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 w be a Fourier series with Fourier coefficients ck, relating to each other as

w(θ)=∑k=−∞∞ckeikθ,θ∈[0,2π],
ck=12π∫02πw(θ)e−ikθdθ,

such that the n×n Toeplitz matrices Tn(w)=(ck−l)0≤k,l≤n−1 are Hermitian, i.e., if Tn(w)=Tn(w)∗ then c−k=ck‾. Then both w and eigenvalues (λm(n))0≤m≤n−1 are real-valued and the determinant of Tn(w) is given by

det⁡Tn(w)=∏m=1n−1λm(n).

## Szegő theorem

Under suitable assumptions the Szegő theorem states that

limn->∞1n∑m=0n−1F(λm(n))=12π∫02πF(w(θ))dθ

for any function F that is continuous on the range of w. In particular

limn->∞1n∑m=0n−1λm(n)=12π∫02πw(θ)dθ<∞

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(1)

such that the arithmetic mean of λ(n) converges to the integral of w.[4]

### First Szegő theorem

The first Szegő theorem[1][3][5] states that, if right-hand side of (1) holds and w≥0, then

limn->∞(det⁡Tn(w))1n=limn->∞det⁡Tn(w)det⁡Tn−1(w)=exp⁡(12π∫02πlog⁡w(θ)dθ)

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(2)

holds for w>0 and w∈L1. The RHS of (2) is the geometric mean of w (well-defined by the arithmetic-geometric mean inequality).

### Second Szegő theorem

Let c^k be the Fourier coefficient of log⁡w∈L1, written as

c^k=12π∫02πlog⁡(w(θ))e−ikθdθ

The second (or strong) Szegő theorem[1][6] states that, if w≥0, then

limn->∞det⁡Tn(w)e(n+1)c^0=exp⁡(∑k=1∞k|c^k|2).

## 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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