Distribution (mathematical analysis)


Distributions, also known as Schwartz distributions are a kind of generalized function in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative.
Distributions are widely used in the theory of partial differential equations, where it may be easier to establish the existence of distributional solutions than classical solutions, or where appropriate classical solutions may not exist. Distributions are also important in physics and engineering where many problems naturally lead to differential equations whose solutions or initial conditions are singular, such as the Dirac delta function.
A function is normally thought of as on the in the function domain by "sending" a point in the domain to the point. Instead of acting on points, distribution theory reinterprets functions such as as acting on in a certain way. In applications to physics and engineering, ' are usually infinitely differentiable complex-valued functions with compact support that are defined on some given non-empty open subset. The set of all such test functions forms a vector space that is denoted by or.
Most commonly encountered functions, including all continuous maps if using can be canonically reinterpreted as acting via "integration against a test function". Explicitly, this means that such a function "acts on" a test function by "sending" it to the number which is often denoted by. This new action of defines a scalar-valued map whose domain is the space of test functions. This functional turns out to have the two defining properties of what is known as a : it is linear, and it is also continuous when is given a certain topology called. The action of this distribution on a test function can be interpreted as a weighted average of the distribution on the support of the test function, even if the values of the distribution at a single point are not well-defined. Distributions like that arise from functions in this way are prototypical examples of distributions, but there exist many distributions that cannot be defined by integration against any function. Examples of the latter include the Dirac delta function and distributions defined to act by integration of test functions against certain measures on. Nonetheless, it is still always possible to continuous functions|reduce any arbitrary distribution] down to a simpler of related distributions that do arise via such actions of integration.
More generally, a is by definition a linear functional on that is linear functional|continuous] when is endowed with the '. The space of all distributions on is usually denoted by.
Definitions of the appropriate topologies on spaces of test functions and distributions are given in the article on spaces of test functions and distributions. This article is primarily concerned with the definition of distributions, together with their properties and some important examples.

History

The practical use of distributions can be traced back to the use of Green's functions in the 1830s to solve ordinary differential equations, but was not formalized until much later. According to, generalized functions originated in the work of on second-order hyperbolic partial differential equations, and the ideas were developed in somewhat extended form by Laurent Schwartz in the late 1940s. According to his autobiography, Schwartz introduced the term "distribution" by analogy with a distribution of electrical charge, possibly including not only point charges but also dipoles and so on. comments that although the ideas in the transformative book by were not entirely new, it was Schwartz's broad attack and conviction that distributions would be useful almost everywhere in analysis that made the difference. A detailed history of the theory of distributions was given by.

Notation

The following notation will be used throughout this article:
  • is a fixed positive integer and is a fixed non-empty open subset of Euclidean space.
  • denotes the natural numbers.
  • will denote a non-negative integer or.
  • If is a function then will denote its domain and the ' of denoted by is defined to be the closure of the set in.
  • For two functions the following notation defines a canonical pairing:
  • A of size is an element in . The ' of a multi-index is defined as and denoted by. Multi-indices are particularly useful when dealing with functions of several variables, in particular, we introduce the following notations for a given multi-index : We also introduce a partial order of all multi-indices by if and only if for all. When we define their multi-index binomial coefficient as:

    Definitions of test functions and distributions

In this section, some basic notions and definitions needed to define real-valued distributions on are introduced. Further discussion of the topologies on the spaces of test functions and distributions is given in the article on spaces of test functions and distributions.
File:Bump.png|thumb|350x350px|The graph of the bump function, where and. This function is a test function on and is an element of. The support of this function is the closed unit disk in. It is non-zero on the open unit disk and it is equal to everywhere outside of it.
For all and any compact subsets and of, we have:
Distributions on are continuous linear functionals on when this vector space is endowed with a particular topology called the '. The following proposition states two necessary and sufficient conditions for the continuity of a linear function on that are often straightforward to verify.
Proposition: A linear functional on is continuous, and therefore a ', if and only if any of the following equivalent conditions is satisfied:
  1. For every compact subset there exist constants and such that for all with support contained in,
  2. For every compact subset and every sequence in whose supports are contained in, if converges uniformly to zero on for every multi-index, then.

    Topology on ''C''''k''(''U'')

We now introduce the seminorms that will define the topology on. Different authors sometimes use different families of seminorms so we list the most common families below. However, the resulting topology is the same no matter which family is used.
All of the functions above are non-negative -valued seminorms on. As explained in this article, every set of seminorms on a vector space induces a locally convex vector topology.
Each of the following sets of seminorms
generate the same locally convex vector topology on .
With this topology, becomes a locally convex Fréchet space that is normable. Every element of is a continuous seminorm on.
Under this topology, a net in converges to if and only if for every multi-index with and every compact, the net of partial derivatives converges uniformly to on For any any bounded subset of is a relatively compact subset of In particular, a subset of is bounded if and only if it is bounded in for all The space is a Montel space if and only if
A subset of is open in this topology if and only if there exists such that is open when is endowed with the subspace topology induced on it by.

Topology on ''C''''k''(''K'')

As before, fix Recall that if is any compact subset of then
If is finite then is a Banach space with a topology that can be defined by the norm

Trivial extensions and independence of ''C''''k''(''K'')'s topology from ''U''

Suppose is an open subset of and is a compact subset. By definition, elements of are functions with domain , so the space and its topology depend on to make this dependence on the open set clear, temporarily denote by
Importantly, changing the set to a different open subset will change the set from to so that elements of will be functions with domain instead of
Despite depending on the open set, the standard notation for makes no mention of it.
This is justified because, as this subsection will now explain, the space is canonically identified as a subspace of .
It is enough to explain how to canonically identify with when one of and is a subset of the other. The reason is that if and are arbitrary open subsets of containing then the open set also contains so that each of and is canonically identified with and now by transitivity, is thus identified with
So assume are open subsets of containing
Given its is the function defined by:
This trivial extension belongs to and it will be denoted by . The assignment thus induces a map that sends a function in to its trivial extension on This map is a linear injection and for every compact subset ,
If is restricted to then the following induced linear map is a homeomorphism :
and thus the next map is a topological embedding:
Using the injection
the vector space is canonically identified with its image in Because through this identification, can also be considered as a subset of
Thus the topology on is independent of the open subset of that contains which justifies the practice of writing instead of

Canonical LF topology

Recall that denotes all functions in that have compact support in where note that is the union of all as ranges over all compact subsets of Moreover, for each is a dense subset of The special case when gives us the space of test functions.
The canonical LF-topology is metrizable and importantly, it is Comparison of topologies| than the subspace topology that induces on However, the canonical LF-topology does make into a complete reflexive nuclear Montel bornological barrelled Mackey space; the same is true of its strong dual space. The canonical LF-topology can be defined in various ways.

Distributions

As discussed earlier, continuous linear functionals on a are known as distributions on Other equivalent definitions are described below.
There is a canonical duality pairing between a distribution on and a test function which is denoted using angle brackets by
One interprets this notation as the distribution acting on the test function to give a scalar, or symmetrically as the test function acting on the distribution

Characterizations of distributions

Proposition: If is a linear functional on then the following are equivalent:
  1. is a distribution;
  2. is continuous;
  3. is continuous at the origin;
  4. is uniformly continuous;
  5. is a bounded operator;
  6. is sequentially continuous;
  7. * explicitly, for every sequence in that converges in to some
  8. is sequentially continuous at the origin; in other words, maps null sequences to null sequences;
  9. * explicitly, for every sequence in that converges in to the origin,
  10. * a is by definition any sequence that converges to the origin;
  11. maps null sequences to bounded subsets;
  12. * explicitly, for every sequence in that converges in to the origin, the sequence is bounded;
  13. maps Mackey convergent null sequences to bounded subsets;
  14. * explicitly, for every Mackey convergent null sequence in the sequence is bounded;
  15. * a sequence is said to be if there exists a divergent sequence of positive real numbers such that the sequence is bounded; every sequence that is Mackey convergent to the origin necessarily converges to the origin ;
  16. The kernel of is a closed subspace of
  17. The graph of is closed;
  18. There exists a continuous seminorm on such that
  19. There exists a constant and a finite subset such that
  20. For every compact subset there exist constants and such that for all
  21. For every compact subset there exist constants and such that for all with support contained in
  22. For any compact subset and any sequence in if converges uniformly to zero for all multi-indices then

    Topology on the space of distributions and its relation to the weak-* topology

The set of all distributions on is the continuous dual space of which when endowed with the strong dual topology is denoted by Importantly, unless indicated otherwise, the topology on is the strong dual topology; if the topology is instead the weak-* topology then this will be indicated. Neither topology is metrizable although unlike the weak-* topology, the strong dual topology makes into a complete nuclear space, to name just a few of its desirable properties.
Neither nor its strong dual is a sequential space and so neither of their topologies can be fully described by sequences.
However, a in converges in the strong dual topology if and only if it converges in the weak-* topology.
More information about the topology that is endowed with can be found in the article on spaces of test functions and distributions and the articles on polar topologies and dual systems.
A map from into another locally convex topological vector space is continuous if and only if it is sequentially continuous at the origin. However, this is no longer guaranteed if the map is not linear or for maps valued in more general topological spaces. The same is true of maps from .

Localization of distributions

There is no way to define the value of a distribution in at a particular point of. However, as is the case with functions, distributions on restrict to give distributions on open subsets of. Furthermore, distributions are in the sense that a distribution on all of can be assembled from a distribution on an open cover of satisfying some compatibility conditions on the overlaps. Such a structure is known as a sheaf.

Extensions and restrictions to an open subset

Let be open subsets of
Every function can be from its domain to a function on by setting it equal to on the complement This extension is a smooth compactly supported function called the and it will be denoted by
This assignment defines the operator
which is a continuous injective linear map. It is used to canonically identify as a vector subspace of .
Its transpose
is called the ' and as the name suggests, the image of a distribution under this map is a distribution on called the restriction of to The defining condition of the restriction is:
If then the trivial extension map is a topological embedding and its range is also dense in its codomain Consequently if then the restriction mapping is neither injective nor surjective. A distribution is said to be ' if it belongs to the range of the transpose of and it is called if it is extendable to
Unless the restriction to is neither injective nor surjective. Lack of surjectivity follows since distributions can blow up towards the boundary of. For instance, if and then the distribution
is in but admits no extension to

Gluing and distributions that vanish in a set

Let be an open subset of. is said to if for all such that we have vanishes in if and only if the restriction of to is equal to 0, or equivalently, if and only if lies in the kernel of the restriction map

Support of a distribution

This last corollary implies that for every distribution on, there exists a unique largest subset of such that vanishes in ; the complement in of this unique largest open subset is called. Thus
If is a locally integrable function on and if is its associated distribution, then the support of is the smallest closed subset of in the complement of which is almost everywhere equal to 0. If is continuous, then the support of is equal to the closure of the set of points in at which does not vanish. The support of the distribution associated with the Dirac measure at a point is the set If the support of a test function does not intersect the support of a distribution then A distribution is 0 if and only if its support is empty. If is identically 1 on some open set containing the support of a distribution then If the support of a distribution is compact then it has finite order and there is a constant and a non-negative integer such that:
If has compact support, then it has a unique extension to a continuous linear functional on ; this function can be defined by where is any function that is identically 1 on an open set containing the support of.
If and then and Thus, distributions with support in a given subset form a vector subspace of Furthermore, if is a differential operator in, then for all distributions on and all we have and

Distributions with compact support

Support in a point set and Dirac measures

For any let denote the distribution induced by the Dirac measure at For any and distribution the support of is contained in if and only if is a finite linear combination of derivatives of the Dirac measure at If in addition the order of is then there exist constants such that:
Said differently, if has support at a single point then is in fact a finite linear combination of distributional derivatives of the function at. That is, there exists an integer and complex constants such that
where is the translation operator.

Distribution with compact support

Distributions of finite order with support in an open subset

Global structure of distributions

The formal definition of distributions exhibits them as a subspace of a very large space, namely the topological dual of . It is not immediately clear from the definition how exotic a distribution might be. To answer this question, it is instructive to see distributions built up from a smaller space, namely the space of continuous functions. Roughly, any distribution is locally a derivative of a continuous function. A precise version of this result, given below, holds for distributions of compact support, tempered distributions, and general distributions. Generally speaking, no proper subset of the space of distributions contains all continuous functions and is closed under differentiation. This says that distributions are not particularly exotic objects; they are only as complicated as necessary.

Distributions as sheaves">Sheaf (mathematics)">sheaves

Decomposition of distributions as sums of derivatives of continuous functions

By combining the above results, one may express any distribution on as the sum of a series of distributions with compact support, where each of these distributions can in turn be written as a finite sum of distributional derivatives of continuous functions on. In other words, for arbitrary we can write:
where are finite sets of multi-indices and the functions are continuous.
Note that the infinite sum above is well-defined as a distribution. The value of for a given can be computed using the finitely many that intersect the support of

Operations on distributions

Many operations which are defined on smooth functions with compact support can also be defined for distributions. In general, if is a linear map that is continuous with respect to the weak topology, then it is not always possible to extend to a map by classic extension theorems of topology or linear functional analysis. The “distributional” extension of the above linear continuous operator A is possible if and only if A admits a Schwartz adjoint, that is another linear continuous operator B of the same type such that, for every pair of test functions. In that condition, B is unique and the extension A’ is the transpose of the Schwartz adjoint B.

Preliminaries: transpose of a linear operator

Operations on distributions and spaces of distributions are often defined using the transpose of a linear operator. This is because the transpose allows for a unified presentation of the many definitions in the theory of distributions and also because its properties are well-known in functional analysis. For instance, the well-known Hermitian adjoint of a linear operator between Hilbert spaces is just the operator's transpose. In general, the transpose of a continuous linear map is the linear map
or equivalently, it is the unique map satisfying for all and all . Since is continuous, the transpose is also continuous when both duals are endowed with their respective strong dual topologies; it is also continuous when both duals are endowed with their respective weak* topologies.
In the context of distributions, the characterization of the transpose can be refined slightly. Let be a continuous linear map. Then by definition, the transpose of is the unique linear operator that satisfies:
Since is dense in it is sufficient that the defining equality hold for all distributions of the form where Explicitly, this means that a continuous linear map is equal to if and only if the condition below holds:
where the right-hand side equals

Differential operators

Differentiation of distributions

Let be the partial derivative operator To extend we compute its transpose:
Therefore Thus, the partial derivative of with respect to the coordinate is defined by the formula
With this definition, every distribution is infinitely differentiable, and the derivative in the direction is a linear operator on
More generally, if is an arbitrary multi-index, then the partial derivative of the distribution is defined by
Differentiation of distributions is a continuous operator on this is an important and desirable property that is not shared by most other notions of differentiation.
If is a distribution in then
where is the derivative of and is a translation by thus the derivative of may be viewed as a limit of quotients.

Differential operators acting on smooth functions

A linear differential operator in with smooth coefficients acts on the space of smooth functions on Given such an operator
we would like to define a continuous linear map, that extends the action of on to distributions on In other words, we would like to define such that the following diagram commutes:
where the vertical maps are given by assigning its canonical distribution which is defined by:
With this notation, the diagram commuting is equivalent to:
To find the transpose of the continuous induced map defined by is considered in the lemma below.
This leads to the following definition of the differential operator on called which will be denoted by to avoid confusion with the transpose map, that is defined by
As discussed above, for any the transpose may be calculated by:
For the last line we used integration by parts combined with the fact that and therefore all the functions have compact support. Continuing the calculation above, for all
The Lemma combined with the fact that the formal transpose of the formal transpose is the original differential operator, that is, enables us to arrive at the correct definition: the formal transpose induces the canonical linear operator defined by We claim that the transpose of this map, can be taken as To see this, for every compute its action on a distribution of the form with :
We call the continuous linear operator the . Its action on an arbitrary distribution is defined via:
If converges to then for every multi-index converges to

Multiplication of distributions by smooth functions

A differential operator of order 0 is just multiplication by a smooth function. And conversely, if is a smooth function then is a differential operator of order 0, whose formal transpose is itself. The induced differential operator maps a distribution to a distribution denoted by We have thus defined the multiplication of a distribution by a smooth function.
We now give an alternative presentation of the multiplication of a distribution on by a smooth function The product is defined by
This definition coincides with the transpose definition since if is the operator of multiplication by the function , then
so that
Under multiplication by smooth functions, is a module over the ring With this definition of multiplication by a smooth function, the ordinary product rule of calculus remains valid. However, some unusual identities also arise. For example, if is the Dirac delta distribution on then and if is the derivative of the delta distribution, then
The bilinear multiplication map given by is continuous; it is however, hypocontinuous.
Example: The product of any distribution with the function that is identically on is equal to
Example: Suppose is a sequence of test functions on that converges to the constant function For any distribution on the sequence converges to
If converges to and converges to then converges to
Problem of multiplying distributions
It is easy to define the product of a distribution with a smooth function, or more generally the product of two distributions whose singular supports are disjoint. With more effort, it is possible to define a well-behaved product of several distributions provided their wave front sets at each point are compatible. A limitation of the theory of distributions is that there is no associative product of two distributions extending the product of a distribution by a smooth function, as has been proved by Laurent Schwartz in the 1950s. For example, if is the distribution obtained by the Cauchy principal value
If is the Dirac delta distribution then
but,
so the product of a distribution by a smooth function cannot be extended to an associative product on the space of distributions.
Thus, nonlinear problems cannot be posed in general and thus are not solved within distribution theory alone. In the context of quantum field theory, however, solutions can be found. In more than two spacetime dimensions the problem is related to the regularization of divergences. Here Henri Epstein and Vladimir Glaser developed the mathematically rigorous . This does not solve the problem in other situations. Many other interesting theories are non-linear, like for example the Navier–Stokes equations of fluid dynamics.
Several not entirely satisfactory theories of algebras of generalized functions have been developed, among which Colombeau's algebra is maybe the most popular in use today.
Inspired by Lyons' rough path theory, Martin Hairer proposed a consistent way of multiplying distributions with certain structures, available in many examples from stochastic analysis, notably stochastic partial differential equations. See also Gubinelli–Imkeller–Perkowski for a related development based on Bony's paraproduct from Fourier analysis.

Composition with a smooth function

Let be a distribution on Let be an open set in and If is a submersion then it is possible to define
This is, and is also called, sometimes written
The pullback is often denoted although this notation should not be confused with the use of '*' to denote the adjoint of a linear mapping.
The condition that be a submersion is equivalent to the requirement that the Jacobian derivative of is a surjective linear map for every A necessary condition for extending to distributions is that be an open mapping. The Inverse function theorem ensures that a submersion satisfies this condition.
If is a submersion, then is defined on distributions by finding the transpose map. The uniqueness of this extension is guaranteed since is a continuous linear operator on Existence, however, requires using the change of variables formula, the inverse function theorem, and a partition of unity argument.
In the special case when is a diffeomorphism from an open subset of onto an open subset of change of variables under the integral gives:
In this particular case, then, is defined by the transpose formula:

Convolution

Under some circumstances, it is possible to define the convolution of a function with a distribution, or even the convolution of two distributions.
Recall that if and are functions on then we denote by defined at to be the integral
provided that the integral exists. If are such that then for any functions and we have and If and are continuous functions on at least one of which has compact support, then and if then the values of on do depend on the values of outside of the Minkowski sum
Importantly, if has compact support then for any the convolution map is continuous when considered as the map or as the map

Translation and symmetry

Given the translation operator sends to defined by This can be extended by the transpose to distributions in the following way: given a distribution is the distribution defined by
Given define the function by Given a distribution let be the distribution defined by The operator is called .

Convolution of a test function with a distribution

Convolution with defines a linear map:
which is continuous with respect to the canonical LF space topology on
Convolution of with a distribution can be defined by taking the transpose of relative to the duality pairing of with the space of distributions. If then by Fubini's theorem
Extending by continuity, the convolution of with a distribution is defined by
An alternative way to define the convolution of a test function and a distribution is to use the translation operator The convolution of the compactly supported function and the distribution is then the function defined for each by
It can be shown that the convolution of a smooth, compactly supported function and a distribution is a smooth function. If the distribution has compact support, and if is a polynomial, then the same is true of If the distribution has compact support as well, then is a compactly supported function, and the Titchmarsh convolution theorem implies that:
where denotes the convex hull and denotes the support.

Convolution of a smooth function with a distribution

Let and and assume that at least one of and has compact support. The of and denoted by or by is the smooth function:
satisfying for all :
Let be the map. If is a distribution, then is continuous as a map. If also has compact support, then is also continuous as the map and continuous as the map
If is a continuous linear map such that for all and all then there exists a distribution such that for all
Example: Let be the Heaviside function on For any
Let be the Dirac measure at 0 and let be its derivative as a distribution. Then and Importantly, the associative law fails to hold:

Convolution of distributions

It is also possible to define the convolution of two distributions and on provided one of them has compact support. Informally, to define where has compact support, the idea is to extend the definition of the convolution to a linear operation on distributions so that the associativity formula
continues to hold for all test functions
It is also possible to provide a more explicit characterization of the convolution of distributions. Suppose that and are distributions and that has compact support. Then the linear maps
are continuous. The transposes of these maps:
are consequently continuous and it can also be shown that
This common value is called and it is a distribution that is denoted by or It satisfies If and are two distributions, at least one of which has compact support, then for any If is a distribution in and if is a Dirac measure then ; thus is the identity element of the convolution operation. Moreover, if is a function then where now the associativity of convolution implies that for all functions and
Suppose that it is that has compact support. For consider the function
It can be readily shown that this defines a smooth function of which moreover has compact support. The convolution of and is defined by
This generalizes the classical notion of convolution of functions and is compatible with differentiation in the following sense: for every multi-index
The convolution of a finite number of distributions, all of which have compact support, is associative.
This definition of convolution remains valid under less restrictive assumptions about and
The convolution of distributions with compact support induces a continuous bilinear map defined by where denotes the space of distributions with compact support. However, the convolution map as a function is continuous although it is separately continuous. The convolution maps and given by both to be continuous. Each of these non-continuous maps is, however, separately continuous and hypocontinuous.

Convolution versus multiplication

In general, regularity is required for multiplication products, and locality is required for convolution products. It is expressed in the following extension of the Convolution Theorem which guarantees the existence of both convolution and multiplication products. Let be a rapidly decreasing tempered distribution or, equivalently, be an ordinary function within the space of tempered distributions and let be the normalized Fourier transform. Then, according to,
hold within the space of tempered distributions. In particular, these equations become the Poisson Summation Formula if is the Dirac Comb. The space of all rapidly decreasing tempered distributions is also called the space of and the space of all ordinary functions within the space of tempered distributions is also called the space of More generally, and A particular case is the Paley-Wiener-Schwartz Theorem which states that and This is because and In other words, compactly supported tempered distributions belong to the space of and
Paley-Wiener functions better known as bandlimited functions, belong to the space of
For example, let be the Dirac comb and be the Dirac delta; then is the function that is constantly one and both equations yield the Dirac-comb identity. Another example is to let be the Dirac comb and be the rectangular function; then is the sinc function and both equations yield the Classical Sampling Theorem for suitable functions. More generally, if is the Dirac comb and is a smooth window function, for example, the Gaussian, then is another smooth window function. They are known as mollifiers, especially in partial differential equations theory, or as regularizers in physics because they allow turning generalized functions into regular functions.

Tensor products of distributions

Let and be open sets. Assume all vector spaces to be over the field where or For define for every and every the following functions:
Given and define the following functions:
where and
These definitions associate every and with the continuous linear map:
Moreover, if either has compact support then it also induces a continuous linear map of (resp.
denoted by or is the distribution in defined by:

Spaces of distributions

For all and all every one of the following canonical injections is continuous and has an image that is a dense subset of its codomain:
where the topologies on are defined as direct limits of the spaces in a manner analogous to how the topologies on were defined. The range of each of the maps above is dense in its codomain.
Suppose that is one of the spaces or or . Because the canonical injection is a continuous injection whose image is dense in the codomain, this map's transpose is a continuous injection. This injective transpose map thus allows the continuous dual space of to be identified with a certain vector subspace of the space of all distributions. This transpose map is continuous but it is necessarily a topological embedding.
A linear subspace of carrying a locally convex topology that is finer than the subspace topology induced on it by is called .
Almost all of the spaces of distributions mentioned in this article arise in this way and any representation theorem about the continuous dual space of may, through the transpose be transferred directly to elements of the space

Radon measures

The inclusion map is a continuous injection whose image is dense in its codomain, so the transpose is also a continuous injection.
Note that the continuous dual space can be identified as the space of Radon measures, where there is a one-to-one correspondence between the continuous linear functionals and integral with respect to a Radon measure; that is,
  • if then there exists a Radon measure on such that for all and
  • if is a Radon measure on then the linear functional on defined by sending to is continuous.
Through the injection every Radon measure becomes a distribution on. If is a locally integrable function on then the distribution is a Radon measure; so Radon measures form a large and important space of distributions.
The following is the theorem of the structure of distributions of Radon measures, which shows that every Radon measure can be written as a sum of derivatives of locally functions on :

Positive Radon measures

A linear function on a space of functions is called if whenever a function that belongs to the domain of is non-negative then One may show that every positive linear functional on is necessarily continuous.
Lebesgue measure is an example of a positive Radon measure.

Locally integrable functions as distributions

One particularly important class of Radon measures are those that are induced locally integrable functions. The function is called if it is Lebesgue integrable over every compact subset of. This is a large class of functions that includes all continuous functions and all functions. The topology on is defined in such a fashion that any locally integrable function yields a continuous linear functional on – that is, an element of – denoted here by whose value on the test function is given by the Lebesgue integral:
Conventionally, one abuses notation by identifying with provided no confusion can arise, and thus the pairing between and is often written
If and are two locally integrable functions, then the associated distributions and are equal to the same element of if and only if and are equal almost everywhere. Similarly, every Radon measure on defines an element of whose value on the test function is As above, it is conventional to abuse notation and write the pairing between a Radon measure and a test function as Conversely, as shown in a theorem by Schwartz, every distribution which is non-negative on non-negative functions is of this form for some Radon measure.

Test functions as distributions

The test functions are themselves locally integrable, and so define distributions. The space of test functions is sequentially dense in with respect to the strong topology on This means that for any there is a sequence of test functions, that converges to when considered as a sequence of distributions. Or equivalently,

Distributions with compact support

The inclusion map is a continuous injection whose image is dense in its codomain, so the transpose map is also a continuous injection. Thus the image of the transpose, denoted by forms a space of distributions.
The elements of can be identified as the space of distributions with compact support. Explicitly, if is a distribution on then the following are equivalent,
  • The support of is compact.
  • The restriction of to when that space is equipped with the subspace topology inherited from , is continuous.
  • There is a compact subset of such that for every test function whose support is completely outside of, we have
Compactly supported distributions define continuous linear functionals on the space ; recall that the topology on is defined such that a sequence of test functions converges to 0 if and only if all derivatives of converge uniformly to 0 on every compact subset of. Conversely, it can be shown that every continuous linear functional on this space defines a distribution of compact support. Thus compactly supported distributions can be identified with those distributions that can be extended from to

Restriction of distributions to compact sets

If then for any compact set there exists a continuous function compactly supported in and a multi-index such that on

Distributions of finite order

Let The inclusion map is a continuous injection whose image is dense in its codomain, so the transpose is also a continuous injection. Consequently, the image of denoted by forms a space of distributions. The elements of are ' The distributions of order which are also called ' are exactly the distributions that are Radon measures.
For a ' is a distribution of order that is not a distribution of order.
A distribution is said to be of ' if there is some integer such that it is a distribution of order and the set of distributions of finite order is denoted by Note that if then so that is a vector subspace of, and furthermore, if and only if

Structure of distributions of finite order

Every distribution with compact support in is a distribution of finite order. Indeed, every distribution in is a distribution of finite order, in the following sense: If is an open and relatively compact subset of and if is [|the restriction mapping] from to, then the image of under is contained in
The following is the theorem of the structure of distributions of finite order, which shows that every distribution of finite order can be written as a sum of derivatives of Radon measures:
Example: Let and for every test function let
Then is a distribution of infinite order on. Moreover, can not be extended to a distribution on ; that is, there exists no distribution on such that the restriction of to is equal to

Tempered distributions and Fourier transform

Defined below are the Schwartz space and its dual; the space of , which forms a proper subspace of the space of distributions on Tempered distributions are useful if one studies the Fourier transform since all tempered distributions have a Fourier transform, which is not true for an arbitrary distribution in

Schwartz space

The Schwartz space is the space of all smooth functions that are rapidly decreasing at infinity along with all partial derivatives. Thus is in the Schwartz space provided that any derivative of multiplied with any power of converges to 0 as These functions form a complete TVS with a suitably defined family of seminorms. More precisely, for any multi-indices and define
Then is in the Schwartz space if all the values satisfy
The family of seminorms defines a locally convex topology on the Schwartz space. For the seminorms are, in fact, norms on the Schwartz space. One can also use the following family of seminorms to define the topology:
Otherwise, one can define a norm on via
The Schwartz space is a Fréchet space. Because the Fourier transform changes into multiplication by and vice versa, this symmetry implies that the Fourier transform of a Schwartz function is also a Schwartz function.
A sequence in converges to 0 in if and only if the functions converge to 0 uniformly in the whole of which implies that such a sequence must converge to zero in
is dense in The subset of all analytic Schwartz functions is dense in as well.
The Schwartz space is nuclear, and the tensor product of two maps induces a canonical surjective TVS-isomorphisms
where represents the completion of the injective tensor product.

Tempered distributions

The inclusion map is a continuous injection whose image is dense in its codomain, so the transpose is also a continuous injection. Thus, the image of the transpose map, denoted by forms a space of distributions.
The space is called the space of. It is the continuous dual space of the Schwartz space. Equivalently, a distribution is a tempered distribution if and only if
The derivative of a tempered distribution is again a tempered distribution. Tempered distributions generalize the bounded locally integrable functions; all distributions with compact support and all square-integrable functions are tempered distributions. More generally, all functions that are products of polynomials with elements of the Lp space for are tempered distributions.
The can also be characterized as, meaning that each derivative of grows at most as fast as some polynomial. This characterization is dual to the behaviour of the derivatives of a function in the Schwartz space, where each derivative of decays faster than every inverse power of An example of a rapidly falling function is for any positive

Fourier transform

To study the Fourier transform, it is best to consider complex-valued test functions and complex-linear distributions. The ordinary continuous Fourier transform is a TVS-automorphism of the Schwartz space, and the is defined to be its transpose which will again be denoted by So the Fourier transform of the tempered distribution is defined by for every Schwartz function is thus again a tempered distribution. The Fourier transform is a TVS isomorphism from the space of tempered distributions onto itself. This operation is compatible with differentiation in the sense that
and also with convolution: if is a tempered distribution and is a smooth function on is again a tempered distribution and
is the convolution of and In particular, the Fourier transform of the constant function equal to 1 is the distribution.

Expressing tempered distributions as sums of derivatives

If is a tempered distribution, then there exists a constant and positive integers and such that for all Schwartz functions
This estimate, along with some techniques from functional analysis, can be used to show that there is a continuous slowly increasing function and a multi-index such that

Using holomorphic functions as test functions

The success of the theory led to an investigation of the idea of hyperfunction, in which spaces of holomorphic functions are used as test functions. A refined theory has been developed, in particular Mikio Sato's algebraic analysis, using sheaf theory and several complex variables. This extends the range of symbolic methods that can be made into rigorous mathematics, for example, Feynman integrals.