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In mathematics, and specifically in
potential theory In mathematics and mathematical physics, potential theory is the study of harmonic functions. The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely grav ...
, the Poisson kernel is an
integral kernel In mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in ...
, used for solving the two-dimensional
Laplace equation In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as \nabla^2\! f = 0 or \Delta f = 0, where \Delta = \nab ...
, given
Dirichlet boundary condition In the mathematical study of differential equations, the Dirichlet (or first-type) boundary condition is a type of boundary condition, named after Peter Gustav Lejeune Dirichlet (1805–1859). When imposed on an ordinary or a partial differential ...
s on the
unit disk In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1: :D_1(P) = \.\, The closed unit disk around ''P'' is the set of points whose ...
. The kernel can be understood as the
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
of the
Green's function In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if \operatorname is the linear differentia ...
for the Laplace equation. It is named for Siméon Poisson. Poisson kernels commonly find applications in
control theory Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a ...
and two-dimensional problems in
electrostatics Electrostatics is a branch of physics that studies electric charges at rest ( static electricity). Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for amb ...
. 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 In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by th ...
, the Poisson kernel for the unit disc is given by P_r(\theta) = \sum_^\infty r^e^ = \frac = \operatorname\left(\frac\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 D = \ is the open
unit disc In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1: :D_1(P) = \.\, The closed unit disk around ''P'' is the set of points whose d ...
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 u(re^) = \frac\int_^\pi P_r(\theta-t)f(e^) \, \mathrmt, \quad 0 \le r < 1 is
harmonic A harmonic is a wave with a frequency that is a positive integer multiple of the '' fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', the ...
in D and has a radial limit that agrees with ''f''
almost everywhere In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
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 In functional analysis and related areas of mathematics, the group algebra is any of various constructions to assign to a locally compact group an operator algebra (or more generally a Banach algebra), such that representations of the algebra ar ...
''L''1(T). As linear operators, they tend to the
Dirac delta function In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
pointwise on ''Lp''(T). By the
maximum principle In the mathematical fields of partial differential equations and geometric analysis, the maximum principle is any of a collection of results and techniques of fundamental importance in the study of elliptic and parabolic differential equations. ...
, ''u'' is the only such harmonic function on ''D''. Convolutions with this approximate unit gives an example of a
summability kernel In mathematics, a summability kernel is a family or sequence of periodic integrable functions satisfying a certain set of properties, listed below. Certain kernels, such as the Fejér kernel, are particularly useful in Fourier analysis. Summability ...
for the
Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
of a function in ''L''1(T) . Let ''f'' ∈ ''L''1(T) have Fourier series . After the
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
, convolution with ''P''''r''(''θ'') becomes multiplication by the sequence ∈ ''ℓ''1(Z). Taking the inverse Fourier transform of the resulting product gives the Abel means ''Arf'' of ''f'': A_r f(e^) = \sum _ f_k r^ e^. Rearranging this
absolutely convergent In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is sa ...
series shows that ''f'' is the boundary value of ''g'' + ''h'', where ''g'' (resp. ''h'') is a
holomorphic In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivati ...
(resp.
antiholomorphic In mathematics, antiholomorphic functions (also called antianalytic functionsEncyclopedia of Mathematics, Springer and The European Mathematical Society, https://encyclopediaofmath.org/wiki/Anti-holomorphic_function, As of 11 September 2020, This ...
) function on ''D''. When one also asks for the harmonic extension to be holomorphic, then the solutions are elements of a
Hardy space In complex analysis, the Hardy spaces (or Hardy classes) ''Hp'' are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, because of the paper . I ...
. 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 In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vect ...
(for 1 ≤ ''p'' ≤ ∞), so is ''Hp''(T).


On the upper half-plane

The
unit disk In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1: :D_1(P) = \.\, The closed unit disk around ''P'' is the set of points whose ...
may be conformally mapped to the
upper half-plane In mathematics, the upper half-plane, \,\mathcal\,, is the set of points in the Cartesian plane with > 0. Complex plane Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds t ...
by means of certain
Möbius transformation In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form f(z) = \frac of one complex variable ''z''; here the coefficients ''a'', ''b'', ''c'', ''d'' are complex numbers satisfying ''ad' ...
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 u(x+iy)=\int_^\infty P_y(x-t)f(t) \, dt, \qquad y > 0. The kernel itself is given by P_y(x)=\frac\frac . Given a function f \in L^p(\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, \, u\, _=\, f\, _ Thus, again, the Hardy space ''Hp'' on the upper half-plane is a
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vect ...
, and, in particular, its restriction to the real axis is a closed subspace of L^p(\R). The situation is only analogous to the case for the unit disk; the
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides w ...
for 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 \R^n, the Poisson kernel takes the form P(x,\zeta) = \frac where x\in B_r, \zeta\in S (the surface of B_), and \omega _ 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 P x) = \int_S u(\zeta)P(x,\zeta) \, d\sigma(\zeta). It can be shown that ''P'' 'u''''x'') is harmonic on the ball B_ 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 \R^ by (t,x) = (t,x_1,\dots,x_n). The upper half-space is the set defined by H^ = \left\. The Poisson kernel for ''H''''n''+1 is given by P(t,x) = c_n\frac where c_n = \frac. The Poisson kernel for the upper half-space appears naturally as the
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
of the
Abel transform In mathematics, the Abel transform,N. H. Abel, Journal für die reine und angewandte Mathematik, 1, pp. 153–157 (1826). named for Niels Henrik Abel, is an integral transform often used in the analysis of spherically symmetric or axially symmetri ...
K(t,\xi) = e^ in which ''t'' assumes the role of an auxiliary parameter. To wit, P(t,x) = \mathcal(K(t,\cdot))(x) = \int_ e^ e^\,d\xi. In particular, it is clear from the properties of the Fourier transform that, at least formally, the convolution P t,x) = (t,\cdot)*ux) is a solution of Laplace's equation in the upper half-plane. One can also show that as , in a suitable sense.


See also

* Schwarz integral formula


References

* *. *. *. *. * *{{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, isbn=3-540-41160-7. Fourier analysis Harmonic functions Potential theory ru:Ядро Пуассона