Determinant of large Toeplitz matrices
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
be a Fourier series with Fourier coefficients
, relating to each other as
![{\displaystyle w(\theta )=\sum _{k=-\infty }^{\infty }c_{k}e^{ik\theta },\qquad \theta \in [0,2\pi ],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c810eeafda39e98041569e2e0e42d4a70ae3148)
![{\displaystyle c_{k}={\frac {1}{2\pi }}\int _{0}^{2\pi }w(\theta )e^{-ik\theta }\,d\theta ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/04314029a99b3bda0917eebcaa71573832eabd1f)
such that the
Toeplitz matrices
are Hermitian, i.e., if
then
. Then both
and eigenvalues
are real-valued and the determinant of
is given by
.
Szegő theorem
Under suitable assumptions the Szegő theorem states that
![{\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 }](https://wikimedia.org/api/rest_v1/media/math/render/svg/70a9afd1d88d97dd89ed5a08fd38d6f8411c45d4)
for any function
that is continuous on the range of
. In particular
![{\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 <\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5c151e5826c29205ca2a5bc66d17b244dba3b9f) | | (1) |
such that the arithmetic mean of
converges to the integral of
.[4]
First Szegő theorem
The first Szegő theorem[1][3][5] states that, if right-hand side of (1) holds and
, then
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c84601653ea5f0e1adc7c7411976c0a6500a42db) | | (2) |
holds for
and
. The RHS of (2) is the geometric mean of
(well-defined by the arithmetic-geometric mean inequality).
Second Szegő theorem
Let
be the Fourier coefficient of
, written as
![{\displaystyle {\widehat {c}}_{k}={\frac {1}{2\pi }}\int _{0}^{2\pi }\log(w(\theta ))e^{-ik\theta }\,d\theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e7a1e53184b5fed50eb84ad0190c42034633070)
The second (or strong) Szegő theorem[1][6] states that, if
, then
![{\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).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4aa49cdaa2467aa2f49241752695a1b830383d5)
See also
References
- ^ a b c Böttcher, Albrecht; Silbermann, Bernd (1990). "Toeplitz determinants". Analysis of Toeplitz operators. Berlin: Springer-Verlag. p. 525. ISBN 3-540-52147-X. MR 1071374.
- ^ Ehrhardt, T.; Silbermann, B. (2001) [1994], "Szegö_limit_theorems", Encyclopedia of Mathematics, EMS Press
- ^ a b 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.
- ^ Gray, Robert M. (2006). "Toeplitz and Circulant Matrices: A Review" (PDF). Foundations and Trends in Signal Processing.
- ^ Szegő, G. (1915). "Ein Grenzwertsatz über die Toeplitzschen Determinanten einer reellen positiven Funktion". Math. Ann. 76 (4): 490–503. doi:10.1007/BF01458220. S2CID 123034653.
- ^ 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.