Szegő limit theorem
E451539
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Szegő limit theorem canonical | 1 |
| Szegő limit theorem in analysis | 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
NERFINISHED
ⓘ
Toeplitz operators NERFINISHED ⓘ 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
NERFINISHED
ⓘ
multidimensional Szegő limit theorem NERFINISHED ⓘ strong Szegő limit theorem NERFINISHED ⓘ weak Szegő limit theorem NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| relatedTo |
Fisher–Hartwig conjecture
NERFINISHED
ⓘ
Szegő–Kolmogorov formula NERFINISHED ⓘ Wiener–Hopf factorization NERFINISHED ⓘ 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 NERFINISHED ⓘ logarithm of the symbol ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Szegő limit theorem Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Szegő limit theorem in analysis