Poisson kernel
E559804
The Poisson kernel is a fundamental function in harmonic analysis and potential theory used to represent harmonic functions inside a domain from their boundary values, especially in the unit disk and upper half-plane.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Poisson kernel canonical | 4 |
| Poisson kernel is positive and integrates to 1 | 1 |
| Poisson kernel on the real line | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5973627 — 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: Poisson kernel Context triple: [Siméon Denis Poisson, notableWork, Poisson kernel]
-
A.
Szegő kernel
The Szegő kernel is a fundamental reproducing kernel in complex analysis and operator theory, associated with Hardy spaces on the boundary of a domain and central to the study of orthogonal polynomials and boundary behavior of analytic functions.
-
B.
Dirichlet kernel
The Dirichlet kernel is a trigonometric polynomial that arises in Fourier series as the summation kernel for partial sums, playing a key role in analyzing convergence properties.
-
C.
Dirichlet problem
The Dirichlet problem is a fundamental boundary value problem in potential theory and partial differential equations, asking for a function that solves a specified PDE inside a domain while taking prescribed values on the domain’s boundary.
-
D.
Khinchin–Pollaczek formula
The Khinchin–Pollaczek formula is a result in probability theory and queueing theory that provides an explicit expression for the stationary waiting-time distribution in certain single-server queues.
-
E.
Koebe quarter theorem
The Koebe quarter theorem is a fundamental result in complex analysis stating that any univalent holomorphic function on the unit disk maps it onto a domain containing a disk of radius one quarter, providing a sharp bound on the size of the image.
- 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: Poisson kernel Target entity description: The Poisson kernel is a fundamental function in harmonic analysis and potential theory used to represent harmonic functions inside a domain from their boundary values, especially in the unit disk and upper half-plane.
-
A.
Szegő kernel
The Szegő kernel is a fundamental reproducing kernel in complex analysis and operator theory, associated with Hardy spaces on the boundary of a domain and central to the study of orthogonal polynomials and boundary behavior of analytic functions.
-
B.
Dirichlet kernel
The Dirichlet kernel is a trigonometric polynomial that arises in Fourier series as the summation kernel for partial sums, playing a key role in analyzing convergence properties.
-
C.
Dirichlet problem
The Dirichlet problem is a fundamental boundary value problem in potential theory and partial differential equations, asking for a function that solves a specified PDE inside a domain while taking prescribed values on the domain’s boundary.
-
D.
Khinchin–Pollaczek formula
The Khinchin–Pollaczek formula is a result in probability theory and queueing theory that provides an explicit expression for the stationary waiting-time distribution in certain single-server queues.
-
E.
Koebe quarter theorem
The Koebe quarter theorem is a fundamental result in complex analysis stating that any univalent holomorphic function on the unit disk maps it onto a domain containing a disk of radius one quarter, providing a sharp bound on the size of the image.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
kernel function
ⓘ
mathematical concept ⓘ |
| appearsIn |
representation formula for harmonic functions in the unit disk
ⓘ
representation formula for harmonic functions in the upper half-plane ⓘ |
| approximationProperty | Poisson integrals approximate boundary data in L^p spaces under suitable conditions ⓘ |
| belongsTo | class of positive harmonic kernels ⓘ |
| category | fundamental solution-type kernel for Laplace equation ⓘ |
| connectedTo |
Brownian motion hitting distribution
ⓘ
Hardy spaces ⓘ
linked to:
Hardy space
boundary behavior of harmonic functions ⓘ conformal mapping theory ⓘ |
| definedOn |
balls in Euclidean space
ⓘ
unit disk ⓘ upper half-plane ⓘ |
| field |
harmonic analysis
ⓘ
potential theory ⓘ |
| generalizationOf | Poisson kernel on the unit circle to higher dimensions ⓘ |
| hasFormula |
P(x,y) = (1/π) · y / (x^2 + y^2) for the upper half-plane
ⓘ
P_r(θ) = (1 - r^2) / (1 - 2r cos θ + r^2) for 0 ≤ r < 1 ⓘ |
| hasGeneralization |
Poisson kernel on smooth bounded domains
ⓘ
Poisson kernel on the unit ball in R^n ⓘ |
| hasIntegralRepresentation | u(x) = ∫_{∂D} P(x,ξ) f(ξ) dσ(ξ) for harmonic u in domain D ⓘ |
| hasProperty |
acts as an approximate identity
ⓘ
harmonic in the interior variable ⓘ integrates to 1 over the boundary ⓘ positive function ⓘ rotationally symmetric in the unit disk ⓘ translation invariant along the boundary of the upper half-plane ⓘ |
| namedAfter | Siméon Denis Poisson ⓘ |
| normalizationCondition | ∫_{∂D} P(x,ξ) dσ(ξ) = 1 for each interior point x ⓘ |
| relatedTo |
Dirichlet problem
ⓘ
Fourier series ⓘ Green function ⓘ Laplace equation ⓘ Poisson integral ⓘ boundary value problem ⓘ harmonic function ⓘ harmonic measure ⓘ |
| satisfies |
lim_{x→boundary} ∫ P(x,ξ) f(ξ) dσ(ξ) = f on suitable function spaces
ⓘ
Δ_x P(x,ξ) = 0 for interior variable x ⓘ |
| usedFor |
constructing harmonic measure
ⓘ
harmonic extension of boundary data ⓘ integral representation of harmonic functions ⓘ representing harmonic functions from boundary values ⓘ solving the Dirichlet problem ⓘ |
| usedIn |
complex analysis
ⓘ
partial differential equations ⓘ probability theory via Brownian motion ⓘ |
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: Poisson kernel Description of subject: The Poisson kernel is a fundamental function in harmonic analysis and potential theory used to represent harmonic functions inside a domain from their boundary values, especially in the unit disk and upper half-plane.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.