Poisson transformation

In mathematics , the Poisson transformation is a method for constructing harmonic functions on the unit disk . The integral that appears in this construction is called the Poisson integral and the integral kernel of this is called the Poisson kernel . The transformation, the integral and the integral kernel are named after the mathematician and physicist Siméon Denis Poisson .

Problem

A (limited) function is given on the unit circle , a (limited) harmonic function is sought on the unit disk whose values on the edge match the given function . ${\ displaystyle S ^ {1} = \ partial D ^ {2}}$ ${\ displaystyle D ^ {2}}$ ${\ displaystyle f}$

In other words: it is supposed to be the Dirichlet problem for the Laplace equation

{\ displaystyle \ Delta \ Phi = 0}

be solved on the circular disc.

construction

The Poisson core is the through

{\ displaystyle P (x, \ xi) = {\ frac {1- \ | x \ | ^ {2}} {\ | \ xi -x \ | ^ {2}}} \ \ forall x \ in D ^ {2}, \ xi \ in S ^ {1}}

given function.

The Poisson transformation is the integral transformation with integral kernel : a function becomes the function defined on ${\ displaystyle P}$ ${\ displaystyle f \ in L ^ {\ infty} (S ^ {1})}$ ${\ displaystyle D ^ {2}}$

{\ displaystyle Pf (x) = \ int _ {S ^ {1}} f (\ xi) P (x, \ xi) d \ sigma (\ xi) \ \ forall x \ in D ^ {2}}

assigned, wherein the uniform probability to designated. ${\ displaystyle d \ sigma}$ ${\ displaystyle S ^ {1}}$

One can show that is a bounded harmonic function. ${\ displaystyle Pf}$

Bijection

The Poisson transform creates a bijection between the set of bounded functions on and the set of bounded harmonic functions on . ${\ displaystyle S ^ {1}}$ ${\ displaystyle D ^ {2}}$

In other words: for every function there is a unique harmonic function with boundary values . ${\ displaystyle f \ in L ^ {\ infty} (S ^ {1})}$ ${\ displaystyle g \ in L ^ {\ infty} (D ^ {2})}$ ${\ displaystyle f}$

The bijection receives the norm. ${\ displaystyle L ^ {\ infty}}$

Generalizations

The Poisson transformation can be generalized to the n-dimensional unit sphere , in this case the Poisson kernel is for . ${\ displaystyle P (x, \ xi) = {\ frac {1- \ | x \ | ^ {2}} {\ | \ xi -x \ | ^ {n}}}}$ ${\ displaystyle x \ in D ^ {n}, \ xi \ in \ partial D ^ {n} = S ^ {n-1}}$

literature

Helgason, Sigurdur: Topics in harmonic analysis on homogeneous spaces. Progress in Mathematics, 13. Birkhauser, Boston, Mass., 1981. ISBN 3-7643-3051-1
Quint, J.-F .: An overview of Patterson-Sullivan theory , Workshop "The barycenter method", FIM, Zurich, May 2006 (Online)

Individual evidence

↑ Poisson integral . In: Guido Walz (Ed.): Lexicon of Mathematics . 1st edition. Spectrum Academic Publishing House, Mannheim / Heidelberg 2000, ISBN 3-8274-0439-8 .
^ Poisson core . In: Guido Walz (Ed.): Lexicon of Mathematics . 1st edition. Spectrum Academic Publishing House, Mannheim / Heidelberg 2000, ISBN 3-8274-0439-8 .

[1] Poisson integral . In: Guido Walz (Ed.): Lexicon of Mathematics . 1st edition. Spectrum Academic Publishing House, Mannheim / Heidelberg 2000, ISBN 3-8274-0439-8 .

[2] Poisson core . In: Guido Walz (Ed.): Lexicon of Mathematics . 1st edition. Spectrum Academic Publishing House, Mannheim / Heidelberg 2000, ISBN 3-8274-0439-8 .