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.

Notation

Let [math]\displaystyle{ w }[/math] be a Fourier series with Fourier coefficients [math]\displaystyle{ c_k }[/math], relating to each other as

[math]\displaystyle{ w(\theta) = \sum_{k=-\infty}^{\infty} c_k e^{i k \theta}, \qquad \theta \in [0,2\pi], }[/math]

[math]\displaystyle{ c_k = \frac{1}{2\pi} \int_0^{2\pi} w(\theta) e^{-ik\theta} \, d\theta, }[/math]

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

[math]\displaystyle{ \det T_n(w) = \prod_{m=1}^{n-1} \lambda_m^{(n)} }[/math].

Szegő theorem

Under suitable assumptions the Szegő theorem states that

[math]\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 }[/math]

for any function [math]\displaystyle{ F }[/math] that is continuous on the range of [math]\displaystyle{ w }[/math]. In particular

[math]\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 \lt \infty }[/math]

(1)

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

First Szegő theorem

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

[math]\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) }[/math]

(2)

holds for [math]\displaystyle{ w \gt 0 }[/math] and [math]\displaystyle{ w\in L_1 }[/math]. The RHS of (2) is the geometric mean of [math]\displaystyle{ w }[/math] (well-defined by the arithmetic-geometric mean inequality).

Second Szegő theorem

Let [math]\displaystyle{ \widehat c_k }[/math] be the Fourier coefficient of [math]\displaystyle{ \log w \in L^{1} }[/math], written as

[math]\displaystyle{ \widehat c_k = \frac{1}{2\pi} \int_0^{2\pi} \log (w(\theta)) e^{-ik\theta} \, d\theta }[/math]

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

[math]\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). }[/math]

See also

• Trigonometric moment problem
• Verblunsky's theorem

References

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