Distribution (mathematics)

A distribution called in mathematics a particular type of functionals, ie an object from the functional analysis.

The theory of distributions makes it possible to define a kind of solutions to differential equations that are not sufficiently often differentiable, or not defined in the classical sense (see distributional solution). In this sense, distributions may be viewed as a generalization of the concept of the function. There are partial differential equations that have no classical solutions, but solutions in the distributional sense. The theory of distributions is therefore particularly in physics and in engineering noting that a lot of there problems under consideration namely lead to differential equations that could be solved only with the help of the theory of distributions.

The mathematician Laurent Schwartz was instrumental in the investigation of the theory of distributions. In 1950 he published the first systematic approach to this theory. For his work on the distributions he received the Fields Medal.

4.1 Continuous function as a producer
4.2 delta distribution
4.3 Dirac comb
4.4 Radon measures
4.5 Cauchy principal value of 1 / x
4.6 Oscillating integral

6.1 Restriction on a subset
6.2 support
6.3 Compact carrier
6.4 Singular carrier

7.1 Multiplication by a function
7.2 differentiation 7.2.1 motivation
7.2.2 definition
7.2.3 example

7.3.1 motivation
7.3.2 definition

7.5.1 definition
7.5.2 example
7.5.3 properties

7.6.1 definition
7.6.2 properties

8.1 Fourier transform
8.2 convolution

9.1 Solutions in the distribution sense
9.2 Example 9.2.1 Constant Functions
9.2.2 Poisson equation

History of Distribution Theory

In 1903 Jacques Hadamard introduced the central notion of the theory for the distribution functional. From today's perspective, a functional is a function that assigns a number of other functions. Hadamard was able to show that every continuous, linear functional as a limit of a sequence of integrals

Can be displayed. In this representation may limit and integral in general not be interchanged. In 1910 it was shown that every continuous, linear functional on the space of p- integrable functions, as

Can be represented by and. In this formulation, no limit must be formed and is uniquely determined. Therefore, the functional is often identified with the "Function". Then has two different meanings: On the one hand is meant as - "Function", on the other hand, it is equated with the functional.

The first, Paul Dirac employed in the 1920s in research in quantum mechanics with distributions. He led this one the important delta distribution. However, he used no mathematically precise definition for this distribution. He left in his investigations, the then functional analysis, ie the theory of functionals in mind. In the 1930s, Sergei Sobolev Lvovitch employed with initial value problems for partial hyperbolic differential equations. For these studies, he introduced the now named after him Sobolev spaces. In 1936 Sobolev examined hyperbolic second order differential equations with analytic coefficients functions. To specify a handier criterion for the existence of a solution of this partial differential equation, Sobolev expanded the question to the space of functionals. He was the first to formulate the current definition of a distribution. However, he developed no comprehensive theory of its definitions, but used it only as a tool for the investigation of partial differential equations.

Finally, Laurent Schwartz developed the theory of distributions in 1944 / 45th At this time it Sobolews works were still unknown, but he also came just as Sobolev through questions in the field of partial differential equations on special functionals, which he now called distributions. From then on, the theory was further developed so rapidly that Schwartz about it in the winter could keep 1945/46 lectures in Paris. Electrical engineer who attended his lectures, urged him to develop his theory towards the Fourier and the Laplacetransformationen. In 1947, Schwartz had defined the space of tempered distributions and thus the Fourier transforms integrated into his theory. 1950/51 he published his monograph Theory of Distributions, which his theory was further strengthened. In 1950 he was awarded the Fields Medal, one of the highest awards in the field of mathematics for his research in the field of distributions.

The theory of distributions was developed from then on in theoretical physics and in the theory of partial differential equations on. The distribution theory is useful to describe singular objects of physics, such as the electromagnetic point charge or point mass mathematically precise. These two physical objects can be described with the help of suitable delta function, because of the spatial density function of a particle with unit mass requiring that it vanishes everywhere except at one point. There, they must be infinite, because to give the space integral of the density function 1 ( unit mass ). There is no function in the usual sense, which meets these requirements. In the theory of partial differential equations and Fourier analysis, distributions are important because a derivation can be associated with this conceptualization, all locally integrable function.

Definitions

Distribution

A distribution is a continuous and linear function of a test function space into the real or complex numbers. Functions that map functions on numbers, traditionally called functionals. Using this concept distributions are continuous, linear functionals on the space of test functions.

The amount of distributions with the corresponding links of addition and scalar multiplication, ie, the topological dual space of test function space and is therefore listed as. The numeral in the functional analysis of the topological dual space. To speak of continuity and topological dual space, the space of test functions with a locally convex topology must be equipped.

Often, therefore, one uses the following characterization as an alternative definition, since this does not require the topology of the test function space and no knowledge of locally convex spaces is required:

Be an open set. A linear functional is called distribution, when one and one for every compact exists, such that for all test functions, the inequality

Applies. This definition is equivalent to that given above, since the continuity of the functional follows from this inequality, although they need not apply to all, because as (LF )-space is bornological.

Order of a distribution

Can be selected from the above alternative definition for all compacta same number, so the smallest possible as the order of is called. The amount of distributions of order is referred to and quoted to the set of all distributions with finite order. This space is smaller than the general distribution space, because there are distributions which are not of finite order.

Regular Distribution

A special subset of the distributions are the normal distributions. These distributions are generated by a locally integrable function. Precisely, this means that a distribution is called regular if there is a representation

Are, in is a locally integrable function. Non - regular distributions are also called singular; the distributions are, for there is no generating function in the sense of this definition.

This integral representation of a regular distribution motivated together with the inner product in the alternative spelling

For all (not just regular ) distributions.

Test functions

In the definition of the term in the distribution of the test function, or the function of the test area is centrally located. This test function space is the space of smooth functions with compact support together with an induced topology. Choosing a topology on the test function space is very important because otherwise the concept of continuity can not be meaningfully defined. The topology is defined on the space with a notion of convergence.

Be an open subset, then referred

The set of all infinitely differentiable functions having compact support, ie outside of a compact set equal to zero. The concept of convergence is determined by defining: A sequence converges to with if there is a compact set with for all and

For all multi- indices. The amount is - with this notion of convergence - a locally convex space, called space of test functions and as quoted.

Two different points of view

As described above in the section on the definition of the distribution, a distribution is a functional, ie a function with certain additional properties. In the history section of the distribution theory has been said, however, that the delta function can not be a function. This is obviously a contradiction in the current literature still finds himself. This contradiction arises from the fact that an attempt is made distributions - to identify with real-valued functions - and even functionals on spaces.

In particular, in theoretical physics is meant by a distribution object, for example, called, with some arising from the context properties. The desired properties often prevent that a function may be, for this reason, one speaks then of a generalized function. Now that the properties are determined by, considering the assignment

Which a test function assigns a real number. However, in general, is not a function, only one meaning must be explained for the expression of each individual case.

Mathematically is a distribution function with a certain abstract properties (linearity and continuity ) that assigns a test function a real number. Is that from previous volume sales is an integrable function, so the expression is mathematically precisely defined. However, here is not the function is called distribution, but the functional is called distribution.

Many mathematics textbooks do not distinguish between the ( distributions ) generating function and the actual distribution in the mathematical sense. In this article, the rigorous mathematical point of view is mainly used.

Examples

Continuous function as a producer

Be and so is defined by a distribution for all.

Delta distribution

The delta distribution is a singular distribution. That is, they can not be produced by an ordinary function, although it is often written as such. The following applies:

That is, the delta function applied to a test function returns a value of the test function at the point 0 Just as any other distribution can also express the delta function as a result of integral terms. The Dirac sequence

Has the limit ( see, eg, the adjacent animation)

Which would lead to the vanishing integral. Because the behavior at only one point is of no consequence at integrals of ordinary functions.

This Dirac sequence but you can with other thresholding, in front of the integral and not behind it, the delta function by

Represent. Mostly, however, the symbolic, to mathematically imprecise interpretation enticing notation

Used for the delta function, which is referred to as a generalized function of the expression, and often the word generalized omits.

Dirac comb

The Dirac comb with is a periodic distribution that is closely related to the Dirac delta function. This distribution is defined for all as

This series converges, as the test function has compact support, and therefore only a finite number of terms are non-zero. An equivalent definition is

Where the equality sign is to be understood as equality between distributions. The series on the right converges with respect to the weak -* topology. On the convergence of distributions convergence is discussed in more detail in the section. The problem encountered in the definition is a real number which is referred to as the period of the Dirac comb. Clearly, the Dirac comb is composed therefore of infinitely many delta distributions that relate to each other in the distance. The Dirac comb in contrast to the delta function is not compact support. What does this mean exactly, is explained in the section compact carrier below.

Radon measures

With the set of all Radon measures is called. Be Now you can using

Assign each a distribution. In this way, you can continuously embed into. An example of a Radon measure the Dirac level. For all it's defined by

If we identify the Dirac measure with the generating distribution

We obtain the delta function, if the following holds.

Cauchy principal value of 1 / x

The Cauchy principal value of the function can also be interpreted as a distribution. For all sets you

This is a singular distribution, since the integral term is not defined in the Lebesgue sense, and exists only as a Cauchy principal value. The abbreviation PV stands for principal value.

This distribution is usually used together with the dispersion relation ( Plemelj - Sokhotsky formula ), all distributions, and in particular as indicated are expressed by generalized functions and the imaginary unit means. This relationship combines the linear response theory, the real and imaginary parts of a response function, see Kramers -Kronig relations. (At this point it is assumed that the test functions are complex, therefore, and also the just -mentioned response functions, but the argument x to be real still, though of course x - iε is complex, and not real. )

Oscillating integral

All symbols are called

An oscillatory integral. This type integral does not converge depending on the choice of the Riemann or Lebesguesinn, but only in the sense of distributions.

Convergence

Since the distribution space is defined as a topological dual space, he also carries a topology. As a dual space of a Montelraums provided with the strong topology, it is itself a Montel space, falls for the consequences of the strong topology together with the weak -* topology. For sequences thus produced the following notion of convergence: A sequence of distributions converges to, if for each test function, the equation

Applies.

Because each test function can be identified with, can be regarded as a topological subspace.

The space is dense in. This means that for any distribution there exists a sequence of test functions with in. One can therefore any distribution by

Represent.

Localization

Limited to a subset of

Be open subsets and is a distribution. The restriction of the subset is defined by

For those wherein it is continued to through zero.

Carrier

Be a distribution. We say that a point belongs to the support of and writes, if for every open neighborhood of a function exists with.

If a regular distribution with a constant f, this definition is equivalent to the definition of the support is a function ( the function f).

Compact carrier

A distribution has a compact support, if a compact space. The amount of distributions with compact support is denoted by. It is a subspace of and the topological dual space to the space of smooth functions. In this room is the family of semi-norms

With arbitrary values takes and generates all compact subsets of passes, a locally convex topology.

Singular carrier

Be a distribution. We say that a point does not belong to singular carrier, if there is an open neighborhood of and a function with

For everyone.

In other words, if and only if there is no open neighborhood of such that the restriction of a smooth function is equal to. In particular, the singular support of a singular distribution is not empty.

Operations on distributions

Since the distribution space with pointwise addition and multiplication by complex numbers is a vector space over the field, the addition of distributions, and the multiplication of a complex number with a distribution are thus already defined.

The following additional operations are explained on distributions such as the derivation of a distribution. Many operations are transferred to distributions by the corresponding operation is applied to the test functions. For example, if a linear mapping that maps a test function to a function, and also there is a adjugierte linear and follow continuous map such that for all test functions and is

Then

A well-defined operation on distributions.

Multiplication by a function

Be and. Then the distribution is defined as

Differentiation

Motivation

Consider a continuously differentiable function and its associated regular distribution, one obtains the calculation rule

This partial integration was used, the boundary terms disappear because of the properties of the selected test function. This corresponds to the weak discharge. The two outer terms are also defined for singular distributions. It uses the definition of the derivative for an arbitrary distribution.

Definition

So be a distribution and a multi- index. Then a distribution is defined as

In the one-dimensional case, this just means

Example

The Heaviside function is by

Defined. Is differentiable everywhere except for the location. You can see them as a regular distribution and the invoice

Shows that its derivative ( as a distribution ) is the delta function:

One can also derive the delta function itself:

The derivatives of the delta function are therefore up to the additional sign factor equal to the derivatives of the test function at the point

Tensor

Motivation

Be the amount as a product space with given. Then you can access the functions and means of provision

Define a tensor product. Analogously, one can define a tensor product between distributions. These first regular distributions are considered. Let and be two locally integrable functions, it follows from the above definition

For all It follows

From this it is deduced from the following definition:

Definition

Let and. Then a distribution is made , which by

Is defined.

Smoothing a distribution

Distributions can be selectively smoothed or smeared or approximated, eg by looking at the distribution replaced by the regular distribution of a smooth approximation functions, such as the distribution by the regular distribution

The above-defined function or the Heaviside distribution by the regular distribution of the integrals of such functions. For three-dimensional differential equations can thus determine, for example, if the boundary conditions of the differential equations fit that apply to the interior. This is useful for many applications, especially as the smoothing functions are not uniquely defined up to the limit, resulting in increased flexibility. Similarly, one can also target distributions regularize as the above PV distribution by the test functions with appropriate factors you know or is going on in other ways, for example.

Convolution with a function

Definition

Be a distribution and a function, then the convolution of with is defined by

Example

Be a Radon measure and was identified as having the Radon measure distribution. Then for the convolution of with

Properties

If a smooth function is so true the definition consistent with the convolution of functions.
The result of the convolution is a smooth function, that applies.
For the convolution and is associative, which means that it applies.
For each multi- index is for the discharge of the folding.

Convolution of two distributions

Definition

Let and be two distributions, each of which has at least one compact support. Then the convolution between these distributions for all defined by

The figure

Is linear, continuous and commutes with shifts. Therefore, there is a unique distribution, so that

Applies to all.

Remark: The condition that a distribution has compact support can be weakened further.

Properties

This definition is a generalization of here already mentioned definitions. If you choose for a regular distribution, that is a function, this corresponds to the definitions listed here. Apply the properties:

The convolution is commutative
For the carrier.
For the singular support is obtained.

Tempered distributions

The tempered distributions are an excellent subset of the thus far observed distributions on the space. On the tempered distributions, it is possible to explain the Fourier and the Laplace transform.

Fourier transformation

In order to define a Fourier transform on distributions, one has to limit the amount of distributions only. Not every function is fouriertransformierbar, by analogy, you can not explain the Fourier transform for each distribution also. For this reason, Laurent Schwartz developed the now named after him, the Schwartz space by defining this space about a family of semi-norms, which then is symmetric with respect to the multiplication by the spatial variable x and the differentiation. Because the Fourier transform of differentiation with respect to x and x multiplication exchanged, this symmetry implies that the Fourier transform of a function is a Schwartz Schwartz function. Therefore, the Fourier transform defined on this space is an automorphism, ie a steady, linear and bijective map to itself The topological dual space, ie the space of continuous linear functionals of, say space of tempered distributions. The set of tempered distributions is larger than the amount of distributions with compact support, which is that the amount of Schwartz functions is a subset of the space of smooth functions. The smaller is a function space, namely, the greater its dual space. Therefore, the amount of the tempered distributions contained in the room. Because the amount of smooth functions with compact support is a subset of the Schwartz space.

The Fourier transform can be used for all by

Be defined. Also on the Fourier transform is an automorphism. The Fourier transform of the delta function is a constant distribution. Another example of a tempered distribution is the already mentioned above Dirac comb.

Convolution theorem

In connection with the above definitions of the convolution of two distributions and the Fourier transform of a distribution, the convolution theorem is interesting, which can be formulated as follows:

Be a tempered distribution and a distribution with compact support, then applies and states the convolution theorem for distributions:

The multiplication of two distributions is not defined in general. In this particular case, however, gives meaning as a smooth function.

Differential equations

Since each locally integrable function, in particular any function that creates a distribution that can be defined for these functions in the weak sense as a distribution derivative. Leaving distributions as a solution of a differential equation, then enlarges the solution space of this equation. The following examines briefly what is a distributional solution of a differential equation and how the fundamental solution is defined.

Solutions in the distribution sense

A differential operator with smooth coefficient functions. A distribution is called distribution solution, if the distributions generated by and consistent. This means

For everyone. If the distribution is regular and even m- times continuously differentiable, then it is a classical solution of the differential equation.

Example

Constant functions

All distributional solutions of the one-dimensional differential equation are the constant functions. That is, for all the equation solved only by a constant.

Poisson equation

A prominent example is the formal identity

From electrostatics. Being indicated by the Laplace operator. Precise means

That is

Is for all a solution of the Poisson equation

It is also said that the consideration here solves Poisson's equation in the distributional sense.

Fundamental solutions

Now let be a linear differential operator. A distribution is called fundamental solution, if the differential equation

Dissolves in the distribution sense.

The set of all fundamental solutions of results from the addition of a special fundamental solution of the general homogeneous solution. The general homogeneous solution is the set of distributions that applies. After a block of Bernard Malgrange, each linear differential operator with constant coefficients has a fundamental solution.

With the help of these fundamental solutions are obtained by convolution of inhomogeneous solutions of corresponding differential equations. Let be a smooth function (or more generally a distribution with compact support ), then results for

A solution in the form of

Which is the same as above, a fundamental solution of the differential operator.

Harmonic distributions

Similar to the harmonic functions are also defined harmonic distributions. Such a distribution is called harmonic if the Laplace equation

Sufficient in the distributional sense. Since the distributional derivative is more general than the ordinary differential, one might also expect more solutions of the Laplace equation. However, this is wrong. For one can prove that there is a smooth function for each harmonic distribution that generates this distribution. So there is no singular distributions that satisfy the equation, in particular the singular support of a harmonious distribution is empty. This statement applies even more generally for elliptic partial differential equations. For physicists, and engineers, this means that they can work in electrodynamics, for example in the theory of Maxwell's equations, without any doubts distributions, even if they are only interested in ordinary functions.

Distributions as integral kernels

Each test function can be carried

Identify with an integral operator. This identification can be extended to distributions. So there is every distribution is a linear operator

Of all and by

Is given. In addition, the reverse direction is true. Thus, for every operator a unique distribution so that applies. This identification between operator and distribution is the statement of the core set of Schwartz. The distribution is also called the Schwartz kernel based on the concept of the integral kernel. However, the operator can not be always represented in the form of an integral term.

Distributions on manifolds

Repatriation

You can distributions with the help of diffeomorphisms on real subsets back and forth transport. Be two real subsets and a diffeomorphism, ie a continuously differentiable, bijective function whose inverse mapping is also continuously differentiable. For true and for all test functions is considered due to the transformation rate equation

This identity motivates the following definition of the concatenation of a distribution with a diffeomorphism. Be, then, for all defined by

Usually you listed as and is called the return of the distribution

Definition

Let be a smooth manifold and is a system of maps and is such that for all

In applies. The system is called a distribution. This distribution is uniquely determined and independent of the choice of the map.

There are other ways to define distributions on manifolds. The definition associated with a density of bundles has the advantage that there is no system of local maps to be selected.

Regular distributions on manifolds

With this definition, one can assign each continuous function by means of the integral representation of a distribution again. Is a continuous function that is on the manifold, it is a continuous function on. By means of the integral representation for regular distributions

One obtains a system that a distribution on forms.

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