Szegő limit theorem

E451539

mathematical theorem result in analysis result in operator theory

The Szegő limit theorem is a fundamental result in analysis and operator theory that describes the asymptotic behavior of determinants of large Toeplitz matrices in terms of the symbol’s integral.

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All labels observed (8)

Label	Occurrences
Szegő limit theorem canonical	4
Szegő limit theorem in analysis	1
Szegő limit theorems	1
Szegő–Kolmogorov formula	1
block Szegő limit theorem	1
multidimensional Szegő limit theorem	1
strong Szegő limit theorem	1
weak Szegő limit theorem	1

How this entity was disambiguated

This entity first appeared as the object of triple T4552551 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Szegő limit theorem
Context triple: [Gábor Szegő, notableWork, Szegő limit theorem]

A. Hadamard three-circle theorem
The Hadamard three-circle theorem is a result in complex analysis that describes how the maximum modulus of a holomorphic function behaves logarithmically between three concentric circles in the complex plane.
B. Bernstein inequalities
Bernstein inequalities are fundamental results in approximation theory and probability that provide bounds on the derivatives or deviations of functions and random variables under certain smoothness or moment conditions.
C. Carathéodory–Fejér interpolation
Carathéodory–Fejér interpolation is a classical result in complex analysis and approximation theory that concerns constructing analytic functions, typically with bounded or positive real part, that match prescribed initial Taylor coefficients.
D. Cauchy–Hadamard theorem
The Cauchy–Hadamard theorem is a fundamental result in complex analysis that characterizes the radius of convergence of a power series in terms of the growth rate of its coefficients.
E. Isserlis’ theorem in probability theory
Isserlis’ theorem in probability theory is a result that expresses higher-order moments of jointly Gaussian random variables in terms of sums of products of their pairwise covariances.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Szegő limit theorem
Target entity description: The Szegő limit theorem is a fundamental result in analysis and operator theory that describes the asymptotic behavior of determinants of large Toeplitz matrices in terms of the symbol’s integral.

A. Hadamard three-circle theorem
The Hadamard three-circle theorem is a result in complex analysis that describes how the maximum modulus of a holomorphic function behaves logarithmically between three concentric circles in the complex plane.
B. Bernstein inequalities
Bernstein inequalities are fundamental results in approximation theory and probability that provide bounds on the derivatives or deviations of functions and random variables under certain smoothness or moment conditions.
C. Carathéodory–Fejér interpolation
Carathéodory–Fejér interpolation is a classical result in complex analysis and approximation theory that concerns constructing analytic functions, typically with bounded or positive real part, that match prescribed initial Taylor coefficients.
D. Cauchy–Hadamard theorem
The Cauchy–Hadamard theorem is a fundamental result in complex analysis that characterizes the radius of convergence of a power series in terms of the growth rate of its coefficients.
E. Isserlis’ theorem in probability theory
Isserlis’ theorem in probability theory is a result that expresses higher-order moments of jointly Gaussian random variables in terms of sums of products of their pairwise covariances.
F. None of above. chosen

Statements (49)

Predicate	Object
instanceOf	mathematical theorem ⓘ result in analysis ⓘ result in operator theory ⓘ
appliesTo	large Toeplitz matrices ⓘ
assumes	integrable symbol on the unit circle ⓘ nonvanishing symbol on the unit circle ⓘ
concerns	Toeplitz matrices ⓘ Toeplitz operators ⓘ linked to: Toeplitz matrices asymptotic behavior of determinants ⓘ determinants of Toeplitz matrices ⓘ symbols of Toeplitz matrices ⓘ
describes	asymptotics of log determinants of Toeplitz matrices ⓘ
domain	infinite-dimensional analysis ⓘ spectral theory ⓘ
field	complex analysis ⓘ functional analysis ⓘ harmonic analysis ⓘ matrix analysis ⓘ operator theory ⓘ probability theory ⓘ
generalizationOf	strong law of large numbers for eigenvalue distributions of Toeplitz matrices ⓘ
hasFormulation	limit of (1/n) log det T_n(f) equals average of log f ⓘ
hasGeneralization	results for block Toeplitz matrices ⓘ results for multidimensional symbols ⓘ results for non-Hermitian Toeplitz matrices ⓘ
hasVariant	block Szegő limit theorem ⓘ linked to: Szegő limit theorem multidimensional Szegő limit theorem ⓘ linked to: Szegő limit theorem strong Szegő limit theorem ⓘ linked to: Szegő limit theorem weak Szegő limit theorem ⓘ linked to: Szegő limit theorem
historicalPeriod	20th century mathematics ⓘ
implies	asymptotic distribution of eigenvalues of Toeplitz matrices ⓘ
influenced	development of Toeplitz operator theory ⓘ
involves	integral of log of the symbol over the unit circle ⓘ limit of normalized log determinants ⓘ
languageOfOriginalPublication	German ⓘ
namedAfter	Gábor Szegő ⓘ
relatedTo	Fisher–Hartwig conjecture ⓘ Szegő–Kolmogorov formula ⓘ linked to: Szegő limit theorem Wiener–Hopf factorization ⓘ linked to: Wiener–Hopf equations prediction theory of stationary processes ⓘ
relates	determinants of Toeplitz matrices to integrals of their symbols ⓘ
usedIn	information theory ⓘ random matrix theory ⓘ signal processing ⓘ statistical mechanics ⓘ time series analysis ⓘ
usesConcept	Fourier coefficients ⓘ Fourier series ⓘ logarithm of the symbol ⓘ

How these facts were elicited