Poisson kernel

In mathematics, and specifically in potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disk. The kernel can be understood as the derivative of the Green's function for the Laplace equation. It is named for Siméon Poisson.

Poisson kernels commonly find applications in control theory and two-dimensional problems in electrostatics. In practice, the definition of Poisson kernels are often extended to n-dimensional problems.

In the complex plane, the Poisson kernel for the unit disc is given by $${\displaystyle P_{r}(\theta )=\sum _{n=-\infty }^{\infty }r^{|n|}e^{in\theta }={\frac {1-r^{2}}{1-2r\cos \theta +r^{2}}}=\operatorname {Re} \left({\frac {1+re^{i\theta }}{1-re^{i\theta }}}\right),\ \ \ 0\leq r<1.}$$

This can be thought of in two ways: either as a function of r and θ, or as a family of functions of θ indexed by r.

If ${\displaystyle D=\{z:|z|<1\}}$ is the open unit disc in C, T is the boundary of the disc, and f a function on T that lies in L¹(T), then the function u given by $${\displaystyle u(re^{i\theta })={\frac {1}{2\pi }}\int _{-\pi }^{\pi }P_{r}(\theta -t)f(e^{it})\,\mathrm {d} t,\quad 0\leq r<1}$$ is harmonic in D and has a radial limit that agrees with f almost everywhere on the boundary T of the disc.

That the boundary value of u is f can be argued using the fact that as r → 1, the functions P_r(θ) form an approximate unit in the convolution algebra L¹(T). As linear operators, they tend to the Dirac delta function pointwise on L^p(T). By the maximum principle, u is the only such harmonic function on D.

Convolutions with this approximate unit gives an example of a summability kernel for the Fourier series of a function in L¹(T) (Katznelson 1976). Let f ∈ L¹(T) have Fourier series {f_k}. After the Fourier transform, convolution with P_r(θ) becomes multiplication by the sequence {r^|k|} ∈ ℓ¹(Z).^{[further explanation needed]} Taking the inverse Fourier transform of the resulting product {r^|k|f_k} gives the Abel means A_rf of f: $${\displaystyle A_{r}f(e^{2\pi ix})=\sum _{k\in \mathbb {Z} }f_{k}r^{|k|}e^{2\pi ikx}.}$$

Rearranging this absolutely convergent series shows that f is the boundary value of g + h, where g (resp. h) is a holomorphic (resp. antiholomorphic) function on D.