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
 * $$ w(\theta) = \sum_{k=-\infty}^{\infty} c_k e^{i k \theta}, \qquad \theta \in [0,2\pi],  $$
 * $$ c_k = \frac{1}{2\pi} \int_0^{2\pi} w(\theta) e^{-ik\theta} \, d\theta, $$

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

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 ($$) holds and $$w \geq 0$$, then

holds for $$w > 0$$ and $$w\in L_1$$. The RHS of ($$) is the geometric mean of $$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).$$