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.

## Contents

* 1 Notation
* 2 Szegő theorem
* 2.1 First Szegő theorem
* 2.2 Second Szegő theorem
* 3 See also
* 4 References

## 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]

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(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]

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(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

1. ↑ 1.0 1.1 1.2 Böttcher, Albrecht; Silbermann, Bernd (1990). "Toeplitz determinants". Analysis of Toeplitz operators. Berlin: Springer-Verlag. p. 525. ISBN 3-540-52147-X.
2. ↑ Hazewinkel, Michiel, ed. (2001), "Szegö_limit_theorems", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Szegö_limit_theorems
3. ↑ 3.0 3.1 Simon, Barry (2011). Szegő's Theorem and Its Descendants: Spectral Theory for L2 Perturbations of Orthogonal Polynomials. Princeton: Princeton University Press. ISBN 978-0-691-14704-8.
4. ↑ Gray, Robert M. (2006). "Toeplitz and Circulant Matrices: A Review". Foundations and Trends in Signal Processing. https://ee.stanford.edu/~gray/toeplitz.pdf.
5. ↑ Szegő, G. (1915). "Ein Grenzwertsatz über die Toeplitzschen Determinanten einer reellen positiven Funktion". Math. Ann. 76 (4): 490-503. doi:10.1007/BF01458220. https://zenodo.org/record/2496405.
6. ↑ Szegő, G. (1952). "On certain Hermitian forms associated with the Fourier series of a positive function". Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.]: 228-238.

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