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

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

where

${\displaystyle {\widehat {\phi }}(k)={\frac {1}{2\pi }}\int _{0}^{2\pi }\phi (e^{i\theta })e^{-ik\theta }\,d\theta }$

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

${\displaystyle \lim _{n\to \infty }{\frac {\det T_{n}(\phi )}{\det T_{n-1}(\phi )}}=\exp \left\{{\frac {1}{2\pi }}\int _{0}^{2\pi }\log \phi (e^{i\theta })\,d\theta \right\}.}$

(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 G. 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

${\displaystyle \lim _{n\to \infty }{\frac {\det T_{n}(\phi )}{G^{n}(\phi )}}=\exp \left\{\sum _{k=1}^{\infty }k\left|{\widehat {(\log \phi )}}(k)\right|^{2}\right\}.}$

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.