Szegő limit theorems

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

Notation

Let φ : T→C be a complex function ("symbol") on the unit circle. Consider the n×n Toeplitz matrices T_n(φ), defined by

$T_n(\phi)_{k,l} = \widehat\phi(k-l), \quad 0 \leq k,l \leq n-1,$

where

$\widehat\phi(k) = \frac{1}{2\pi} \int_0^{2\pi} \phi(e^{i\theta}) e^{-ik\theta} \, d\theta$

are the Fourier coefficients of φ.

First Szegő theorem

The first Szegő theorem^[1][3] states that, if φ > 0 and φ ∈ L₁(T), then

$\lim_{n \to \infty} \frac{\det T_n(\phi)}{\det T_{n-1}(\phi)} = \exp \left\{ \frac{1}{2\pi} \int_0^{2\pi} \log \phi(e^{i\theta}) \, d\theta \right\}.$

(1)

The right-hand side of (1) is the geometric mean of φ (well-defined by the arithmetic-geometric mean inequality).

Second Szegő theorem

Denote the right-hand side of (1) by Gⁿ. The second (or strong) Szegő theorem^[1][4] asserts that if, in addition, the derivative of φ is Hölder continuous of order α > 0, then

$\lim_{n \to \infty} \frac{\det T_n(\phi)}{G^n(\phi)} = \exp \left\{ \sum_{k=1}^\infty k \left| \widehat{(\log \phi)}(k)\right|^2 \right\}.$

References

1. Böttcher, Albrecht; Silbermann, Bernd (1990). "Toeplitz determinants". Analysis of Toeplitz operators. Berlin: Springer-Verlag. p. 525. ISBN 3-540-52147-X. MR 1071374.
2. Ehrhardt, T.; Silbermann, B. (2001), "Szegö_limit_theorems", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
3. Szegő, G. (1915). "Ein Grenzwertsatz über die Toeplitzschen Determinanten einer reellen positiven Funktion". Math. Ann. 76 (4): 490–503. doi:10.1007/BF01458220.
4. 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. MR 0051961.