Stieltjes transform
E898466
The Stieltjes transform is an integral transform that encodes a measure or distribution via a complex-analytic function, widely used in random matrix theory to study limiting spectral distributions and resolvents.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Stieltjes transform canonical | 3 |
| Cauchy–Stieltjes transform | 1 |
| Stieltjes inversion formula | 1 |
| Stieltjes–Cauchy transform | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10991198 — 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: Stieltjes transform Context triple: [random matrix theory, hasKeyConcept, Stieltjes transform]
-
A.
Mellin transforms
Mellin transforms are integral transforms that convert functions into complex-variable representations, playing a central role in analytic number theory by linking arithmetic functions to Dirichlet series and zeta functions.
-
B.
Hilbert transform
The Hilbert transform is an integral transform that produces the harmonic conjugate of a real-valued function, playing a central role in signal processing, harmonic analysis, and the theory of analytic signals.
-
C.
Szegő limit theorem
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.
-
D.
Mittag-Leffler function
The Mittag-Leffler function is a complex function that generalizes the exponential function and plays a central role in fractional calculus and the theory of differential and integral equations.
-
E.
Sommerfeld-Watson transform
The Sommerfeld-Watson transform is a complex-analysis technique that converts discrete sums over angular momentum into contour integrals, widely used in scattering theory and Regge theory to study analytic properties of amplitudes.
- 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: Stieltjes transform Target entity description: The Stieltjes transform is an integral transform that encodes a measure or distribution via a complex-analytic function, widely used in random matrix theory to study limiting spectral distributions and resolvents.
-
A.
Mellin transforms
Mellin transforms are integral transforms that convert functions into complex-variable representations, playing a central role in analytic number theory by linking arithmetic functions to Dirichlet series and zeta functions.
-
B.
Hilbert transform
The Hilbert transform is an integral transform that produces the harmonic conjugate of a real-valued function, playing a central role in signal processing, harmonic analysis, and the theory of analytic signals.
-
C.
Szegő limit theorem
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.
-
D.
Mittag-Leffler function
The Mittag-Leffler function is a complex function that generalizes the exponential function and plays a central role in fractional calculus and the theory of differential and integral equations.
-
E.
Sommerfeld-Watson transform
The Sommerfeld-Watson transform is a complex-analysis technique that converts discrete sums over angular momentum into contour integrals, widely used in scattering theory and Regge theory to study analytic properties of amplitudes.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
integral transform
ⓘ
mathematical concept ⓘ |
| alsoKnownAs |
Cauchy–Stieltjes transform
ⓘ
linked to:
Stieltjes transform
Stieltjes–Cauchy transform ⓘ
linked to:
Stieltjes transform
|
| appliesTo |
empirical spectral distributions of random matrices
ⓘ
spectral measures of self-adjoint operators ⓘ |
| associatedWith |
Nevanlinna–Pick functions
ⓘ
Stieltjes moment problem ⓘ |
| category | integral transforms in complex analysis ⓘ |
| centralIn |
analysis of Wigner matrices
ⓘ
analysis of sample covariance matrices ⓘ derivation of the Marchenko–Pastur law ⓘ derivation of the semicircle law ⓘ |
| definedOn | measures of bounded variation on the real line ⓘ |
| domain | complex plane minus support of the measure ⓘ |
| field |
analysis
ⓘ
complex analysis ⓘ functional analysis ⓘ measure theory ⓘ probability theory ⓘ random matrix theory ⓘ |
| generalizationOf | classical Cauchy integral of a density ⓘ |
| hasInverseOperation |
Stieltjes inversion formula
ⓘ
linked to:
Stieltjes transform
|
| input |
Borel measure on the real line
ⓘ
probability measure on the real line ⓘ |
| namedAfter | Thomas Joannes Stieltjes ⓘ |
| output |
analytic function
ⓘ
complex-valued function ⓘ |
| property |
admits inversion formulas
ⓘ
analytic off the support of the measure ⓘ decays as 1 over z at infinity for probability measures ⓘ determines the measure under mild conditions ⓘ |
| relatedTo |
Cauchy transform
ⓘ
Fourier transform ⓘ Hilbert transform ⓘ Laplace transform ⓘ resolvent of an operator ⓘ |
| usedFor |
encoding distributions as analytic functions
ⓘ
encoding measures as analytic functions ⓘ free probability theory ⓘ inversion to recover measures ⓘ moment problems ⓘ studying limiting spectral distributions ⓘ studying resolvents of operators ⓘ studying resolvents of random matrices ⓘ |
| usedIn |
free convolution calculations
ⓘ
local laws in random matrix theory ⓘ proofs of convergence of empirical spectral distributions ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Stieltjes transform Description of subject: The Stieltjes transform is an integral transform that encodes a measure or distribution via a complex-analytic function, widely used in random matrix theory to study limiting spectral distributions and resolvents.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Stieltjes transform
linked to: Stieltjes transform
linked to: Stieltjes transform