View source for Szegő limit theorems

In [[Mathematical analysis|mathematical analysis]], the '''Szegő limit theorems''' describe the asymptotic behaviour of the [[Determinant|determinant]]s of large [[Toeplitz matrix|Toeplitz matrices]].<ref name=bs>{{cite book|mr=1071374|last1=Böttcher|first1=Albrecht|last2=Silbermann|first2=Bernd|title=Analysis of Toeplitz operators|publisher=Springer-Verlag|location=Berlin|year=1990|isbn=3-540-52147-X|chapter=Toeplitz determinants|page=525}}</ref><ref>{{eom|id=Szegö_limit_theorems|first=T.|last=Ehrhardt|first2=B.|last2=Silbermann}}</ref><ref name=Simon2010>{{cite book|last1=Simon|first1=Barry|title=Szegő's Theorem and Its Descendants: Spectral Theory for L<sup>2</sup> Perturbations of Orthogonal Polynomials|publisher=Princeton University Press|location=Princeton|year=2011|isbn=978-0-691-14704-8}}</ref> They were first proved by [[Biography:Gábor Szegő|Gábor Szegő]].
==Notation==

Let <math>w</math> be a [[Fourier series]] with Fourier coefficients <math>c_k</math>, relating to each other as
:<math> w(\theta) = \sum_{k=-\infty}^{\infty}  c_k e^{i k \theta}, \qquad \theta \in [0,2\pi],  </math>
:<math> c_k = \frac{1}{2\pi} \int_0^{2\pi} w(\theta) e^{-ik\theta} \, d\theta, </math>

such that the <math>n\times n</math> Toeplitz matrices <math>T_n(w) = \left(c_{k-l}\right)_{0\leq k,l \leq n-1}</math>  are [[Hermitian matrix|Hermitian]], i.e., if <math>T_n(w)=T_n(w)^\ast</math> then <math>c_{-k}=\overline{c_k}</math>. Then both <math>w</math> and eigenvalues <math>(\lambda_m^{(n)})_{0\leq m \leq n-1}</math> are real-valued and the determinant of <math>T_n(w)</math> is given by
:<math>\det T_n(w) = \prod_{m=1}^{n-1} \lambda_m^{(n)}</math>.

==Szegő theorem==
Under suitable assumptions the Szegő theorem states that

:<math>\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</math>

for any function <math>F</math> that is continuous on the range of <math>w</math>. In particular
{{NumBlk|:|<math>\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</math>|{{EquationRef|1}}}}

such that the [[Arithmetic mean|arithmetic mean]] of <math>\lambda^{(n)}</math> converges to the integral of <math>w</math>.<ref>{{cite journal |first=Robert M.|last=Gray|title=Toeplitz and Circulant Matrices: A Review|journal = Foundations and Trends in Signal Processing|year=2006|url=https://ee.stanford.edu/~gray/toeplitz.pdf}}</ref>

===First Szegő theorem===
The first Szegő theorem<ref name=bs/><ref name=Simon2010/><ref>{{cite journal|first=G.|last=Szegő|title=Ein Grenzwertsatz über die Toeplitzschen Determinanten einer reellen positiven Funktion|journal=Math. Ann.|volume=76|year=1915|pages=490&ndash;503|doi=10.1007/BF01458220|issue=4|s2cid=123034653 |url=https://zenodo.org/record/2496405}}</ref> states that, if right-hand side of ({{EquationNote|1}}) holds and <math>w \geq 0</math>, then 
{{NumBlk|:|<math> \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)</math>|{{EquationRef|2}}}}

holds for <math>w > 0</math> and <math>w\in L_1</math>. The RHS of ({{EquationNote|2}}) is the [[Geometric mean|geometric mean]] of <math>w</math> (well-defined by the arithmetic-geometric mean inequality).

===Second Szegő theorem===

Let <math>\widehat c_k</math> be the Fourier coefficient of <math>\log w \in L^{1}</math>, written as
:<math>\widehat c_k = \frac{1}{2\pi} \int_0^{2\pi} \log (w(\theta)) e^{-ik\theta} \, d\theta</math>

The second (or strong) Szegő theorem<ref name=bs/><ref>{{cite journal|mr=0051961|last=Szegő|first=G.|title=On certain Hermitian forms associated with the Fourier series of a positive function|journal=Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.]|year=1952|pages=228&ndash;238}}</ref> states that, if <math>w \geq 0</math>, then

:<math> \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).</math>

==See also==
* [[Trigonometric moment problem]]
* [[Orthogonal polynomials on the unit circle#Szegő's theorem|Verblunsky's theorem]]

==References==

{{Reflist}}

{{DEFAULTSORT:Szego limit theorems}}
[[Category:Theorems in analysis]]
[[Category:Matrices]]
[[Category:Linear algebra]]

{{Sourceattribution|Szegő limit theorems}}