Distribution (mathematics)


Distributions are objects that generalize the classical notion of functions 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 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 distributions, such as the Dirac delta function.
The practical use of distributions can be traced back to the use of Green 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.
Distribution theory reinterprets functions as linear functionals acting on a space of test functions. Standard functions act by integration against a test function, but many other linear functionals do not arise in this way, and these are the "generalized functions". There are different possible choices for the space of test functions, leading to different spaces of distributions. The basic space of test function consists of smooth functions with compact support, leading to standard distributions. Use of the space of smooth, rapidly decreasing test functions gives instead the tempered distributions, which are important because they have a well-defined distributional Fourier transform. Every tempered distribution is a distribution in the normal sense, but the converse is not true: in general the larger the space of test functions, the more restrictive the notion of distribution. On the other hand, the use of spaces of analytic test functions leads to Sato's theory of hyperfunctions; this theory has a different character from the previous ones because there are no analytic functions with non-empty compact support.

Basic idea

Distributions are a class of linear functionals that map a set of test functions into the set of real numbers. In the simplest case, the set of test functions considered is D, which is the set of functions ' : R → R having two properties:
A distribution T is a linear mapping T : D → R. Instead of writing T, it is conventional to write for the value of T acting on a test function '. A simple example of a distribution is the Dirac delta δ, defined by
meaning that δ evaluates a test function at 0. Its physical interpretation is as the density of a point source.
As described next, there are straightforward mappings from both locally integrable functions and Radon measures to corresponding distributions, but not all distributions can be formed in this manner.

Functions and measures as distributions

Suppose that f : R → R is a locally integrable function. Then a corresponding distribution Tf may be defined by
This integral is a real number which depends linearly and continuously on. Conversely, the values of the distribution Tf on test functions in D determine the pointwise almost everywhere values of the function f on R. In a conventional abuse of notation, f is often used to represent both the original function f and the corresponding distribution Tf. This example suggests the definition of a distribution as a linear and, in an appropriate sense, continuous functional on the space of test functions D.
Similarly, if μ is a Radon measure on R, then a corresponding distribution Rμ may be defined by
This integral also depends linearly and continuously on, so that Rμ is a distribution. If μ is absolutely continuous with respect to Lebesgue measure with density f and dμ = f dx, then this definition for Rμ is the same as the previous one for Tf, but if μ is not absolutely continuous, then Rμ is a distribution that is not associated with a function. For example, if P is the point-mass measure on R that assigns measure one to the singleton set and measure zero to sets that do not contain zero, then
so that RP = δ is the Dirac delta.

Adding and multiplying distributions

Distributions may be multiplied by real numbers and added together, so they form a real vector space.
A distribution may also be multiplied by a rapidly decreasing infinitely differentiable function to get another distribution, but
it is not possible to define a product of general distributions that extends the usual pointwise product of functions and has the same algebraic properties. This result was shown by, and is usually referred to as the
Schwartz Impossibility Theorem.

Derivatives of distributions

It is desirable to choose a definition for the derivative of a distribution which, at least for distributions derived from smooth functions, has the property that. If is a test function, we can use integration by parts to see that
where the last equality follows from the fact that has compact support, so is zero outside of a bounded set. This suggests that if is a distribution, we should define its derivative by
It turns out that this is the proper definition; it extends the ordinary definition of derivative, every distribution becomes infinitely differentiable and the usual properties of derivatives hold.
Example: Recall that the Dirac delta is the distribution defined by the equation
It is the derivative of the distribution corresponding to the Heaviside step function H: For any test function ,
so H′ = δ. Note, = 0 because has compact support by our definition of a test function. Similarly, the derivative of the Dirac delta is the distribution defined by the equation
This latter distribution is an example of a distribution that is not derived from a function or a measure. Its physical interpretation is the density of a dipole source. Just as the Dirac impulse can be realized in the weak limit as a sequence of various kinds of constant norm bump functions of ever increasing amplitude and narrowing support, its derivative can by definition be realized as the weak limit of the negative derivatives of said functions, which are now antisymmetric about the eventual distribution's point of singular support.

Test functions and distributions

Throughout, n is a fixed positive integer, U is a fixed non-empty open subset of Euclidean space, and.
In the following, real-valued distributions on an open subset U will be formally defined.
With minor modifications, one can also define complex-valued distributions, and one can replace Rn by any smooth manifold.
For any let denote the vector space of all k-times continuously differentiable complex-valued functions on.
Recall that the support of a function f, denoted by, is the closure in of the set.
For any compact subset, let denote all those functions such that .
Note that if f is a real-valued function on U, then f is an element of if and only if it is a bump function.
Distributions on U are defined to be the continuous linear functionals on the space of test functions on U when this space is endowed with a particular topology.
The space of test functions on U is also denoted is also D or.
So to define the space of distributions, which is denoted by, we must first define the topology on the space, which requires that several other topological vector spaces be defined first.
To define the various special subspaces of , we construct the spaces and for arbitrary, which adds almost no additional complexity to the presentation and allows us to easily define additional spaces of distributions later.
Once the topology on is defined, we can then immediately place a topology on the space of distributions.
Throughout, we will let be any collection of compact subsets of such that , and for any compact there exists some such that.
The most common choices for are:
We make into a directed set by defining if and only if.
Note that although the definitions of the subsequently defined topologies explicitly reference, in reality they do not depend on the choice of ; that is, if and are any two such collections of compact subsets of, then the topologies defined on and by using in place of are the same as those defined by using in place of.

Topology on the space of Ck maps

We give the Fréchet space topology defined by the family of seminorms
as i ranges over and K ranges over, where is a multi-index of non-negative integers and denotes its length. Under this topology, a net of functions in converges to a function if and only if for every multi-index p with and every, converges to uniformly on K.
Note that is a Montel space if and only if.

Topology on the space of Ck maps with support in a compact set

For any compact, is a closed subspace of and we give the subspace topology induced by, thereby making it into a Fréchet space.
For all compact with, denote the natural inclusion by where note that this map is a linear embedding of TVSs whose range is closed in its codomain.
When then is a Banach space and when, it is even a Hilbert space. If K is non-trivial then is not a Banach space but is a Fréchet space for all.

Topology on the space of test functions

Let denote all those functions in that have compact support in, where note that is the union of all as K ranges over.
Moreover, for every k, is a dense subset of.
If then is called the space of test functions and it may also be denoted by.
For all compact with, there are natural inclusions, which form a direct system in the category of locally convex TVSs that is directed by .
This system's direct limit is the locally convex TVS together with, where is the natural inclusion and where the topology on is the strongest locally convex topology making all of these inclusion maps continuous.
This topology, called the canonical LF topology, makes into a complete Hausdorff locally convex LF-space and this topology is finer than the subspace topology that inherits from .
From basic category theory, we know that this topology is independent of the particular choice of the directed collection of compact subsets .
The spaces,, and the strong duals and are also barreled nuclear Montel bornological Mackey spaces.
It may be shown that for any compact subset, the natural inclusion is an embedding of TVSs.
Furthermore, a sequence in converges in if and only if there exists some such that contains this sequence and this sequence converges in.
From the universal property of direct limits, we know that if is a linear map into a locally convex space Y, then u is continuous u is bounded for every, 's restriction to,, is continuous.
A subset B of is bounded in if and only if there exists some such that and B is a bounded subset of.
Moreover, if is compact and then S is bounded in if and only if it is bounded in.
For any, any bounded subset of is a relatively compact subset of , where.
For all compact K ⊆ U, the interior of in is empty so is of the first category in itself.
By Baire's theorem, is not metrizable.
The bilinear multiplication map given by is not continuous; it is however, hypocontinuous.

Topology defined via neighborhoods

The canonical LF topology may also be defined by defining the neighborhoods of the origin as follows: if U is a convex subset of, then U is a neighborhood of the origin in the canonical LF topology if and only if for all compact, is a neighborhood of the origin in.

Topology defined via differential operators

A linear differential operator in U with C∞ coefficients is a sum where all but finitely many of the C∞ functions are identically 0.
The integer is called the order of the differential operator, where when.
If is a linear differential operator of order k then it induces a canonical linear map defined by, where we shall reuse notation and also denote this map by.
For any, the canonical LF topology on is the weakest locally convex TVS topology making all linear differential operators in U of order < k + 1 into continuous maps from into.

Sequentional definition of the topology on D(''U'')

The space D of test functions on U, which is a real vector space, can be given a topology by defining the limit of a sequence of elements of D.
A sequence in D is said to converge to ' ∈ D if the following two conditions hold:
With this definition, D becomes a complete locally convex topological vector space satisfying the Heine–Borel property.
This topology can be placed in the context of the following general construction: let
be a countable increasing union of locally convex topological vector spaces and ιi : Xi → X be the inclusion maps.
In this context, the inductive limit topology, or final topology, τ on X is the finest locally convex vector space topology making all the inclusion maps continuous.
The topology τ can be explicitly described as follows: let β be the collection of convex balanced subsets W of X such that W ∩ Xi is open for all i.
A base for the inductive limit topology τ then consists of the sets of the form x + W, where x in X and W in β.
The proof that τ is a vector space topology makes use of the assumption that each Xi is locally convex. By construction, β is a local base for τ.
That any locally convex vector space topology on X must necessarily contain τ means it is the weakest one.
One can also show that, for each i, the subspace topology Xi inherits from τ coincides with its original topology.
When each Xi is a Fréchet space, is called an LF space.
Now let U be the union of Ui where is a countable nested family of open subsets of U with compact closures Ki = i. Then we have the countable increasing union
where is the set of all smooth functions on U with support lying in Ki.
On each, consider the topology given by the seminorms
i.e. the topology of uniform convergence of derivatives of arbitrary order. This makes each a Fréchet space.
The resulting LF space structure on D is the topology described in the beginning of the section.
On D, one can also consider the topology given by the seminorms
However, this topology has the disadvantage of not being complete. On the other hand, because of the particular features of the 's, a set is bounded with respect to τ if and only if it lies in some 's.
The completeness of then follow from that of DKi's.
The topology τ is not metrizable by the Baire category theorem, since D is the union of subspaces of the first category in D.

Distributions

A distribution is a continuous linear functional on.
If T is a linear functional on then the following are equivalent:
for all test functions f with support contained in K.
We have the canonical duality pairing between a distribution and a test function ), which is denoted using angle brackets by
so that T,' = T.
One interprets this notation as the distribution T acting on the test function ' to give a scalar, or symmetrically as the test function acting on the distribution T.

Topology on the space of distributions

The space of distributions, denoted by, is the continuous dual space of with the topology of uniform convergence on bounded subsets of , where this topology is also called the strong dual topology.
This topology is chosen because it is with this topology that D′ becomes a nuclear Montel space and it is with this topology that the kernels theorem of Schwartz holds.
No matter what dual topology is placed on D′, a sequence of distributions converges in this topology if and only if it converges pointwise, which is why the topology is sometimes defined to be the weak-* topology.
No matter which topology is chosen, D′ will be a non-metrizable, locally convex topological vector space.
Each of,,, and are nuclear Montel spaces.
One reason for giving the canonical LF topology is because it is with this topology that and its continuous dual space both become nuclear spaces, which have many nice properties and which may be viewed as a generalization of finite-dimensional spaces.
It is precisely because is a nuclear space that the Schwartz kernel theorem holds, as Alexander Grothendieck discovered when he investigated why the theorem works for the space of distributions but not for other "nice" spaces like the Hilbert space .
One of the primary results of the Schwartz kernel theorem is that for any open subsets and, the canonical map is an isomorphism of TVSs ;
this result is false if one replaces the space with and replaces with the dual of this space.
Each of and is a nuclear Montel space, which implies that they are each reflexive, barreled, Mackey, and have the Heine-Borel property. Other consequences are that:
A sequence of distributions converges with respect to the weak-* topology on D′ to a distribution T if and only if
for every test function ' in D.
For example, if fk : R → R' is the function
and Tk is the distribution corresponding to fk, then
as k → ∞, so Tk → δ in D′.
Thus, for large k, the function fk'' can be regarded as an approximation of the Dirac delta distribution.

Localization of distributions

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

Restrictions to an open subset

Let U and V be open subsets of Rn with V ⊂ U.
Let InVU : D → D be the operator which extends by zero a given smooth function compactly supported in V to a smooth function compactly supported in the larger set U.
The transpose of InVU is called the restriction mapping and is denoted by.
The map InVU : D → D is a continuous injection where if then it is not a topological embedding and its range is not dense in D, which implies that this map's transpose is neither injective nor surjective and that the topology that InVU transfers from D onto its image is strictly finer than the subspace topology that D induces on this same set.
A distribution S ∈ D′ is said to be extendible to U if it belongs to the range of the transpose of InVU and it is called extendible if it is extendable to.
For any distribution T ∈ D′, the restriction ρVU is a distribution in the dual space D′ defined by
for all test functions ' ∈ D.
Unless U = V, the restriction to V is neither injective nor surjective. Lack of surjectivity follows since distributions can blow up towards the boundary of V. For instance, if U = R' and V'' = , then the distribution
is in D′ but admits no extension to D′.

Gluing and distributions that vanish in a set

If V is an open subset of U then we say that a distribution vanishes in V if for each test function, if then.
T vanishes in V if and only if the restriction of T to V is equal to 0, or equivalently, if and only if T lies in the kernel of the restriction map ρVU.

Support of a distribution

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

Support in a point set and Dirac measures

For any, let denote the distribution induced by the Dirac measure at x.
For any and distribution, the support of T is contained in if and only if T is a finite linear combination of derivatives of the Dirac measure at x, where if in addition the order of T is then so some scalars .

Decomposition of distributions

Operations on distributions

Many operations which are defined on smooth functions with compact support can also be defined for distributions. In general, if A : D → D is a linear mapping of vector spaces which is continuous with respect to the weak-* topology, then it is possible to extend A to a mapping A : D′ → D′ by passing to the limit.
In practice, however, it is more convenient to define operations on distributions by means of the transpose. If A : D → D is a continuous linear operator, then the transpose is an operator At : D → D such that for, is the map satisfying

If such an operator At exists and is continuous on D, then the original operator A may be extended to D′ by defining AT for a distribution T as

Differential operators

Differentiation

Suppose A : D → D is the partial derivative operator
If and ψ are in D, then an integration by parts gives
so that At = −A. This operator is a continuous linear transformation on D. So, if T ∈ D′ is a distribution, then the partial derivative of T with respect to the coordinate xk is defined by the formula
With this definition, every distribution is infinitely differentiable, and the derivative in the direction xk is a linear operator on D′.
More generally, if α = is an arbitrary multi-index and ∂α is the associated partial derivative operator, then the partial derivative ∂αT of the distribution T ∈ D′ is defined by
Differentiation of distributions is a continuous operator on D′;
this is an important and desirable property that is not shared by most other notions of differentiation.
If T is a distribution in then in where D T is the derivative of T and is translation by x;
thus the derivative of T may be viewed as a limit of quotients.

Differential operators

A linear differential operator in U with C∞ coefficients is a sum
where all but finitely many of the C∞ functions are identically 0.
The integer is called the order of the differential operator, where when.
If is a linear differential operator of order k then it deduces a canonical linear map defined by, where we shall reuse notation and again denote this map by.
The restriction of the canonical map to induces a continuous linear map
as well as another continuous linear map
;
the transposes of these two map are consequently continuous linear maps and, respectively, where denotes the continuous dual space of.
Recall that the transpose of a continuous linear map is the map defined by, or equivalently, it is the unique map satisfying for all and all.

Differential operators acting on distributions and formal transposes

For any two complex-valued functions f and g on U, define and if and then define.
Given any, let
be the canonical distribution defined by.
We want to extend the action of a differential linear operator
to distributions, where this extension will be the map denoted by.
One property that this extension should reasonably be required to have is that for any, where is the smooth function that results from applying the differential operator to.
It is natural to consider the transpose map
that canonically induces but this is not the extension of to distributions that we want because it doesn't have the aforementioned property.
But by investigating this map's action on, we will be led to the appropriate definitions.
Note that for all
Since has compact support, so too does every function so when we integrate by parts we get
.
We now define the formal transpose of, denoted by, to be the differential operator in U defined by
where
with and with if and only if for all i.
This definition stems from the fact that by using the Leibniz rule, we obtain
We've thus shown that
for all from which it follows that
.
Note that the formal transpose of the formal transpose is the original differential operator, i.e..
The formal transpose of induces a continuous linear map from into, whose transpose will be denoted by or or simply and called a differential operator, where this map is a linear map.
Explicitly, for any and,.
For any, we have, which justifies our definition.
If converges to T in then for every multi-index, converges to in.
Multiplication of a distribution by smooth function
Observe that a differential operator of degree 0 is just multiplication by a C∞ function.
And conversely, if then is a differential operator of degree 0, whose formal transpose is itself.
The induced differential operator maps a distribution T to a distribution denoted by.
We have thus defined the multiplication of a distribution by a function.
If converges to T in and if converges to f in then converges to f T in.
We now give an alternative presentation of multiplication by a smooth function.
; Multiplication by a smooth function
If m : U → R is an infinitely differentiable function and T is a distribution on U, then the product mT is defined by
This definition coincides with the transpose definition since if M : D → D is the operator of multiplication by the function m, then
so that Mt = M.
Under multiplication by smooth functions, D′ is a module over the ring C∞. With this definition of multiplication by a smooth function, the ordinary product rule of calculus remains valid. However, a number of unusual identities also arise. For example, if δ is the Dirac delta distribution on R, then mδ = m'δ, and if δ′ is the derivative of the delta distribution, then
These definitions of differentiation and multiplication also make it possible to define the operation of a linear differential operator with smooth coefficients on a distribution. A linear differential operator P takes a distribution T ∈ D′ to another distribution PT given by a sum of the form
where the coefficients pα are smooth functions on U. The action of the distribution PT on a test function ' is given by
The minimum integer k for which such an expansion holds for every distribution T is called the order of P.
The space D′ is a D-module with respect to the action of the ring of linear differential operators.
The bilinear multiplication map given by is not continuous; it is however, hypocontinuous.

Composition with a smooth function

Let T be a distribution on an open set U ⊂ Rn. Let V be an open set in Rn, and F : V → U. Then provided F is a submersion, it is possible to define
This is the composition of the distribution T with F, and is also called the pullback of T along F, sometimes written
The pullback is often denoted F*, although this notation should not be confused with the use of '*' to denote the adjoint of a linear mapping.
The condition that F be a submersion is equivalent to the requirement that the Jacobian derivative dF of F is a surjective linear map for every x ∈ V. A necessary condition for extending F# to distributions is that F be an open mapping. The inverse function theorem ensures that a submersion satisfies this condition.
If F is a submersion, then F# is defined on distributions by finding the transpose map. Uniqueness of this extension is guaranteed since F# is a continuous linear operator on D. Existence, however, requires using the change of variables formula, the inverse function theorem and a partition of unity argument.
In the special case when F is a diffeomorphism from an open subset V of Rn onto an open subset U of Rn change of variables under the integral gives
In this particular case, then, F# 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 f and g are functions on then we denote by f ∗ g the convolution of f and g, defined at to be the integral
provided that the integral exists.
If are such that 1/r = + - 1 then for any functions and we have and
.
If f and g are continuous functions on, at least one of which has compact support, then and if then the value of f ∗ g in the set A do not depend on the values of f 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 τa is sends a function to the function defined by.
This can be extended by the transpose to distributions in the following way: given a distribution T, the translation of T by a is the distribution defined by.
Given a function, define the function by.
Given a distribution T, let be the distribution defined by.
The operator is called the symmetry with respect to the origin.

Convolution of a smooth function with a distribution

Let and and assume that at least one of f and T has compact support.
For any, note that is the function given by.
The convolution of f and T, denoted by or by, is the smooth function defined by.
We have.
For all multi-indices p, it satisfies
and
.
If T is a distribution then the map is continuous as a map where if in addition T has compact support then it 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 H be the Heavyside function on and be the Dirac measure at 0.
Then, where is the derivative of, and.
Moreover, for any,.
Importantly, the associative law fails to hold:

Convolution of a test function with a distribution

If is a compactly supported smooth test function, then convolution with f defines a linear map given by, which is continuous with respect to the canonical LF space topology on.
Convolution of f with a distribution can be defined by taking the transpose of Cf relative to the duality pairing of with the space of distributions. If f, g, , then by Fubini's theorem
where.
Extending by continuity, the convolution of f with a distribution T is defined by
for all test functions.
An alternative way to define the convolution of a function f and a distribution T is to use the translation operator τa.
The convolution of the compactly supported function f and the distribution T 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 T has compact support then if f is a polynomial then the same is true of.
If the distribution T has compact support as well, then f∗T is a compactly supported function, and the Titchmarsh convolution theorem implies that
where ch denotes the convex hull and supp denotes the support.

Convolution of distributions

It is also possible to define the convolution of two distributions S and T on, provided one of them has compact support.
Informally, in order to define S∗T where T 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 S and T are distributions and that S has compact support.
Then the linear map defined by is continuous as is the linear map defined by.
The transposes of these maps,
and
,
are consequently continuous and one may show that
.
This common value is called the convolution' of S and T and it is a distribution that is denoted by or.
It satisfies.
If S and T are two distributions, at least one of which has compact support, then for any,.
If T is a distribution in and if is a Dirac measure then.
Suppose that it is T that has compact support.
For any test function, consider the function
It can be readily shown that this defines a smooth function of x, which moreover has compact support. The convolution of S and T 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 S and T.
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 not continuous although it is separately continuous.
The convolution maps and given by both fail'' to be continuous.
Each of these non-continuous maps is, however, separately continuous and hypocontinuous.

Tensor product and Fubini's theorem

Let U be an open subset of .
And suppose that all vector fields are over the field, where is either the real numbers or else the complex numbers.
For any, if then define to be the map ;
if then define to be the map.
If S is a distribution in U, then define a map by and if T is a distribution in V then define a map by.
Note that and.
The distribution S on U thus produces a continuous linear map defined by, where if in addition the support of S is compact then it also induces a continuous linear map of.
Similarly, a distribution T on V produces a continuous linear map defined by, where if in addition the support of T is compact then it also induces a continuous linear map of.
Let S be a distribution on U and let T be a distribution on V. Then the tensor product of S and T is the distribution in, denoted by or, defined by.
There is thus a bilinear map given by ;
the span of the range of this map is a dense subspace of its codomain.
Furthermore,.
The tensor product of distributions induces a continuous bilinear map, where denotes the space of distributions with compact support.
The tensor product as the map is also continuous, where is the Schwartz space of rapidly decreasing functions.
These maps induce canonical surjective TVS-isomorphisms
and
,
where represents the completion of the injective tensor product.
Furthermore, there are canonical TVS-isomorphisms
and

Schwartz kernel theorem

The Schwartz kernel theorem states that:
where all of these TVS-isomorphisms are canonical.
This result is false if one replaces the space with a Hilbert space and replaces with the dual of this space.
Why does such a nice result hold for the space of distributions and test functions but not for the Hilbert space ?
This question led Alexander Grothendieck to discover nuclear spaces, nuclear maps, and the injective tensor product.

Spaces of distributions

For all and all, all of the following canonical injections are continuous and have a range that is dense in their codomain:
as are the following
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 the codomain.
Indeed, is even sequentially dense in every.
All of the canonical injections are continuous and the range of this injection is dense in the codomain if and only if .
Suppose that X is one of the spaces or or .
Since the canonical injection is a continuous injection whose image is dense in the codomain, the transpose is a continuous injection.
This transpose thus allows us to identify with a certain vector subspace of the space of distributions.
This transpose map is not necessarily a TVS-embedding so that topology that this map transfers to the image is finer than the subspace topology that this space inherits from.
A linear subspace of carrying a locally convex topology that is finer than the subspace topology induced by is called a space of distributions.
Almost all of the spaces of distributions mentioned in this article arise in this way and any representation theorem about the dual space of X may, through the transpose, be transferred directly to elements of the space.

Radon measures

The natural inclusion 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,
Through the injection, every Radon measure becomes a distribution on U.
If f is a locally integrable function on U 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 in U:
Theorem:
Suppose that T is a distribution in that is a Radon measure, is a neighborhood of the support of T, and, where is the set of non-negative integers.
There exists is a family of locally functions in U such that
and for very,.
;Positive Radon measures
A linear function on a space of functions is called positive if whenever a function f that belongs to the domain of T is non-negative then.
One may show that every positive linear functional on is necessarily continuous.
Note that 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 f : U → R is called locally integrable if it is Lebesgue integrable over every compact subset K of U. This is a large class of functions which includes all continuous functions and all Lp functions. The topology on D is defined in such a fashion that any locally integrable function f yields a continuous linear functional on D – that is, an element of D′ – denoted here by Tf, whose value on the test function ' is given by the Lebesgue integral:
Conventionally, one abuses notation by identifying Tf with f, provided no confusion can arise, and thus the pairing between Tf and ' is often written
If f and g are two locally integrable functions, then the associated distributions Tf and Tg are equal to the same element of D′ if and only if f and g are equal almost everywhere. In a similar manner, every Radon measure μ on U defines an element of D′ whose value on the test function ' is ∫' dμ. As above, it is conventional to abuse notation and write the pairing between a Radon measure μ 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.
The test functions are themselves locally integrable, and so define distributions. As such they are dense in D′ with respect to the topology on D′ in the sense that for any distribution T ∈ D′, there is a net 'i ∈ D such that
for all Ψ ∈ D. This fact follows from the Hahn–Banach theorem, since the dual of D′ with its weak-* topology is the space D. A stronger result of sequential density can be proven more constructively by a convolution argument.

Distributions with compact support

The natural inclusion is a continuous injection whose image is dense in its codomain, so the transpose is also a continuous injection.
Thus, the image of, denoted by, forms a space of distributions when it is endowed with the strong dual topology of .
The elements of can be identified as the space of distributions with compact support.
Explicitly, if T is a distribution on U then the following are equivalent,
Compactly supported distributions define continuous linear functionals on the space C∞;
recall that the topology on C∞ is defined such that a sequence of test functions 'k converges to 0 if and only if all derivatives of k converge uniformly to 0 on every compact subset of U.
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 Cc∞ to C∞.
Suppose that T is a distribution in U with compact support K and let V be an open subset of U containing K.
Since every distribution with compact support has finite order, let N be the order of T and let.
Then there exists a family of continuous function defined on U with support in V such that .

Distributions of finite order

Suppose that k is a non-negative integer.
The natural inclusion is a continuous injection whose image is dense in its codomain, so the transpose is also a continuous injection.
Thus, the image of, denoted by, forms a space of distributions when it is endowed with the strong dual topology of .
The elements of are the distributions of order ≤ k.
Note that the distributions of order ≤ 0, which are also called distributions of order 0, are exactly the distributions that are Radon measures.
For, a distribution of order k is a distribution of order ≤ k that is not a distribution of order ≤ k - 1.
A distribution is said to be of finite order if there is some integer k such that it is a distribution of order ≤ k', 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 U is a distribution of finite order.
Indeed, every distribution in U is locally a distribution of finite order, in the following sense:
If V is an open and relatively compact subset of U and if is the restriction mapping from U to V, 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:
Theorem:
Suppose that, T is a distribution of order ≤ k in, and, where is the set of non-negative integers.
Given any open subset V of U containing the support of T, there is a family of Radon measures in U'',, such that
and for very,.

Tempered distributions and Fourier transform

By using a larger space of test functions S,
one can define the space of tempered distributions
S′, a subspace of D′. These distributions are useful if one studies the Fourier transform: all tempered distributions have a Fourier transform, but not all distributions in D′ have one.
;Schwartz space
The space of test functions employed here, the so-called Schwartz space S, is the function space of all infinitely differentiable 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 |x|, converges towards 0 for |x| → ∞. These functions form a complete topological vector space with a suitably defined family of seminorms. More precisely, let
for α, β multi-indices of size n. Then ' is a Schwartz function if all the values satisfy
The family of seminorms pα, β defines a locally convex topology on the Schwartz space.
When n is equal to 1, the seminorms are, in fact, norms on the Schwartz space.
One can also use the following family of seminorms to define the topology:
for.
Otherwise, one can define a norm on S via
The Schwartz space is a Fréchet space.
Because the Fourier transform changes differentiation by xα into multiplication by xα and vice versa, this symmetry implies that the Fourier transform of a Schwartz function is also a Schwartz function.
A sequence of functions in S converges to 0 in S 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.
The space of test functions is dense in S are is the set of all analytic functions that belong to S.
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 natural inclusion is a continuous injection whose image is dense in its codomain, so the transpose is also a continuous injection.
Thus, the image of, denoted by, forms a space of distributions when it is endowed with the strong dual topology of .
The space is called the space of tempered distributions is it is the continuous dual of the Schwartz space.
Equivalently, a distribution T is a tempered distribution if and only if
is true whenever
holds for all multi-indices α, β.
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 Lp for p ≥ 1 are tempered distributions.
The tempered distributions can also be characterized as slowly growing, meaning that each derivative of T grows at most as fast as some polynomial.
This characterization is dual to the rapidly falling behaviour of the derivatives of a function in the Schwartz space, where each derivative of ' decays faster than every inverse power of |x|.
An example of a rapidly falling function is for any positive n, λ, β.
;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 function space, and we will define the Fourier transform to be its transpose, which will again be denoted by F.
So the Fourier transform of the tempered distribution T is defined by = T for every Schwartz function Ψ.
FT is thus again a tempered distribution.
The Fourier transform is an isomorphism of TVSs from the space of tempered distributions onto itself.
This operation is compatible with differentiation in the sense that
and also with convolution: if T is a tempered distribution and Ψ is a slowly increasing infinitely differentiable function on R'n, then ψT is again a tempered distribution and
is the convolution of FT and Fψ. In particular, the Fourier transform of the constant function equal to 1 is the δ'' distribution.

Distributions as derivatives of continuous functions

The formal definition of distributions exhibits them as a subspace of a very large space, namely the topological dual of D. 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.

Tempered distributions

If f ∈ S′ is a tempered distribution, then there exists a constant C > 0, and positive integers M and N such that for all Schwartz functions ∈ S
This estimate along with some techniques from functional analysis can be used to show that there is a continuous slowly increasing function F and a multi-index α such that

Restriction of distributions to compact sets

If f ∈ D′, then for any compact set K ⊂ Rn, there exists a continuous function F compactly supported
in Rn and a multi-index α such that f = DαF on Cc∞.
This follows from the previously quoted result on tempered distributions by means of a localization argument.

Distributions with point support

If f has support at a single point, then f is in fact a finite linear combination of distributional derivatives of the δ function at P. That is, there exists an integer m and complex constants aα for multi-indices |α| ≤ m such that
where τP is the translation operator.

General distributions

A version of the above theorem holds locally in the following sense. Let T be a distribution on U, then one can find for every multi-index α a continuous function gα such that
and that any compact subset K of U intersects the supports of only finitely many gα; therefore, to evaluate the value of T for a given smooth function f compactly supported in U, we only need finitely many gα; hence the infinite sum above is well-defined as a distribution. If the distribution T is of finite order, then one can choose gα in such a way that only finitely many of them are nonzero.

Using holomorphic functions as test functions

The success of the theory led to 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.

Problem of multiplication

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 p.v. 1/x is the distribution obtained by the Cauchy principal value
for all ∈ S, and δ 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 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 causal perturbation theory. 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 structure, 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.

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 convolution operators and
the space of all ordinary functions within the space of tempered distributions is also
called the space of multiplication operators.
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 convolution operators and
Paley-Wiener functions, better known as bandlimited functions,
belong to the space of multiplication operators.
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,
e.g. 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.