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{{Short description|Mathematical concept}}
{{More footnotes|date=February 2021}}

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 condition]]s 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.

==Two-dimensional Poisson kernels==

=== On the unit disc ===

In the [[complex plane]], the Poisson kernel for the unit disc <ref>{{Cite web |title=complex analysis - Deriving the Poisson Integral Formula from the Cauchy Integral Formula |url=https://math.stackexchange.com/q/1197966 |access-date=2022-08-21 |website=Mathematics Stack Exchange |language=en}}</ref> is given by
<math display="block">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 \le r < 1.</math>

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 <math>D = \{z:|z|<1\}</math> is the open [[unit disc]] in '''C''', '''T''' is the boundary of the disc, and ''f'' a function on '''T''' that lies in ''L''1('''T'''), then the function ''u'' given by
<math display="block">u(re^{i\theta}) = \frac{1}{2\pi}\int_{-\pi}^\pi P_r(\theta-t)f(e^{it}) \, \mathrm{d}t, \quad 0 \le r < 1 </math>
is [[Harmonic function|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 {{math|''r'' → 1}}, the functions {{math|''P''''r''(''θ'')}} form an [[approximate identity|approximate unit]] in the [[convolution algebra]] ''L''1('''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 ''L''1('''T''') {{harv|Katznelson|1976}}. Let ''f'' ∈ ''L''1('''T''') have Fourier series {''fk''}. After the [[Fourier transform]], convolution with ''P''''r''(''θ'') becomes multiplication by the sequence {''r|k|''} ∈ ''ℓ''1('''Z''').{{elucidate|date=January 2015}} Taking the inverse Fourier transform of the resulting product {''r|k|fk''} gives the [[Abel's theorem|Abel means]] ''Arf'' of ''f'':
<math display="block"> A_r f(e^{2 \pi i x}) = \sum _{k \in \Z} f_k r^{|k|} e^{2 \pi i k x}.</math>

Rearranging this [[absolutely convergent]] series shows that ''f'' is the boundary value of ''g'' + ''h'', where ''g'' (resp. ''h'') is a [[Holomorphic function|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 ''Hp''(''z'') may be called ''Hp''('''T'''). It is a closed subspace of ''Lp''('''T''') (at least for ''p'' ≥ 1). Since ''Lp''('''T''') is a [[Banach space]] (for 1 ≤ ''p'' ≤ ∞), so is ''Hp''('''T''').

===On the upper half-plane===
The [[unit disk]] may be [[conformal map|conformally mapped]] to the [[upper half-plane]] by means of certain [[Möbius transformation]]s. 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
<math display="block">u(x+iy)=\int_{-\infty}^\infty P_y(x-t)f(t) \, dt, \qquad y > 0.</math>

The kernel itself is given by
<math display="block">P_y(x)=\frac{1}{\pi}\frac {y}{x^2 + y^2}.</math>

Given a function <math>f \in L^p(\R)</math>, the [[Lp space|''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, <math>H^p,</math> and in particular,
<math display="block">\|u\|_{H^p}=\|f\|_{L^p}</math>

Thus, again, the Hardy space ''Hp'' on the upper half-plane is a [[Banach space]], and, in particular, its restriction to the real axis is a closed subspace of <math>L^p(\R).</math> 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.

==On the ball==
For the ball of radius <math>r, B_r \subset \R^n,</math> the Poisson kernel takes the form
<math display="block">P(x,\zeta) = \frac{r^2-|x|^2}{r\omega _{n-1}|x-\zeta|^n}</math>
where <math>x\in B_r, \zeta\in S</math> (the surface of <math>B_{r}</math>), and <math>\omega _{n-1}</math> is the [[Unit sphere#Volume and area|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
<math display="block">P[u](x) = \int_S u(\zeta)P(x,\zeta) \, d\sigma(\zeta).</math>

It can be shown that ''P''[''u''](''x'') is harmonic on the ball <math>B_{r}</math> and that ''P''[''u''](''x'') extends to a continuous function on the closed ball of radius ''r'', and the boundary function coincides with the original function ''u''.

==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 <math>\R^{n+1}</math> by
<math display="block">(t,x) = (t,x_1,\dots,x_n).</math>
The upper half-space is the set defined by
<math display="block">H^{n+1} = \left\{ (t;x) \in \R^{n+1} \mid t>0 \right\}.</math>
The Poisson kernel for ''H''''n''+1 is given by
<math display="block">P(t,x) = c_n\frac{t}{\left(t^2 + \left\|x\right\|^2\right)^{(n+1)/2}}</math>
where
<math display="block">c_n = \frac{\Gamma[(n+1)/2]}{\pi^{(n+1)/2}}.</math>

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,
<math display="block">P(t,x) = \mathcal{F}(K(t,\cdot))(x) = \int_{\R^n} e^{-2\pi t|\xi|} e^{-2\pi i \xi\cdot x}\,d\xi.</math>
In particular, it is clear from the properties of the Fourier transform that, at least formally, the convolution
<math display="block">P[u](t,x) = [P(t,\cdot)*u](x)</math>
is a solution of Laplace's equation in the upper half-plane. One can also show that as {{math|''t'' → 0}}, {{math|''P''[''u''](''t'',''x'') → ''u''(''x'')}} in a suitable sense.

== See also ==

* [[Schwarz integral formula]]

==References==
{{Reflist}}
*{{citation | first=Yitzhak | last=Katznelson | authorlink=Yitzhak Katznelson | title=An introduction to Harmonic Analysis | year=1976 | publisher=Dover | isbn=0-486-63331-4}}
*{{citation | first = John B. |last=Conway | authorlink = John B. Conway | title = Functions of One Complex Variable I | publisher = Springer-Verlag | year = 1978 | isbn=0-387-90328-3 }}.
*{{citation | first1 = S. |last1=Axler|authorlink1=Sheldon Axler|first2=Paul | last2=Bourdon|first3=Wade|last3=Ramey | title = Harmonic Function Theory | publisher = Springer-Verlag | year = 1992 | isbn=0-387-95218-7 }}.
*{{citation | first = Frederick W. |last = King | title = Hilbert Transforms Vol. I |publisher = Cambridge University Press | year = 2009 | isbn = 978-0-521-88762-5 }}.
*{{citation | first1=Elias | last1=Stein | authorlink1=Elias Stein | first2=Guido | last2=Weiss | authorlink2=Guido Weiss | title=Introduction to Fourier Analysis on Euclidean Spaces | publisher=Princeton University Press | year=1971 | isbn= 0-691-08078-X | url-access=registration | url=https://archive.org/details/introductiontofo0000stei}}.
* {{MathWorld | urlname=PoissonKernel | title=Poisson Kernel}}
*{{citation|author1-link=David Gilbarg|author2-link=Neil Trudinger|first1=D.|last1=Gilbarg|first2=N.|last2=Trudinger| title=Elliptic Partial Differential Equations of Second Order|date=12 January 2001 |publisher=Springer | isbn=3-540-41160-7}}.

[[Category:Fourier analysis]]
[[Category:Harmonic functions]]
[[Category:Potential theory]]

[[ru:Ядро Пуассона]]

Poisson kernel - Revision history

imported>D.Lazard: /* On the ball */ fixing brocken anchor