Poisson kernel
Encyclopedia
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 disc. The kernel can be understood as the derivative
of the Green's function
for the Laplace equation. It is named for Siméon Poisson.
The Poisson kernel is important in complex analysis
because its integral against a function defined on the unit circle — the Poisson integral — gives the extension of a function defined on the unit circle to a harmonic function
on the unit disk. By definition, harmonic functions are solutions to Laplace's equation, and, in two dimensions, harmonic functions are equivalent to meromorphic function
s. Thus, the two-dimensional Dirichlet problem is essentially the same problem as that of finding a meromorphic extension of a function defined on a boundary.
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.
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 unit disc in C, and if f is a continuous function from the unit circle 
into R, then the function u given by
or equivalently by
is harmonic in D, and extends to a continuous function on
that agrees with f on the boundary of the disc.
It is common to restrict oneself to functions which are either square integrable or p-integrable
on the unit circle. When one also asks for the harmonic extension to be holomorphic, then the solutions are elements of a Hardy space
. In particular, the Poisson kernel is commonly used to demonstrate the equivalence of the Hardy space
s on the unit disk, and the unit circle.
In the study of Fourier series the Poisson kernel arises in the study of Abel means
for a Fourier series, and gives an example of a summability kernel
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
for
. The kernel itself is given by
Given a function
, the Lp space
of integrable functions on the real line, then 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
, and, in particular,
Thus, again, the Hardy space Hp on the upper half-place is a Banach space
, and, in particular, a closed subspace of
. 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.
, in Rn, the Poisson kernel takes the form
where
, 
(the surface of 
), and 
is the surface area of the unit sphere.
Then, if u(x) is a continuous function defined on S, the corresponding Poisson integral is the function P[u](x) defined by
It can be shown that P[u](x) is harmonic on the ball
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.
The upper half-space is the set defined by
The Poisson kernel for Hn+1 is given by
where
The Poisson kernel for the upper half-space appears naturally as the Fourier transform
of the Abel kernel
in which t assumes the role of an auxiliary parameter. To wit,
In particular, it is clear from the properties of the Fourier transform that, at least formally, the convolution
is a solution of Laplace's equation in the upper half-plane. One can also show easily that as t → 0, P[u](t,x) → u(x) in a weak sense.
Potential theory
In mathematics and mathematical physics, potential theory may be defined as the study of harmonic functions.- Definition and comments :The term "potential theory" was coined in 19th-century physics, when it was realized that the fundamental forces of nature could be modeled using potentials which...
, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary condition
Dirichlet boundary condition
In mathematics, the Dirichlet boundary condition is a type of boundary condition, named after Johann Peter Gustav Lejeune Dirichlet who studied under Cauchy and succeeded Gauss at University of Göttingen. When imposed on an ordinary or a partial differential equation, it specifies the values a...
s on the unit disc. The kernel can be understood as the derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
of the Green's function
Green's function
In mathematics, a Green's function is a type of function used to solve inhomogeneous differential equations subject to specific initial conditions or boundary conditions...
for the Laplace equation. It is named for Siméon Poisson.
The Poisson kernel is important in complex analysis
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
because its integral against a function defined on the unit circle — the Poisson integral — gives the extension of a function defined on the unit circle to a harmonic function
Harmonic function
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R which satisfies Laplace's equation, i.e....
on the unit disk. By definition, harmonic functions are solutions to Laplace's equation, and, in two dimensions, harmonic functions are equivalent to meromorphic function
Meromorphic function
In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function...
s. Thus, the two-dimensional Dirichlet problem is essentially the same problem as that of finding a meromorphic extension of a function defined on a boundary.
Poisson kernels commonly find applications in control theory
Control theory
Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...
and two-dimensional problems in electrostatics
Electrostatics
Electrostatics is the branch of physics that deals with the phenomena and properties of stationary or slow-moving electric charges....
.
In practice, the definition of Poisson kernels are often extended to n-dimensional problems.
On the unit disc
In the complex plane, the Poisson kernel for the unit disc is given byThis 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 unit disc in C, and if f is a continuous function from the unit circle into R, then the function u given byor equivalently by
is harmonic in D, and extends to a continuous function on
that agrees with f on the boundary of the disc.It is common to restrict oneself to functions which are either square integrable or p-integrable
Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
on the unit circle. When one also asks for the harmonic extension to be holomorphic, then the solutions are elements of a Hardy space
Hardy space
In complex analysis, the Hardy spaces 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...
. In particular, the Poisson kernel is commonly used to demonstrate the equivalence of the Hardy space
Hardy space
In complex analysis, the Hardy spaces 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...
s on the unit disk, and the unit circle.
In the study of Fourier series the Poisson kernel arises in the study of Abel means
Abel's theorem
In mathematics, Abel's theorem for power series relates a limit of a power series to the sum of its coefficients. It is named after Norwegian mathematician Niels Henrik Abel.-Theorem:...
for a Fourier series, and gives an example of a summability kernel
On the upper half-plane
The unit disk may be conformally mappedConformal map
In mathematics, a conformal map is a function which preserves angles. In the most common case the function is between domains in the complex plane.More formally, a 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
for
. The kernel itself is given byGiven a function
, the Lp spaceLp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
of integrable functions on the real line, then 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
, and, in particular,Thus, again, the Hardy space Hp on the upper half-place is a Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
, and, in particular, a closed subspace of
. The situation is only analogous to the case for the unit disk; the Lebesgue measureLebesgue measure
In measure theory, 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 with the standard measure of length, area, or volume. In general, it is also called...
for the unit circle is finite, whereas that for the real line is not.
On the ball
For the ball of radius r,, in Rn, the Poisson kernel takes the formwhere
, (the surface of ), and is the surface area of the unit sphere.Then, if u(x) is a continuous function defined on S, the corresponding Poisson integral is the function P[u](x) defined by
It can be shown that P[u](x) is harmonic on the ball
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 Rn+1 byThe upper half-space is the set defined by
The Poisson kernel for Hn+1 is given by
where
The Poisson kernel for the upper half-space appears naturally as the Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
of the Abel kernel
in which t assumes the role of an auxiliary parameter. To wit,
In particular, it is clear from the properties of the Fourier transform that, at least formally, the convolution
is a solution of Laplace's equation in the upper half-plane. One can also show easily that as t → 0, P[u](t,x) → u(x) in a weak sense.