Szegő limit theorems describe the asymptotic behaviour of the determinants of large Toeplitz matrices.^[1][2][3] They were first proved by Gábor Szegő.

Let ${\displaystyle w}$ be a Fourier series with Fourier coefficients ${\displaystyle c_{k}}$, relating to each other as

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

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

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

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

Under suitable assumptions the Szegő theorem states that

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

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

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

1

such that the arithmetic mean of ${\displaystyle \lambda ^{(n)}}$ converges to the integral of ${\displaystyle w}$.^[4]

The first Szegő theorem^[1][3][5] states that, if right-hand side of (1) holds and ${\displaystyle w\geq 0}$, then

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

2

holds for ${\displaystyle w>0}$ and ${\displaystyle w\in L_{1}}$. The RHS of (2) is the geometric mean of ${\displaystyle w}$ (well-defined by the arithmetic-geometric mean inequality).

Let ${\displaystyle {\widehat {c}}_{k}}$ be the Fourier coefficient of ${\displaystyle \log w\in L^{1}}$, written as

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

The second (or strong) Szegő theorem^[1][6] states that, if ${\displaystyle w\geq 0}$, then

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

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