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 (1)
| Label | Occurrences |
|---|---|
| Stieltjes transform canonical | 1 |
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
integral transform
ⓘ
mathematical concept ⓘ |
| alsoKnownAs |
Cauchy–Stieltjes transform
NERFINISHED
ⓘ
Stieltjes–Cauchy transform NERFINISHED ⓘ |
| appliesTo |
empirical spectral distributions of random matrices
ⓘ
spectral measures of self-adjoint operators ⓘ |
| associatedWith |
Nevanlinna–Pick functions
NERFINISHED
ⓘ
Stieltjes moment problem NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| input |
Borel measure on the real line
ⓘ
probability measure on the real line ⓘ |
| namedAfter | Thomas Joannes Stieltjes NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
Fourier transform NERFINISHED ⓘ Hilbert transform ⓘ Laplace transform NERFINISHED ⓘ 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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.