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 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 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 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 Pr(θ) form an approximate unit in the convolution algebra L1(T). As linear operators, they tend to the Dirac delta function pointwise on Lp(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) . Let f ∈ L1(T) have Fourier series {fk}. After the Fourier transform, convolution with Pr(θ) becomes multiplication by the sequence {r|k|} ∈ l1(Z). Taking the inverse Fourier transform of the resulting product {r|k|fk} gives the Abel means Arf of f: 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. When one also asks for the harmonic extension to be holomorphic, then the solutions are elements of a Hardy space. This is true when the negative Fourier coefficients of f all vanish.