Poisson kernel

In mathematics, and specifically inpotential theory,thePoisson kernelis anintegral kernel,used for solving the two-dimensionalLaplace equation,givenDirichlet boundary conditionson theunit disk.The kernel can be understood as thederivativeof theGreen's functionfor the Laplace equation. It is named forSiméon Poisson.

Poisson kernels commonly find applications incontrol theoryand two-dimensional problems inelectrostatics. In practice, the definition of Poisson kernels are often extended ton-dimensional problems.

Two-dimensional Poisson kernels

On the unit disc

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

This can be thought of in two ways: either as a function ofrandθ,or as a family of functions ofθindexed byr.

If$D=\{z:|z|<1\}$is the openunit discinC,Tis the boundary of the disc, andfa function onTthat lies inL¹(T), then the functionugiven by $$u(re^{i\theta })={\frac {1}{2\pi }}\int _{-\pi }^{\pi }P_{r}(\theta -t)f(e^{it})\,\mathrm {d} t,\quad 0\leq r<1$$ isharmonicinDand has a radial limit that agrees withfalmost everywhereon the boundaryTof the disc.

That the boundary value ofuisfcan be argued using the fact that asr→ 1,the functionsP_r(θ)form anapproximate unitin theconvolution algebraL¹(T). As linear operators, they tend to theDirac delta functionpointwise onL^p(T). By themaximum principle,uis the only such harmonic function onD.

Convolutions with this approximate unit gives an example of asummability kernelfor theFourier seriesof a function inL¹(T) (Katznelson 1976). Letf∈L¹(T) have Fourier series {f_k}. After theFourier transform,convolution withP_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 theAbel meansA_rfoff: $$A_{r}f(e^{2\pi ix})=\sum _{k\in \mathbb {Z} }f_{k}r^{|k|}e^{2\pi ikx}.$$

Rearranging thisabsolutely convergentseries shows thatfis the boundary value ofg+h,whereg(resp.h) is aholomorphic(resp.antiholomorphic) function onD.

When one also asks for the harmonic extension to be holomorphic, then the solutions are elements of aHardy space.This is true when the negative Fourier coefficients offall 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 onTof functions inH^p(z) may be calledH^p(T). It is a closed subspace ofL^p(T) (at least forp≥ 1). SinceL^p(T) is aBanach space(for 1 ≤p≤ ∞), so isH^p(T).

On the upper half-plane

Theunit diskmay beconformally mappedto theupper half-planeby means of certainMöbius 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 $$u(x+iy)=\int _{-\infty }^{\infty }P_{y}(x-t)f(t)\,dt,\qquad y>0.$$

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

Given a function$f\in L^{p}(\mathbb {R} )$,theL^pspaceof integrable functions on the real line,ucan be understood as a harmonic extension offinto the upper half-plane. In analogy to the situation for the disk, whenuis holomorphic in the upper half-plane, thenuis an element of the Hardy space,$H^{p},$and in particular, $$\|u\|_{H^{p}}=\|f\|_{L^{p}}$$

Thus, again, the Hardy spaceH^pon the upper half-plane is aBanach space,and, in particular, its restriction to the real axis is a closed subspace of$L^{p}(\mathbb {R} ).$The situation is only analogous to the case for the unit disk; theLebesgue measurefor the unit circle is finite, whereas that for the real line is not.

On the ball

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

Then, ifu(x) is a continuous function defined onS,the corresponding Poisson integral is the functionP[u](x) defined by $$P[u](x)=\int _{S}u(\zeta )P(x,\zeta )\,d\sigma (\zeta ).$$

It can be shown thatP[u](x) is harmonic on the ball$B_{r}$and thatP[u](x) extends to a continuous function on the closed ball of radiusr,and the boundary function coincides with the original functionu.

On the upper half-space

An expression for the Poisson kernel of anupper half-spacecan also be obtained. Denote the standard Cartesian coordinates of$\mathbb {R} ^{n+1}$by $$(t,x)=(t,x_{1},\dots,x_{n}).$$ The upper half-space is the set defined by $$H^{n+1}=\left\{(t;x)\in \mathbb {R} ^{n+1}\mid t>0\right\}.$$ The Poisson kernel forHⁿ⁺¹is given by $$P(t,x)=c_{n}{\frac {t}{\left(t^{2}+\left\|x\right\|^{2}\right)^{(n+1)/2}}}$$ where $$c_{n}={\frac {\Gamma [(n+1)/2]}{\pi ^{(n+1)/2}}}.$$

The Poisson kernel for the upper half-space appears naturally as theFourier transformof theAbel transform in whichtassumes the role of an auxiliary parameter. To wit, $$P(t,x)={\mathcal {F}}(K(t,\cdot ))(x)=\int _{\mathbb {R} ^{n}}e^{-2\pi t|\xi |}e^{-2\pi i\xi \cdot x}\,d\xi.$$ In particular, it is clear from the properties of the Fourier transform that, at least formally, the convolution $$P[u](t,x)=[P(t,\cdot )*u](x)$$ is a solution of Laplace's equation in the upper half-plane. One can also show that ast→ 0,P[u](t,x) →u(x)in a suitable sense.

See also

• Schwarz integral formula

References

• Katznelson, Yitzhak(1976),An introduction to Harmonic Analysis,Dover,ISBN0-486-63331-4
• Conway, John B.(1978),Functions of One Complex Variable I,Springer-Verlag,ISBN0-387-90328-3.
• Axler, S.;Bourdon, Paul; Ramey, Wade (1992),Harmonic Function Theory,Springer-Verlag,ISBN0-387-95218-7.
• King, Frederick W. (2009),Hilbert Transforms Vol. I,Cambridge University Press,ISBN978-0-521-88762-5.
• Stein, Elias;Weiss, Guido(1971),Introduction to Fourier Analysis on Euclidean Spaces,Princeton University Press,ISBN0-691-08078-X.
• Weisstein, Eric W."Poisson Kernel".MathWorld.
• Gilbarg, D.;Trudinger, N.(12 January 2001),Elliptic Partial Differential Equations of Second Order,Springer,ISBN3-540-41160-7.