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 .
Let w {\displaystyle w} be a Fourier series with Fourier coefficients c k {\displaystyle c_{k}} , relating to each other as
w ( θ ) = ∑ k = − ∞ ∞ c k e i k θ , θ ∈ [ 0 , 2 π ] , {\displaystyle w(\theta )=\sum _{k=-\infty }^{\infty }c_{k}e^{ik\theta },\qquad \theta \in [0,2\pi ],} c k = 1 2 π ∫ 0 2 π w ( θ ) e − i k θ d θ , {\displaystyle c_{k}={\frac {1}{2\pi }}\int _{0}^{2\pi }w(\theta )e^{-ik\theta }\,d\theta ,} such that the n × n {\displaystyle n\times n} Toeplitz matrices T n ( w ) = ( c k − l ) 0 ≤ k , l ≤ n − 1 {\displaystyle T_{n}(w)=\left(c_{k-l}\right)_{0\leq k,l\leq n-1}} are Hermitian , i.e., if T n ( w ) = T n ( w ) ∗ {\displaystyle T_{n}(w)=T_{n}(w)^{\ast }} then c − k = c k ¯ {\displaystyle c_{-k}={\overline {c_{k}}}} . Then both w {\displaystyle w} and eigenvalues ( λ m ( n ) ) 0 ≤ m ≤ n − 1 {\displaystyle (\lambda _{m}^{(n)})_{0\leq m\leq n-1}} are real-valued and the determinant of T n ( w ) {\displaystyle T_{n}(w)} is given by
det T n ( w ) = ∏ m = 1 n − 1 λ m ( n ) {\displaystyle \det T_{n}(w)=\prod _{m=1}^{n-1}\lambda _{m}^{(n)}} . Under suitable assumptions the Szegő theorem states that
lim n → ∞ 1 n ∑ m = 0 n − 1 F ( λ m ( n ) ) = 1 2 π ∫ 0 2 π F ( w ( θ ) ) d θ {\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 } for any function F {\displaystyle F} that is continuous on the range of w {\displaystyle w} . In particular
lim n → ∞ 1 n ∑ m = 0 n − 1 λ m ( n ) = 1 2 π ∫ 0 2 π w ( θ ) d θ < ∞ {\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 } (1 )
such that the arithmetic mean of λ ( n ) {\displaystyle \lambda ^{(n)}} converges to the integral of w {\displaystyle w} .[ 4]
First Szegő theorem[ edit ] The first Szegő theorem[ 1] [ 3] [ 5] states that, if right-hand side of (1 ) holds and w ≥ 0 {\displaystyle w\geq 0} , then
lim n → ∞ ( det T n ( w ) ) 1 n = lim n → ∞ det T n ( w ) det T n − 1 ( w ) = exp ( 1 2 π ∫ 0 2 π log w ( θ ) d θ ) {\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)} (2 )
holds for w > 0 {\displaystyle w>0} and w ∈ L 1 {\displaystyle w\in L_{1}} . The RHS of (2 ) is the geometric mean of w {\displaystyle w} (well-defined by the arithmetic-geometric mean inequality ).
Second Szegő theorem[ edit ] Let c ^ k {\displaystyle {\widehat {c}}_{k}} be the Fourier coefficient of log w ∈ L 1 {\displaystyle \log w\in L^{1}} , written as
c ^ k = 1 2 π ∫ 0 2 π log ( w ( θ ) ) e − i k θ d θ {\displaystyle {\widehat {c}}_{k}={\frac {1}{2\pi }}\int _{0}^{2\pi }\log(w(\theta ))e^{-ik\theta }\,d\theta } The second (or strong) Szegő theorem[ 1] [ 6] states that, if w ≥ 0 {\displaystyle w\geq 0} , then
lim n → ∞ det T n ( w ) e ( n + 1 ) c ^ 0 = exp ( ∑ k = 1 ∞ k | c ^ k | 2 ) . {\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).} ^ 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 .