Poisson kernel

 ( Fourier Analysis ) 

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.

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Two-dimensional Poisson kernels

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On the unit disc In the complex plane, the Poisson kernel for the unit disc is given by $$P\_r(\backslash theta)\; =\; \backslash sum\_\{n=-\backslash infty\}^\backslash infty\; r^$$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 \le 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 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 , the functions 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) . 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). Taking the inverse Fourier transform of the resulting product { r^|k|f_k} gives the Abel means A_rf of f: $$A\_r\; f(e^\{2\; \backslash pi\; i\; x\})\; =\; \backslash sum\; \_\{k\; \backslash in\; \backslash Z\}\; f\_k\; r^$$

e^{2 \pi i k x}.

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. In particular, the Poisson kernel is commonly used to demonstrate the equivalence of the Hardy spaces on the unit disk, and the unit circle.

The space of functions that are the limits on T of functions in H^p( z) may be called H^p( T). It is a closed subspace of L^p( T) (at least for p ≥ 1). Since L^p( T) is a Banach space (for 1 ≤  p ≤ ∞), so is H^p( T).

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On the upper half-plane The unit disk may be conformal map to the upper half-plane by means of certain Möbius transformations. Since the conformal map of a harmonic function is also harmonic, the Poisson kernel carries over to the upper half-plane. In this case, the Poisson integral equation takes the form $$u(x+iy)=\backslash int\_\{-\backslash infty\}^\backslash infty\; P\_y(x-t)f(t)\; \backslash ,\; dt,\; \backslash qquad\; y\; >\; 0.$$

The kernel itself is given by $$P\_y(x)=\backslash frac\{1\}\{\backslash pi\}\backslash frac\; \{y\}\{x^2\; +\; y^2\}.$$

Given a function $f\; \backslash in\; L^p(\backslash R)$, the Lp space of integrable functions on the real line, u can be understood as a harmonic extension of f into the upper half-plane. In analogy to the situation for the disk, when u is holomorphic in the upper half-plane, then u is an element of the Hardy space, $H^p,$ and in particular, $$\backslash |u\backslash |\_\{H^p\}=\backslash |f\backslash |\_\{L^p\}$$

Thus, again, the Hardy space H^p on the upper half-plane is a Banach space, and, in particular, its restriction to the real axis is a closed subspace of $L^p(\backslash R).$ The situation is only analogous to the case for the unit disk; the Lebesgue measure for the unit circle is finite, whereas that for the real line is not.

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On the ball For the ball of radius $r,\; B\_r\; \backslash subset\; \backslash R^n,$ the Poisson kernel takes the form $$P(x,\backslash zeta)\; =\; \backslash frac\{r^2-|x|^2\}\{r\backslash omega\; \_\{n-1\}|x-\backslash zeta|^n\}$$ where $x\backslash in\; B\_r,\; \backslash zeta\backslash in\; S$ (the surface of $B\_\{r\}$), and $\backslash omega\; \_\{n-1\}$ is the surface area of the unit ( n − 1)-sphere.

Then, if u( x) is a continuous function defined on S, the corresponding Poisson integral is the function P u( x) defined by $$Pu(x)\; =\; \backslash int\_S\; u(\backslash zeta)P(x,\backslash zeta)\; \backslash ,\; d\backslash sigma(\backslash zeta).$$

It can be shown that P u( x) is harmonic on the ball $B\_\{r\}$ and that Pu( x) extends to a continuous function on the closed ball of radius r, and the boundary function coincides with the original function  u.

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On the upper half-space An expression for the Poisson kernel of an upper half-space can also be obtained. Denote the standard Cartesian coordinates of $\backslash R^\{n+1\}$ by $$(t,x)\; =\; (t,x\_1,\backslash dots,x\_n).$$ The upper half-space is the set defined by $$H^\{n+1\}\; =\; \backslash left\backslash \{\; (t;x)\; \backslash in\; \backslash R^\{n+1\}\; \backslash mid\; t>0\; \backslash right\backslash \}.$$ The Poisson kernel for H ⁿ⁺¹ is given by $$P(t,x)\; =\; c\_n\backslash frac\{t\}\{\backslash left(t^2\; +\; \backslash left\backslash |x\backslash right\backslash |^2\backslash right)^\{(n+1)/2\}\}$$ where $$c\_n\; =\; \backslash frac\{\backslash Gamma(n+1)/2\}\{\backslash pi^\{(n+1)/2\}\}.$$

The Poisson kernel for the upper half-space appears naturally as the Fourier transform of the Abel transform in which t assumes the role of an auxiliary parameter. To wit, $$P(t,x)\; =\; \backslash mathcal\{F\}(K(t,\backslash cdot))(x)\; =\; \backslash int\_\{\backslash R^n\}\; e^\{-2\backslash pi\; t|\backslash xi|\}\; e^\{-2\backslash pi\; i\; \backslash xi\backslash cdot\; x\}\backslash ,d\backslash xi.$$ In particular, it is clear from the properties of the Fourier transform that, at least formally, the convolution $$Pu(t,x)\; =\; P(t,\backslash cdot)*u(x)$$ is a solution of Laplace's equation in the upper half-plane. One can also show that as , in a suitable sense.

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See also

• Schwarz integral formula

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