Poisson kernel

In mathematics, the Poisson transform is a process for the construction of harmonic functions on the unit disk. The integral, which appears in this construction is, Poisson integral and the integral core which is called Poisson core. Named both the transformation, the integral and the integral kernel after Siméon Denis Poisson mathematician and physicist.

Problem

Given a (limited ) function on the unit circle, the system searches a (limited ) harmonic function on the unit disk that have the same values ​​on the border with the given function.

In other words, it is the Dirichlet problem of the Laplace equation

Be achieved on the disc.

Construction

The Poisson kernel is obtained by

Given function.

The Poisson transform is the integral transform with integral kernel: a function is the function defined on

Associated with the uniform probability measure referred to.

It can be shown that a limited harmonic function.

Bijection

Poisson transform is a bijection between the amount of the limited functions and the limited amount of harmonic functions forth.

In other words: for each function there is a unique harmonic function with boundary values ​​.

The bijection given the standard.

Generalizations

The Poisson transform can be generalized to the n -dimensional unit sphere, in this case, is the Poisson kernel for.