Szegő limit theorems explained

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

w be a Fourier series with Fourier coefficients c _{k} , relating to each other asw(\theta) = \sum ^{inf ty} _{k =-inf ty} c _{k} e ^{i} , \theta \in [ 0 , 2 \pi], c _{k} = (1)/(2 \pi) \int ^{2 \pi} _{0} w(\theta) e ^{-ik\theta} d\theta,

such that the

n x n Toeplitz matrices T _{n(w)} = \left(c _{k-l} \right) _{0\leq} are Hermitian, i.e., if T _{n(w)=T} ^{\ast} ^{n(w)} then c _{-k} = \overline{c _{k}} . Then both w and eigenvalues ( λ ^{(n)} _{m} ) _{0\leq} are real-valued and the determinant of T _{n(w)} is given by\det T _{n(w)} = \prod ^{n -1} _{m =1} λ ^{(n)} _{m} .

Szegő theorem

Under suitable assumptions the Szegő theorem states that

\lim _{n →} (1)/(n) \sum ^{n -1} _{m =0} F( λ ^{(n)} _{m} ) = (1)/(2 \pi) \int ^{2 \pi} _{0} F(w(\theta)) d\theta

for any function

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

such that the arithmetic mean of

λ ^{(n)} converges to the integral of w .^[3]

First Szegő theorem

The first Szegő theorem^{[1] [2] [4]} states that, if right-hand side of holds and

w \geq 0 , then

holds for

w > 0 and w\in L _{1} . The RHS of is the geometric mean of w (well-defined by the arithmetic-geometric mean inequality).

Second Szegő theorem

Let

\widehat c _{k} be the Fourier coefficient of log w \in L ^{1} , written as\widehat c _{k} = (1)/(2 \pi) \int ^{2 \pi} _{0} log (w(\theta)) e ^{-ik\theta} d\theta

The second (or strong) Szegő theorem^{[1] [5]} states that, if

w \geq 0 , then\lim _{n} (\det T _{n(w)})/(e ^{(n +1 ) \widehat c _{0}}) = \exp \left( \sum ^{inf ty} _{k =1} k \left| \widehat c ^{2} _{k\right|} \right).

See also

• Trigonometric moment problem
• Verblunsky's theorem

Notes and References

1. Book: 1071374. Böttcher. Albrecht. Silbermann. Bernd. Analysis of Toeplitz operators. Springer-Verlag. Berlin. 1990. 3-540-52147-X. Toeplitz determinants. 525.
2. Book: Simon. Barry. Szegő's Theorem and Its Descendants: Spectral Theory for L² Perturbations of Orthogonal Polynomials. Princeton University Press. Princeton. 2011. 978-0-691-14704-8.
3. Robert M.. Gray. Toeplitz and Circulant Matrices: A Review. Foundations and Trends in Signal Processing. 2006.
4. G.. Szegő. Ein Grenzwertsatz über die Toeplitzschen Determinanten einer reellen positiven Funktion. Math. Ann.. 76. 1915. 490 - 503. 10.1007/BF01458220. 4. 123034653 .
5. 0051961. Szegő. G.. On certain Hermitian forms associated with the Fourier series of a positive function. Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.]. 1952. 228 - 238.