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.
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 NERFINISHED ⓘ 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 NERFINISHED ⓘ |
| normalizationCondition | ∫_{∂D} P(x,ξ) dσ(ξ) = 1 for each interior point x ⓘ |
| relatedTo |
Dirichlet problem
NERFINISHED
ⓘ
Fourier series NERFINISHED ⓘ Green function ⓘ Laplace equation NERFINISHED ⓘ Poisson integral NERFINISHED ⓘ 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 ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.