Distribution (mathematics) explained

Distributions, also known as Schwartz distributions or generalized functions, 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 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.

f

is normally thought of as on the in the function domain by "sending" a point in its domain to the point

f(x).

Instead of acting on points, distribution theory reinterprets functions such as

f

as acting on in a certain way. In applications to physics and engineering, are usually infinitely differentiable complex-valued (or real-valued) functions with compact support that are defined on some given non-empty open subset

U\subseteq\Rn

. (Bump functions are examples of test functions.) The set of all such test functions forms a vector space that is denoted by
 infty(U) C c
or

l{D}(U).

Most commonly encountered functions, including all continuous maps

f:\R\to\R

if using

U:=\R,

can be canonically reinterpreted as acting via "integration against a test function." Explicitly, this means that

f

"acts on" a test function

\psi\inl{D}(\R)

by "sending" it to the number $\int_\R f \, \psi \, dx,$ which is often denoted by

Df(\psi).

This new action $\psi \mapsto D_f(\psi)$ of

f

is a scalar-valued map, denoted by

Df,

whose domain is the space of test functions

l{D}(\R).

This functional

Df

turns out to have the two defining properties of what is known as a : it is linear and also continuous when

l{D}(\R)

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

Df

that arise from functions in this way are prototypical examples of distributions, but there are many 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. It is nonetheless still possible to 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

 infty(U) C c
that is continuous when
 infty(U) C c
is given a topology called the . This leads to space of (all) distributions on

U

, usually denoted by

l{D}'(U)

(note the prime), which by definition is the space of all distributions on

U

(that is, it is the continuous dual space of
 infty(U) C c
); it is these distributions that are the main focus of this article.

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 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.

Notation

n

is a fixed positive integer and

U

is a fixed non-empty open subset of Euclidean space

\Rn.

\N=\{0,1,2,\ldots\}

denotes the natural numbers.

k

will denote a non-negative integer or

infty.

• If

f

is a function then

\operatorname{Dom}(f)

will denote its domain and the of

f,

denoted by

\operatorname{supp}(f),

is defined to be the closure of the set

\{x\in\operatorname{Dom}(f):f(x)0\}

in

\operatorname{Dom}(f).

• For two functions

f,g:U\to\Complex,

the following notation defines a canonical pairing: $\langle f, g\rangle := \int_U f(x) g(x) \,dx.$
• A of size

n

is an element in

\Nn

(given that

n

is fixed, if the size of multi-indices is omitted then the size should be assumed to be

n

). The of a multi-index

\alpha=(\alpha1,\ldots,\alphan)\in\Nn

is defined as

\alpha1+ … +\alphan

and denoted by

|\alpha|.

Multi-indices are particularly useful when dealing with functions of several variables, in particular we introduce the following notations for a given multi-index

\alpha=(\alpha1,\ldots,\alphan)\in\Nn

: $\beginx^\alpha &= x_1^ \cdots x_n^ \\\partial^\alpha &= \frac\end$ We also introduce a partial order of all multi-indices by

\beta\ge\alpha

if and only if

\betai\ge\alphai

for all

1\lei\len.

When

\beta\ge\alpha

we define their multi-index binomial coefficient as: $\binom := \binom \cdots \binom.$

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 are given in the article on spaces of test functions and distributions.

For all

j,k\in\{0,1,2,\ldots,infty\}

and any compact subsets and of, we have:$\beginC^k(K) &\subseteq C^k_c(U) \subseteq C^k(U) \\C^k(K) &\subseteq C^k(L) && \text K \subseteq L \\C^k(K) &\subseteq C^j(K) && \text j \le k \\C_c^k(U) &\subseteq C^j_c(U) && \text j \le k \\C^k(U) &\subseteq C^j(U) && \text j \le k \\\end$

Distributions on are continuous linear functionals on

 infty(U) C c
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 functional on
 infty(U) C c
that are often straightforward to verify.

Proposition: A linear functional on

 infty(U) C c
is continuous, and therefore a distribution, if and only if either of the following equivalent conditions are satisfied:
1. For every compact subset

K\subseteqU

there exist constants

C>0

and

N\in\N

(dependent on

K

) such that for all

f\in

 infty(U) C c
with support contained in

K

, $|T(f)| \leq C \sup \$
 x \in U, \alpha \leq N\
.
1. For every compact subset

K\subseteqU

and every sequence

\{fi\}

 infty i=1
in
 infty(U) C c
whose supports are contained in

K

, if

\{\partial\alphafi\}

 infty i=1
converges uniformly to zero on

U

for every multi-index

\alpha

, then

T(fi)\to0.

Topology on Ck(U)

We now introduce the seminorms that will define the topology on

Ck(U).

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

\R

-valued[1] seminorms on

Ck(U).

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 $\beginA ~:= \quad &\ \\B ~:= \quad &\ \\C ~:= \quad &\ \\D ~:= \quad &\\end$generate the same locally convex vector topology on

Ck(U)

(so for example, the topology generated by the seminorms in

A

is equal to the topology generated by those in

C

).

With this topology,

Ck(U)

becomes a locally convex Fréchet space that is normable. Every element of

A\cupB\cupC\cupD

is a continuous seminorm on

Ck(U).

Under this topology, a net

(fi)i\in

in

Ck(U)

converges to

f\inCk(U)

if and only if for every multi-index

p

with

|p|<k+1

and every compact

K,

the net of partial derivatives

\left(\partialpfi\right)i

converges uniformly to

\partialpf

on

K.

For any

k\in\{0,1,2,\ldots,infty\},

any (von Neumann) bounded subset of

Ck+1(U)

is a relatively compact subset of

Ck(U).

In particular, a subset of

Cinfty(U)

is bounded if and only if it is bounded in

Ci(U)

for all

i\in\N.

The space

Ck(U)

is a Montel space if and only if

k=infty.

A subset

W

of

Cinfty(U)

is open in this topology if and only if there exists

i\in\N

such that

W

is open when

Cinfty(U)

is endowed with the subspace topology induced on it by

Ci(U).

Topology on Ck(K)

As before, fix

k\in\{0,1,2,\ldots,infty\}.

Recall that if

K

is any compact subset of

U

then

Ck(K)\subseteqCk(U).

If

k

is finite then

Ck(K)

is a Banach space with a topology that can be defined by the norm$r_K(f) := \sup_$ \left(\sup_ \left|\partial^p f(x_0)\right| \right).And when

k=2,

then

Ck(K)

is even a Hilbert space.

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

Suppose

U

is an open subset of

\Rn

and

K\subseteqU

is a compact subset. By definition, elements of

Ck(K)

are functions with domain

U

(in symbols,

Ck(K)\subseteqCk(U)

) so the space

Ck(K)

and its topology depends on

U;

to make this dependence on the open set

U

clear, temporarily denote

Ck(K)

by

Ck(K;U).

Importantly, changing the set

U

to a different open subset

U'

(with

K\subseteqU'

) will change the set

Ck(K)

from

Ck(K;U)

to

Ck(K;U'),

[2] so that elements of

Ck(K)

will be functions with domain

U'

U.

Despite

Ck(K)

depending on the open set (

UorU'

), the standard notation for

Ck(K)

makes no mention of it. This is justified because, as this subsection will now explain, the space

Ck(K;U)

is canonically identified as a subspace of

Ck(K;U')

(both algebraically and topologically).

It is enough to explain how to canonically identify

Ck(K;U)

with

Ck(K;U')

when one of

U

and

U'

is a subset of the other. The reason is because if

V

and

W

are arbitrary open subset of

\Rn

containing

K

then the open set

U:=V\capW

also contains

K,

so that each of

Ck(K;V)

and

Ck(K;W)

is canonically identified with

Ck(K;V\capW)

and now by transitivity,

Ck(K;V)

is thus identified with

Ck(K;W).

So assume

U\subseteqV

are open subset of

\Rn

containing

K.

Given

f\in

 k(U), C c
its is the function

F:V\to\Complex

defined by:$F(x) = \beginf(x) & x \in U, \\0 & \text.\end$This trivial extension belongs to

Ck(V)

(because

f\in

 k(U) C c
has compact support) and it will be denoted by

I(f)

(that is,

I(f):=F

). The assignment

f\mapstoI(f)

thus induces a map

I:

 k(U) C c

\toCk(V)

that sends a function in
 k(U) C c
to its trivial extension on

V.

This map is a linear injection and for every compact subset

K\subseteqU

(where

K

is also a compact subset of

V

since

K\subseteqU\subseteqV

), $\beginI\left(C^k(K; U)\right) &~=~ C^k(K; V) \qquad \text \\I\left(C_c^k(U)\right) &~\subseteq~ C_c^k(V).\end$If

I

is restricted to

Ck(K;U)

then the following induced linear map is a homeomorphism (linear homeomorphisms are called):$\begin \,& C^k(K; U) && \to \,&& C^k(K;V) \\ & f && \mapsto\,&& I(f) \\\end$and thus the next map is a topological embedding:$\begin \,& C^k(K; U) && \to \,&& C^k(V) \\ & f && \mapsto\,&& I(f). \\\end$Using the injection$I : C_c^k(U) \to C^k(V)$the vector space
 k(U) C c
is canonically identified with its image in
 k(V) C c

\subseteqCk(V).

Because

Ck(K;U)\subseteq

 k(U), C c
through this identification,

Ck(K;U)

can also be considered as a subset of

Ck(V).

Thus the topology on

Ck(K;U)

is independent of the open subset

U

of

\Rn

that contains

K,

which justifies the practice of writing

Ck(K)

Ck(K;U).

Canonical LF topology

See main article: Spaces of test functions and distributions.

Recall that

 k(U) C c
denote all those functions in

Ck(U)

that have compact support in

U,

where note that
 k(U) C c
is the union of all

Ck(K)

as

K

ranges over all compact subsets of

U.

Moreover, for every

k,

 k(U) C c
is a dense subset of

Ck(U).

The special case when

k=infty

gives us the space of test functions.

The canonical LF-topology is metrizable and importantly, it is than the subspace topology that

Cinfty(U)

induces on
 infty(U). C c
However, the canonical LF-topology does make
 infty(U) C c
into a complete reflexive nuclear Montel bornological barrelled Mackey space; the same is true of its strong dual space (that is, the space of all distributions with its usual topology). The canonical LF-topology can be defined in various ways.

Distributions

As discussed earlier, continuous linear functionals on a

 infty(U) C c
are known as distributions on

U.

Other equivalent definitions are described below.

There is a canonical duality pairing between a distribution

T

on

U

and a test function

f\in

 infty(U), C c
which is denoted using angle brackets by$\begin\mathcal'(U) \times C_c^\infty(U) \to \R \\(T, f) \mapsto \langle T, f \rangle := T(f)\end$

One interprets this notation as the distribution

T

acting on the test function

f

to give a scalar, or symmetrically as the test function

f

acting on the distribution

T.

Characterizations of distributions

Proposition. If

T

is a linear functional on
 infty(U) C c
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;
• explicitly, for every sequence

\left(fi\right)

 infty i=1
in
 infty(U) C c
that converges in
 infty(U) C c
to some

f\in

 infty(U), C c
$\lim_ T\left(f_i\right) = T(f);$[3]
1. is sequentially continuous at the origin; in other words, maps null sequences[4] to null sequences;
• explicitly, for every sequence

\left(fi\right)

 infty i=1
in
 infty(U) C c
that converges in
 infty(U) C c
to the origin (such a sequence is called a), $\lim_ T\left(f_i\right) = 0;$
• a is by definition any sequence that converges to the origin;
1. maps null sequences to bounded subsets;
• explicitly, for every sequence

\left(fi\right)

 infty i=1
in
 infty(U) C c
that converges in
 infty(U) C c
to the origin, the sequence

\left(T\left(fi\right)\right)

 infty i=1
is bounded;
1. maps Mackey convergent null sequences to bounded subsets;
• explicitly, for every Mackey convergent null sequence

\left(fi\right)

 infty i=1
in
 infty(U), C c
the sequence

\left(T\left(fi\right)\right)

 infty i=1
is bounded;
• a sequence

f\bull=\left(fi\right)

 infty i=1
is said to be if there exists a divergent sequence

r\bull=\left(ri\right)

 infty i=1

\toinfty

of positive real number such that the sequence

\left(rifi\right)

 infty i=1
is bounded; every sequence that is Mackey convergent to the origin necessarily converges to the origin (in the usual sense);
1. The kernel of is a closed subspace of
 infty(U); C c
1. The graph of is closed;
1. There exists a continuous seminorm

g

on
 infty(U) C c
such that

|T|\leqg;

1. There exists a constant

C>0

and a finite subset

\{g1,\ldots,gm\}\subseteql{P}

(where

l{P}

is any collection of continuous seminorms that defines the canonical LF topology on
 infty(U) C c
) such that

|T|\leqC(g1++gm);

[5]
1. For every compact subset

K\subseteqU

there exist constants

C>0

and

N\in\N

such that for all

f\inCinfty(K),

$|T(f)| \leq C \sup \$
 x \in U, \alpha \leq N\
;
1. For every compact subset

K\subseteqU

there exist constants

CK>0

and

NK\in\N

such that for all

f\in

 infty(U) C c
with support contained in

K,

[6] $|T(f)| \leq C_K \sup \$
 x \in K, \alpha \leq N_K\
;
1. For any compact subset

K\subseteqU

and any sequence

\{fi\}

 infty i=1
in

Cinfty(K),

if

\{\partialpfi\}

 infty i=1
converges uniformly to zero for all multi-indices

p,

then

T(fi)\to0;

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

The set of all distributions on

U

is the continuous dual space of
 infty(U), C c
which when endowed with the strong dual topology is denoted by

l{D}'(U).

Importantly, unless indicated otherwise, the topology on

l{D}'(U)

is the strong dual topology; if the topology is instead the weak-* topology then this will be clearly indicated. Neither topology is metrizable although unlike the weak-* topology, the strong dual topology makes

l{D}'(U)

into a complete nuclear space, to name just a few of its desirable properties.

Neither

 infty(U) C c
nor its strong dual

l{D}'(U)

is a sequential space and so neither of their topologies can be fully described by sequences (in other words, defining only what sequences converge in these spaces is enough to fully/correctly define their topologies).However, a in

l{D}'(U)

converges in the strong dual topology if and only if it converges in the weak-* topology (this leads many authors to use pointwise convergence to actually the convergence of a sequence of distributions; this is fine for sequences but this is guaranteed to extend to the convergence of nets of distributions because a net may converge pointwise but fail to converge in the strong dual topology).More information about the topology that

l{D}'(U)

is endowed with can be found in the article on spaces of test functions and distributions and in the articles on polar topologies and dual systems.

A map from

l{D}'(U)

into another locally convex topological vector space (such as any normed 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 (for example, that are not also locally convex topological vector spaces). The same is true of maps from
 infty(U) C c
(more generally, this is true of maps from any locally convex bornological space).

Localization of distributions

There is no way to define the value of a distribution in

l{D}'(U)

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

V\subseteqU

be open subsets of

\Rn.

Every function

f\inl{D}(V)

can be from its domain to a function on by setting it equal to

0

on the complement

U\setminusV.

This extension is a smooth compactly supported function called the and it will be denoted by

EVU(f).

This assignment

f\mapstoEVU(f)

defines the operator

EVU:l{D}(V)\tol{D}(U),

which is a continuous injective linear map. It is used to canonically identify

l{D}(V)

as a vector subspace of

l{D}(U)

(although as a topological subspace). Its transpose (explained here) $\rho_ := ^E_ : \mathcal'(U) \to \mathcal'(V),$ is called the and as the name suggests, the image

\rhoVU(T)

of a distribution

T\inl{D}'(U)

under this map is a distribution on

V

called the restriction of

T

to

V.

The defining condition of the restriction

\rhoVU(T)

is:$\langle \rho_ T, \phi \rangle = \langle T, E_ \phi \rangle \quad \text \phi \in \mathcal(V).$If

VU

then the (continuous injective linear) trivial extension map

EVU:l{D}(V)\tol{D}(U)

is a topological embedding (in other words, if this linear injection was used to identify

l{D}(V)

as a subset of

l{D}(U)

then

l{D}(V)

's topology would strictly finer than the subspace topology that

l{D}(U)

induces on it; importantly, it would be a topological subspace since that requires equality of topologies) and its range is also dense in its codomain

l{D}(U).

Consequently, if

VU

then the restriction mapping is neither injective nor surjective. A distribution

S\inl{D}'(V)

is said to be if it belongs to the range of the transpose of

EVU

and it is called if it is extendable to

\Rn.

Unless

U=V,

the restriction to is neither injective nor surjective. Lack of surjectivity follows since distributions can blow up towards the boundary of . For instance, if

U=\R

and

V=(0,2),

then the distribution$T(x) = \sum_^\infty n \, \delta\left(x-\frac\right)$is in

l{D}'(V)

l{D}'(U).

Gluing and distributions that vanish in a set

Let be an open subset of .

T\inl{D}'(U)

is said to if for all

f\inl{D}(U)

such that

\operatorname{supp}(f)\subseteqV

we have

Tf=0.

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

\rhoVU.

Support of a distribution

This last corollary implies that for every distribution on, there exists a unique largest subset of such that vanishes in (and does not vanish in any open subset of that is not contained in); the complement in of this unique largest open subset is called . Thus$\operatorname(T) = U \setminus \bigcup \.$

If

f

is a locally integrable function on and if

Df

is its associated distribution, then the support of

Df

is the smallest closed subset of in the complement of which

f

is almost everywhere equal to 0. If

f

is continuous, then the support of

Df

is equal to the closure of the set of points in at which

f

does not vanish. The support of the distribution associated with the Dirac measure at a point

x0

is the set

\{x0\}.

If the support of a test function

f

does not intersect the support of a distribution then

Tf=0.

A distribution is 0 if and only if its support is empty. If

f\inCinfty(U)

is identically 1 on some open set containing the support of a distribution then

fT=T.

If the support of a distribution is compact then it has finite order and furthermore, there is a constant

C

and a non-negative integer

N

such that:$|T \phi| \leq C\|\phi\|_N := C \sup \left\ \quad \text \phi \in \mathcal(U).$

If has compact support then it has a unique extension to a continuous linear functional

\widehat{T}

on

Cinfty(U)

; this functional can be defined by

\widehat{T}(f):=T(\psif),

where

\psi\inl{D}(U)

is any function that is identically 1 on an open set containing the support of .

If

S,T\inl{D}'(U)

and

λ0

then

\operatorname{supp}(S+T)\subseteq\operatorname{supp}(S)\cup\operatorname{supp}(T)

and

\operatorname{supp}(λT)=\operatorname{supp}(T).

Thus, distributions with support in a given subset

A\subseteqU

form a vector subspace of

l{D}'(U).

Furthermore, if

P

is a differential operator in, then for all distributions on and all

f\inCinfty(U)

we have

\operatorname{supp}(P(x,\partial)T)\subseteq\operatorname{supp}(T)

and

\operatorname{supp}(fT)\subseteq\operatorname{supp}(f)\cap\operatorname{supp}(T).

Distributions with compact support

Support in a point set and Dirac measures

For any

x\inU,

let

\deltax\inl{D}'(U)

denote the distribution induced by the Dirac measure at

x.

For any

x0\inU

and distribution

T\inl{D}'(U),

the support of is contained in

\{x0\}

if and only if is a finite linear combination of derivatives of the Dirac measure at

x0.

If in addition the order of is

\leqk

then there exist constants

\alphap

such that:$T = \sum_$
 \leq k
\alpha_p \partial^p \delta_.

Said differently, if has support at a single point

\{P\},

then is in fact a finite linear combination of distributional derivatives of the

\delta

function at . That is, there exists an integer and complex constants

a\alpha

such that$T = \sum_$
 \leq m
a_\alpha \partial^\alpha(\tau_P\delta)where

\tauP

is the translation operator.

Global structure of distributions

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

l{D}(U)

(or the Schwartz space

l{S}(\Rn)

for tempered distributions). 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 (multiple) 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.

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

T\inl{D}'(U)

we can write:$T = \sum_^\infty \sum_ \partial^p f_,$where

P1,P2,\ldots

are finite sets of multi-indices and the functions

fip

are continuous.

Note that the infinite sum above is well-defined as a distribution. The value of for a given

f\inl{D}(U)

can be computed using the finitely many

g\alpha

that intersect the support of

f.

Operations on distributions

Many operations which are defined on smooth functions with compact support can also be defined for distributions. In general, if

A:l{D}(U)\tol{D}(U)

is a linear map which is continuous with respect to the weak topology, then it is possible to extend

A

to a map

A:l{D}'(U)\tol{D}'(U)

by passing to the limit.[7]

Preliminaries: Transpose of a linear operator

See main article: Transpose of a linear map.

Operations on distributions and spaces of distributions are often defined by means of 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.[8] For instance, the well-known Hermitian adjoint of a linear operator between Hilbert spaces is just the operator's transpose (but with the Riesz representation theorem used to identify each Hilbert space with its continuous dual space). In general the transpose of a continuous linear map

A:X\toY

is the linear map $^A : Y' \to X' \qquad \text \qquad ^A(y') := y' \circ A,$ or equivalently, it is the unique map satisfying

\langley',A(x)\rangle=\left\langle{}tA(y'),x\right\rangle

for all

x\inX

and all

y'\inY'

(the prime symbol in

y'

does not denote a derivative of any kind; it merely indicates that

y'

is an element of the continuous dual space

Y'

). Since

A

is continuous, the transpose

{}tA:Y'\toX'

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 (see the articles polar topology and dual system for more details).

In the context of distributions, the characterization of the transpose can be refined slightly. Let

A:l{D}(U)\tol{D}(U)

be a continuous linear map. Then by definition, the transpose of

A

is the unique linear operator

At:l{D}'(U)\tol{D}'(U)

that satisfies:$\langle ^A(T), \phi \rangle = \langle T, A(\phi) \rangle \quad \text \phi \in \mathcal(U) \text T \in \mathcal'(U).$

Since

l{D}(U)

is dense in

l{D}'(U)

(here,

l{D}(U)

actually refers to the set of distributions

\left\{D\psi:\psi\inl{D}(U)\right\}

) it is sufficient that the defining equality hold for all distributions of the form

T=D\psi

where

\psi\inl{D}(U).

Explicitly, this means that a continuous linear map

B:l{D}'(U)\tol{D}'(U)

is equal to

{}tA

if and only if the condition below holds:$\langle B(D_\psi), \phi \rangle = \langle ^A(D_\psi), \phi \rangle \quad \text \phi, \psi \in \mathcal(U)$where the right hand side equals

\langle{}tA(D\psi),\phi\rangle=\langleD\psi,A(\phi)\rangle=\langle\psi,A(\phi)\rangle=\intU\psiA(\phi)dx.

Differential operators

Differentiation of distributions

Let

A:l{D}(U)\tol{D}(U)

be the partial derivative operator

\tfrac{\partial}{\partialxk}.

In order to extend

A

we compute its transpose:$\begin\langle ^A(D_\psi), \phi \rangle&= \int_U \psi (A\phi) \,dx && \text \\&= \int_U \psi \frac \, dx \\[4pt]&= -\int_U \phi \frac\, dx && \text \\[4pt]&= -\left\langle \frac, \phi \right\rangle \\[4pt]&= -\langle A \psi, \phi \rangle = \langle - A \psi, \phi \rangle\end$

Therefore

{}tA=-A.

Therefore, the partial derivative of

T

with respect to the coordinate

xk

is defined by the formula$\left\langle \frac, \phi \right\rangle = - \left\langle T, \frac \right\rangle \qquad \text \phi \in \mathcal(U).$

With this definition, every distribution is infinitely differentiable, and the derivative in the direction

xk

is a linear operator on

l{D}'(U).

More generally, if

\alpha

is an arbitrary multi-index, then the partial derivative

\partial\alphaT

of the distribution

T\inl{D}'(U)

is defined by$\langle \partial^\alpha T, \phi \rangle = (-1)^$
\langle T, \partial^\alpha \phi \rangle \qquad \text \phi \in \mathcal(U).

Differentiation of distributions is a continuous operator on

l{D}'(U);

this is an important and desirable property that is not shared by most other notions of differentiation.

If

T

is a distribution in

\R

then$\lim_ \frac = T'\in \mathcal'(\R),$where

T'

is the derivative of

T

and

\taux

is translation by

x;

thus the derivative of

T

may be viewed as a limit of quotients.

Differential operators acting on smooth functions

A linear differential operator in

U

with smooth coefficients acts on the space of smooth functions on

U.

Given such an operator$P := \sum_\alpha c_\alpha \partial^\alpha,$we would like to define a continuous linear map,

DP

that extends the action of

P

on

Cinfty(U)

to distributions on

U.

In other words, we would like to define

DP

such that the following diagram commutes:$\begin\mathcal'(U) & \stackrel & \mathcal'(U) \\[2pt]\uparrow & & \uparrow \\[2pt]C^\infty(U) & \stackrel & C^\infty(U)\end$where the vertical maps are given by assigning

f\inCinfty(U)

its canonical distribution

Df\inl{D}'(U),

which is defined by: $D_f(\phi) = \langle f, \phi \rangle := \int_U f(x) \phi(x) \,dx \quad \text \phi \in \mathcal(U).$ With this notation the diagram commuting is equivalent to:$D_ = D_PD_f \qquad \text f \in C^\infty(U).$

In order to find

DP,

the transpose

{}tP:l{D}'(U)\tol{D}'(U)

of the continuous induced map

P:l{D}(U)\tol{D}(U)

defined by

\phi\mapstoP(\phi)

is considered in the lemma below. This leads to the following definition of the differential operator on

U

called which will be denoted by

P*

to avoid confusion with the transpose map, that is defined by$P_* := \sum_\alpha b_\alpha \partial^\alpha \quad \text \quad b_\alpha := \sum_ (-1)^$
\binom \partial^ c_\beta.

As discussed above, for any

\phi\inl{D}(U),

the transpose may be calculated by:$\begin\left\langle ^P(D_f), \phi \right\rangle &= \int_U f(x) P(\phi)(x) \,dx \\&= \int_U f(x) \left[\sum\nolimits_\alpha c_\alpha(x) (\partial^\alpha \phi)(x) \right] \,dx \\&= \sum\nolimits_\alpha \int_U f(x) c_\alpha(x) (\partial^\alpha \phi)(x) \,dx \\&= \sum\nolimits_\alpha (-1)^$
\int_U \phi(x) (\partial^\alpha(c_\alpha f))(x) \,d x\end

For the last line we used integration by parts combined with the fact that

\phi

and therefore all the functions

f(x)c\alpha(x)\partial\alpha\phi(x)

have compact support.[9] Continuing the calculation above, for all

\phi\inl{D}(U):

$\begin\left\langle ^P(D_f), \phi \right\rangle &=\sum\nolimits_\alpha (-1)^$
\int_U \phi(x) (\partial^\alpha(c_\alpha f))(x) \,dx && \text \\[4pt]&= \int_U \phi(x) \sum\nolimits_\alpha (-1)^
(\partial^\alpha(c_\alpha f))(x)\,dx \\[4pt]&= \int_U \phi(x) \sum_\alpha \left[\sum_{\gamma \le \alpha} \binom{\alpha}{\gamma} (\partial^{\gamma}c_\alpha)(x) (\partial^{\alpha-\gamma}f)(x) \right] \,dx && \text\\&= \int_U \phi(x) \left[\sum_\alpha \sum_{\gamma \le \alpha} (-1)^{|\alpha|} \binom{\alpha}{\gamma} (\partial^{\gamma}c_\alpha)(x) (\partial^{\alpha-\gamma}f)(x)\right] \,dx \\&= \int_U \phi(x) \left[\sum_\alpha \left[ \sum_{\beta \geq \alpha} (-1)^{|\beta|} \binom{\beta}{\alpha} \left(\partial^{\beta-\alpha}c_{\beta}\right)(x) \right] (\partial^\alpha f)(x)\right] \,dx && \text f \\&= \int_U \phi(x) \left[\sum\nolimits_\alpha b_\alpha(x) (\partial^\alpha f)(x) \right] \, dx && b_\alpha:=\sum_ (-1)^
\binom \partial^c_ \\&= \left\langle \left(\sum\nolimits_\alpha b_\alpha \partial^\alpha \right) (f), \phi \right\rangle\end

The Lemma combined with the fact that the formal transpose of the formal transpose is the original differential operator, that is,

P**=P,

enables us to arrive at the correct definition: the formal transpose induces the (continuous) canonical linear operator

P*:

 infty(U) C c

\to

 infty(U) C c
defined by

\phi\mapstoP*(\phi).

We claim that the transpose of this map,

{}tP*:l{D}'(U)\tol{D}'(U),

can be taken as

DP.

To see this, for every

\phi\inl{D}(U),

compute its action on a distribution of the form

Df

with

f\inCinfty(U)

:

$\begin\left\langle ^P_*\left(D_f\right),\phi \right\rangle &= \left\langle D_, \phi \right\rangle && \text P_* \text P\\&= \left\langle D_, \phi \right\rangle && P_ = P\end$

We call the continuous linear operator

DP:={}tP*:l{D}'(U)\tol{D}'(U)

the . Its action on an arbitrary distribution

S

is defined via:$D_P(S)(\phi) = S\left(P_*(\phi)\right) \quad \text \phi \in \mathcal(U).$

If

(Ti)

 infty i=1
converges to

T\inl{D}'(U)

then for every multi-index

\alpha,(\partial\alphaTi)

 infty i=1
converges to

\partial\alphaT\inl{D}'(U).

Multiplication of distributions by smooth functions

A differential operator of order 0 is just multiplication by a smooth function. And conversely, if

f

is a smooth function then

P:=f(x)

is a differential operator of order 0, whose formal transpose is itself (that is,

P*=P

). The induced differential operator

DP:l{D}'(U)\tol{D}'(U)

maps a distribution

T

to a distribution denoted by

fT:=DP(T).

We have thus defined the multiplication of a distribution by a smooth function.

We now give an alternative presentation of multiplication of a distribution

T

on

U

by a smooth function

m:U\to\R.

The product

mT

is defined by$\langle mT, \phi \rangle = \langle T, m\phi \rangle \qquad \text \phi \in \mathcal(U).$

This definition coincides with the transpose definition since if

M:l{D}(U)\tol{D}(U)

is the operator of multiplication by the function

m

(that is,

(M\phi)(x)=m(x)\phi(x)

), then$\int_U (M \phi)(x) \psi(x)\,dx = \int_U m(x) \phi(x) \psi(x)\,d x = \int_U \phi(x) m(x) \psi(x) \,d x = \int_U \phi(x) (M \psi)(x)\,d x,$so that

{}tM=M.

Under multiplication by smooth functions,

l{D}'(U)

is a module over the ring

Cinfty(U).

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

\delta

is the Dirac delta distribution on

\R,

then

m\delta=m(0)\delta,

and if

\delta'

is the derivative of the delta distribution, then$m\delta' = m(0) \delta' - m' \delta = m(0) \delta' - m'(0) \delta.$

The bilinear multiplication map

Cinfty(\Rn) x l{D}'(\Rn)\tol{D}'\left(\Rn\right)

given by

(f,T)\mapstofT

is continuous; it is however, hypocontinuous.

Example. The product of any distribution

T

with the function that is identically on

U

is equal to

T.

Example. Suppose

(fi)

 infty i=1
is a sequence of test functions on

U

that converges to the constant function

1\inCinfty(U).

For any distribution

T

on

U,

the sequence

(fi

 infty T) i=1
converges to

T\inl{D}'(U).

If

(Ti)

 infty i=1
converges to

T\inl{D}'(U)

and

(fi)

 infty i=1
converges to

f\inCinfty(U)

then

(fiTi)

 infty i=1
converges to

fT\inl{D}'(U).

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 (and hyperfunctions) 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

\operatorname{p.v.}

 1 x
is the distribution obtained by the Cauchy principal value$\left(\operatorname \frac\right)(\phi) = \lim_ \int_$
 \geq \varepsilon
\frac\, dx \quad \text \phi \in \mathcal(\R).

If

\delta

is the Dirac delta distribution then$(\delta \times x) \times \operatorname \frac = 0$but,$\delta \times \left(x \times \operatorname \frac\right) = \delta$so the product of a distribution by a smooth function (which is always well defined) 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 (but extremely technical) . 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 (simplified) algebra is maybe the most popular in use today.

Inspired by Lyons' rough path theory,[10] Martin Hairer proposed a consistent way of multiplying distributions with certain structure (regularity structures[11]), available in many examples from stochastic analysis, notably stochastic partial differential equations. See also Gubinelli–Imkeller–Perkowski (2015) for a related development based on Bony's paraproduct from Fourier analysis.

Composition with a smooth function

Let

T

be a distribution on

U.

Let

V

be an open set in

\Rn

and

F:V\toU.

If

F

is a submersion then it is possible to define$T \circ F \in \mathcal'(V).$

This is, and is also called, sometimes written$F^\sharp : T \mapsto F^\sharp T = T \circ F.$

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(x)

of

F

is a surjective linear map for every

x\inV.

A necessary (but not sufficient) condition for extending

F\#

to distributions is that

F

be an open mapping.[12] 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

l{D}(U).

Existence, however, requires using the change of variables formula, the inverse function theorem (locally) and a partition of unity argument.[13]

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:$\int_V \phi\circ F(x) \psi(x)\,dx = \int_U \phi(x) \psi \left(F^(x) \right) \left|\det dF^(x) \right|\,dx.$

In this particular case, then,

F\#

is defined by the transpose formula:$\left\langle F^\sharp T, \phi \right\rangle = \left\langle T, \left|\det d(F^) \right|\phi\circ F^ \right\rangle.$

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

\Rn

then we denote by

f\astg

defined at

x\in\Rn

to be the integral$(f \ast g)(x) := \int_ f(x-y) g(y) \,dy = \int_ f(y)g(x-y) \,dy$provided that the integral exists. If

1\leqp,q,r\leqinfty

are such that $\frac = \frac + \frac - 1$ then for any functions

f\inLp(\Rn)

and

g\inLq(\Rn)

we have

f\astg\inLr(\Rn)

and

\|f\ast

 g\| Lr

\leq

 \|f\| Lp
 \|g\| Lq

.

If

f

and

g

are continuous functions on

\Rn,

at least one of which has compact support, then

\operatorname{supp}(f\astg)\subseteq\operatorname{supp}(f)+\operatorname{supp}(g)

and if

A\subseteq\Rn

then the value of

f\astg

on

A

do depend on the values of

f

outside of the Minkowski sum

A-\operatorname{supp}(g)=\{a-s:a\inA,s\in\operatorname{supp}(g)\}.

Importantly, if

g\inL1(\Rn)

has compact support then for any

0\leqk\leqinfty,

the convolution map

f\mapstof\astg

is continuous when considered as the map

Ck(\Rn)\toCk(\Rn)

or as the map
 k(\R C c

n)\to

 k(\R C c

n).

Translation and symmetry

Given

a\in\Rn,

the translation operator

\taua

sends

f:\Rn\to\Complex

to

\tauaf:\Rn\to\Complex,

defined by

\tauaf(y)=f(y-a).

This can be extended by the transpose to distributions in the following way: given a distribution

T,

is the distribution

\tauaT:l{D}(\Rn)\to\Complex

defined by

\tauaT(\phi):=\left\langleT,\tau-a\phi\right\rangle.

[14]

Given

f:\Rn\to\Complex,

define the function

\tilde{f}:\Rn\to\Complex

by

\tilde{f}(x):=f(-x).

Given a distribution

T,

let

\tilde{T}:l{D}(\Rn)\to\Complex

be the distribution defined by

\tilde{T}(\phi):=T\left(\tilde{\phi}\right).

The operator

T\mapsto\tilde{T}

is called .

Convolution of a test function with a distribution

Convolution with

f\inl{D}(\Rn)

defines a linear map:$\beginC_f : \,& \mathcal(\R^n) && \to \,&& \mathcal(\R^n) \\ & g && \mapsto\,&& f \ast g \\\end$which is continuous with respect to the canonical LF space topology on

l{D}(\Rn).

Convolution of

f

with a distribution

T\inl{D}'(\Rn)

can be defined by taking the transpose of

Cf

relative to the duality pairing of

l{D}(\Rn)

with the space

l{D}'(\Rn)

of distributions. If

f,g,\phi\inl{D}(\Rn),

then by Fubini's theorem$\langle C_fg, \phi \rangle = \int_\phi(x)\int_f(x-y) g(y) \,dy \,dx = \left\langle g,C_\phi \right\rangle.$

Extending by continuity, the convolution of

f

with a distribution

T

is defined by$\langle f \ast T, \phi \rangle = \left\langle T, \tilde \ast \phi \right\rangle, \quad \text \phi \in \mathcal(\R^n).$

An alternative way to define the convolution of a test function

f

and a distribution

T

is to use the translation operator

\taua.

The convolution of the compactly supported function

f

and the distribution

T

is then the function defined for each

x\in\Rn

by$(f \ast T)(x) = \left\langle T, \tau_x \tilde \right\rangle.$

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 (resp. an exponential function, an analytic function, the restriction of an entire analytic function on

\Complexn

to

\Rn,

the restriction of an entire function of exponential type in

\Complexn

to

\Rn

) then the same is true of

T\astf.

If the distribution

T

has compact support as well, then

f\astT

is a compactly supported function, and the Titchmarsh convolution theorem implies that:$\operatorname(\operatorname(f \ast T)) = \operatorname(\operatorname(f)) + \operatorname (\operatorname(T))$where

\operatorname{ch}

denotes the convex hull and supp denotes the support.

Convolution of a smooth function with a distribution

Let

f\inCinfty(\Rn)

and

T\inl{D}'(\Rn)

and assume that at least one of

f

and

T

has compact support. The of

f

and

T,

denoted by

f\astT

or by

T\astf,

is the smooth function:$\beginf \ast T : \,& \R^n && \to \,&& \Complex \\ & x && \mapsto\,&& \left\langle T, \tau_x \tilde \right\rangle \\\end$satisfying for all

p\in\Nn

:$\begin&\operatorname(f \ast T) \subseteq \operatorname(f)+ \operatorname(T) \\[6pt]&\text p \in \N^n: \quad\begin\partial^p \left\langle T, \tau_x \tilde \right\rangle = \left\langle T, \partial^p \tau_x \tilde \right\rangle \\\partial^p (T \ast f) = (\partial^p T) \ast f = T \ast (\partial^p f).\end\end$

If

T

is a distribution then the map

f\mapstoT\astf

is continuous as a map

l{D}(\Rn)\toCinfty(\Rn)

T

has compact support then it is also continuous as the map

Cinfty(\Rn)\toCinfty(\Rn)

and continuous as the map

l{D}(\Rn)\tol{D}(\Rn).

If

L:l{D}(\Rn)\toCinfty(\Rn)

is a continuous linear map such that

L\partial\alpha\phi=\partial\alphaL\phi

for all

\alpha

and all

\phi\inl{D}(\Rn)

then there exists a distribution

T\inl{D}'(\Rn)

such that

L\phi=T\circ\phi

for all

\phi\inl{D}(\Rn).

Example. Let

H

be the Heaviside function on

\R.

For any

\phi\inl{D}(\R),

$(H \ast \phi)(x) = \int_^x \phi(t) \, dt.$

Let

\delta

be the Dirac measure at 0 and

\delta'

its derivative as a distribution. Then

\delta'\astH=\delta

and

1\ast\delta'=0.

Importantly, the associative law fails to hold:$1 = 1 \ast \delta = 1 \ast (\delta' \ast H) \neq (1 \ast \delta') \ast H = 0 \ast H = 0.$

Convolution of distributions

It is also possible to define the convolution of two distributions

S

and

T

on

\Rn,

provided one of them has compact support. Informally, in order to define

S\astT

where

T

has compact support, the idea is to extend the definition of the convolution

\ast

to a linear operation on distributions so that the associativity formula$S \ast (T \ast \phi) = (S \ast T) \ast \phi$continues to hold for all test functions

\phi.

[15]

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 maps$\begin\bullet \ast \tilde : \,& \mathcal(\R^n) && \to \,&& \mathcal(\R^n) && \quad \text \quad && \bullet \ast \tilde : \,&& \mathcal(\R^n) && \to \,&& \mathcal(\R^n) \\ & f && \mapsto\,&& f \ast \tilde && && && f && \mapsto\,&& f \ast \tilde \\\end$are continuous. The transposes of these maps:$^\left(\bullet \ast \tilde\right) : \mathcal'(\R^n) \to \mathcal'(\R^n) \qquad ^\left(\bullet \ast \tilde\right) : \mathcal'(\R^n) \to \mathcal'(\R^n)$are consequently continuous and it can also be shown that$^\left(\bullet \ast \tilde\right)(T) = ^\left(\bullet \ast \tilde\right)(S).$

This common value is called and it is a distribution that is denoted by

S\astT

or

T\astS.

It satisfies

\operatorname{supp}(S\astT)\subseteq\operatorname{supp}(S)+\operatorname{supp}(T).

If

S

and

T

are two distributions, at least one of which has compact support, then for any

a\in\Rn,

\taua(S\astT)=\left(\tauaS\right)\astT=S\ast\left(\tauaT\right).

If

T

is a distribution in

\Rn

and if

\delta

is a Dirac measure then

T\ast\delta=T=\delta\astT

; thus

\delta

is the identity element of the convolution operation. Moreover, if

f

is a function then

f\ast\delta\prime=f\prime=\delta\prime\astf

where now the associativity of convolution implies that

f\prime\astg=g\prime\astf

for all functions

f

and

g.

Suppose that it is

T

that has compact support. For

\phi\inl{D}(\Rn)

consider the function$\psi(x) = \langle T, \tau_ \phi \rangle.$

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$\langle S \ast T, \phi \rangle = \langle S, \psi \rangle.$

This generalizes the classical notion of convolution of functions and is compatible with differentiation in the following sense: for every multi-index

\alpha.

$\partial^\alpha(S \ast T) = (\partial^\alpha S) \ast T = S \ast (\partial^\alpha T).$

The convolution of a finite number of distributions, all of which (except possibly one) have compact support, is associative.

This definition of convolution remains valid under less restrictive assumptions about

S

and

T.

[16]

The convolution of distributions with compact support induces a continuous bilinear map

l{E}' x l{E}'\tol{E}'

defined by

(S,T)\mapstoS*T,

where

l{E}'

denotes the space of distributions with compact support. However, the convolution map as a function

l{E}' x l{D}'\tol{D}'

is continuous although it is separately continuous. The convolution maps

l{D}(\Rn) x l{D}'\tol{D}'

and

l{D}(\Rn) x l{D}'\tol{D}(\Rn)

given by

(f,T)\mapstof*T

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

F(\alpha)=f\inl{O}'C

be a rapidly decreasing tempered distribution or, equivalently,

F(f)=\alpha\inl{O}M

be an ordinary (slowly growing, smooth) function within the space of tempered distributions and let

F

be the normalized (unitary, ordinary frequency) Fourier transform.[17] Then, according to,$F(f * g) = F(f) \cdot F(g) \qquad \text \qquad F(\alpha \cdot g) = F(\alpha) * F(g)$hold within the space of tempered distributions.[18] [19] [20] In particular, these equations become the Poisson Summation Formula if

g\equiv\operatorname{Ш

} is the Dirac Comb.[21] The space of all rapidly decreasing tempered distributions is also called the space of

l{O}'C

and the space of all ordinary functions within the space of tempered distributions is also called the space of

l{O}M.

More generally,

F(l{O}'C)=l{O}M

and

F(l{O}M)=l{O}'C.

[22] A particular case is the Paley-Wiener-Schwartz Theorem which states that

F(l{E}')=\operatorname{PW}

and

F(\operatorname{PW})=l{E}'.

This is because

l{E}'\subseteql{O}'C

and

\operatorname{PW}\subseteql{O}M.

In other words, compactly supported tempered distributions

l{E}'

belong to the space of

l{O}'C

andPaley-Wiener functions

\operatorname{PW},

better known as bandlimited functions, belong to the space of

l{O}M.

For example, let

g\equiv\operatorname{Ш

} \in \mathcal' be the Dirac comb and

f\equiv\delta\inl{E}'

be the Dirac delta;then

\alpha\equiv1\in\operatorname{PW}

is the function that is constantly one and both equations yield the Dirac-comb identity. Another example is to let

g

be the Dirac comb and

f\equiv\operatorname{rect}\inl{E}'

be the rectangular function; then

\alpha\equiv\operatorname{sinc}\in\operatorname{PW}

is the sinc function and both equations yield the Classical Sampling Theorem for suitable

\operatorname{rect}

functions. More generally, if

g

is the Dirac comb and

f\inl{S}\subseteql{O}'C\capl{O}M

is a smooth window function (Schwartz function), for example, the Gaussian, then

\alpha\inl{S}

is another smooth window function (Schwartz 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

U\subseteq\Rm

and

V\subseteq\Rn

be open sets. Assume all vector spaces to be over the field

F,

where

F=\R

or

\Complex.

For

f\inl{D}(U x V)

define for every

u\inU

and every

v\inV

the following functions:$\beginf_u : \,& V && \to \,&& \mathbb && \quad \text \quad && f^v : \,&& U && \to \,&& \mathbb \\ & y && \mapsto\,&& f(u, y) && && && x && \mapsto\,&& f(x, v) \\\end$

Given

S\inl{D}\prime(U)

and

T\inl{D}\prime(V),

define the following functions:$\begin\langle S, f^\rangle : \,& V && \to \,&& \mathbb && \quad \text \quad && \langle T, f_\rangle : \,&& U && \to \,&& \mathbb \\ & v && \mapsto\,&& \langle S, f^v \rangle && && && u && \mapsto\,&& \langle T, f_u \rangle \\\end$where

\langleT,f\bullet\rangle\inl{D}(U)

and

\langleS,f\bullet\rangle\inl{D}(V).

These definitions associate every

S\inl{D}'(U)

and

T\inl{D}'(V)

with the (respective) continuous linear map:$\begin \,&& \mathcal(U \times V) & \to \,&& \mathcal(V) && \quad \text \quad && \,& \mathcal(U \times V) && \to \,&& \mathcal(U) \\ && f \ & \mapsto\,&& \langle S, f^ \rangle && && & f \ && \mapsto\,&& \langle T, f_ \rangle \\\end$

Moreover, if either

S

(resp.

T

) has compact support then it also induces a continuous linear map of

Cinfty(U x V)\toCinfty(V)

(resp.

denoted by

ST

or

TS,

is the distribution in

U x V

defined by:$(S \otimes T)(f) := \langle S, \langle T, f_ \rangle \rangle = \langle T, \langle S, f^\rangle \rangle.$

Spaces of distributions

For all

0<k<infty

and all

1<p<infty,

every one of the following canonical injections is continuous and has an image (also called the range) that is a dense subset of its codomain:$\beginC_c^\infty(U) & \to & C_c^k(U) & \to & C_c^0(U) & \to & L_c^\infty(U) & \to & L_c^p(U) & \to & L_c^1(U) \\\downarrow & &\downarrow && \downarrow \\C^\infty(U) & \to & C^k(U) & \to & C^0(U) \\\end$where the topologies on
 q(U) L c
(

1\leqq\leqinfty

) are defined as direct limits of the spaces
 q(K) L c
in a manner analogous to how the topologies on
 k(U) C c
were defined (so in particular, they are not the usual norm topologies). The range of each of the maps above (and of any composition of the maps above) is dense in its codomain.

Suppose that

X

is one of the spaces
 k(U) C c
(for

k\in\{0,1,\ldots,infty\}

) or
 p L c(U)
(for

1\leqp\leqinfty

) or

Lp(U)

(for

1\leqp<infty

). Because the canonical injection

\operatorname{In}X:

 infty(U) C c

\toX

is a continuous injection whose image is dense in the codomain, this map's transpose

{}t\operatorname{In}X:X'b\tol{D}'(U)=

 infty(U)\right)' \left(C b
is a continuous injection. This injective transpose map thus allows the continuous dual space

X'

of

X

to be identified with a certain vector subspace of the space

l{D}'(U)

of all distributions (specifically, it is identified with the image of this transpose map). This transpose map is continuous but it is necessarily a topological embedding.A linear subspace of

l{D}'(U)

carrying a locally convex topology that is finer than the subspace topology induced on it by

l{D}'(U)=

 infty(U)\right)' \left(C b
is called .Almost all of the spaces of distributions mentioned in this article arise in this way (for example, tempered distribution, restrictions, distributions of order

\leq

some integer, distributions induced by a positive Radon measure, distributions induced by an

Lp

-function, etc.) and any representation theorem about the continuous dual space of

X

may, through the transpose

{}t\operatorname{In}X:X'b\tol{D}'(U),

be transferred directly to elements of the space

\operatorname{Im}\left({}t\operatorname{In}X\right).

The inclusion map

\operatorname{In}:

 infty(U) C c

\to

 0(U) C c
is a continuous injection whose image is dense in its codomain, so the transpose

{}t\operatorname{In}:

 0(U))' (C b

\tol{D}'(U)=

 infty(U))' (C b
is also a continuous injection.

Note that the continuous dual space

 0(U))' (C b
can be identified as the space of Radon measures, where there is a one-to-one correspondence between the continuous linear functionals

T\in

 0(U))' (C b
and integral with respect to a Radon measure; that is,
• if

T\in

 0(U))' (C b
then there exists a Radon measure

\mu

on such that for all $f \in C_c^0(U), T(f) = \int_U f \, d\mu,$ and
• if

\mu

is a Radon measure on then the linear functional on
 0(U) C c
defined by sending $f \in C_c^0(U)$ to $\int_U f \, d\mu$ is continuous.

Through the injection

{}t\operatorname{In}:

 0(U))' (C b

\tol{D}'(U),

every Radon measure becomes a distribution on . If

f

is a locally integrable function on then the distribution $\phi \mapsto \int_U f(x) \phi(x) \, dx$ 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

Linfty

functions on :

A linear function

T

on a space of functions is called if whenever a function

f

that belongs to the domain of

T

is non-negative (that is,

f

is real-valued and

f\geq0

) then

T(f)\geq0.

One may show that every positive linear functional on
 0(U) C c
is necessarily continuous (that is, necessarily a Radon measure). 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\to\R

is called if it is Lebesgue integrable over every compact subset of . This is a large class of functions which includes all continuous functions and all Lp space

Lp

functions. The topology on

l{D}(U)

is defined in such a fashion that any locally integrable function

f

yields a continuous linear functional on

l{D}(U)

– that is, an element of

l{D}'(U)

– denoted here by

Tf,

whose value on the test function

\phi

is given by the Lebesgue integral:$\langle T_f, \phi \rangle = \int_U f \phi\,dx.$

Conventionally, one abuses notation by identifying

Tf

with

f,

provided no confusion can arise, and thus the pairing between

Tf

and

\phi

is often written$\langle f, \phi \rangle = \langle T_f, \phi \rangle.$

If

f

and

g

are two locally integrable functions, then the associated distributions

Tf

and

Tg

are equal to the same element of

l{D}'(U)

if and only if

f

and

g

are equal almost everywhere (see, for instance,). In a similar manner, every Radon measure

\mu

on

U

defines an element of

l{D}'(U)

whose value on the test function

\phi

is $\int\phi \,d\mu.$ As above, it is conventional to abuse notation and write the pairing between a Radon measure

\mu

and a test function

\phi

as

\langle\mu,\phi\rangle.

Conversely, as shown in a theorem by Schwartz (similar to the Riesz representation theorem), every distribution which is non-negative on non-negative functions is of this form for some (positive) Radon measure.

Test functions as distributions

The test functions are themselves locally integrable, and so define distributions. The space of test functions

 infty(U) C c
is sequentially dense in

l{D}'(U)

with respect to the strong topology on

l{D}'(U).

This means that for any

T\inl{D}'(U),

there is a sequence of test functions,

(\phii)

 infty, i=1
that converges to

T\inl{D}'(U)

(in its strong dual topology) when considered as a sequence of distributions. Or equivalently,$\langle \phi_i, \psi \rangle \to \langle T, \psi \rangle \qquad \text \psi \in \mathcal(U).$

Distributions with compact support

The inclusion map

\operatorname{In}:

 infty(U) C c

\toCinfty(U)

is a continuous injection whose image is dense in its codomain, so the transpose map

{}t\operatorname{In}:

 infty(U))' (C b

\tol{D}'(U)=

 infty(U))' (C b
is also a continuous injection. Thus the image of the transpose, denoted by

l{E}'(U),

forms a space of distributions.

The elements of

l{E}'(U)=

 infty(U))' (C b
can be identified as the space of distributions with compact support. Explicitly, if

T

is a distribution on then the following are equivalent,

T\inl{E}'(U).

• The support of

T

is compact.
• The restriction of

T

to
 infty(U), C c
when that space is equipped with the subspace topology inherited from

Cinfty(U)

(a coarser topology than the canonical LF topology), is continuous.
• There is a compact subset of such that for every test function

\phi

whose support is completely outside of, we have

T(\phi)=0.

Compactly supported distributions define continuous linear functionals on the space

Cinfty(U)

; recall that the topology on

Cinfty(U)

is defined such that a sequence of test functions

\phik

converges to 0 if and only if all derivatives of

\phik

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
 infty(U) C c
to

Cinfty(U).

Distributions of finite order

Let

k\in\N.

The inclusion map

\operatorname{In}:

 infty(U) C c

\to

 k(U) C c
is a continuous injection whose image is dense in its codomain, so the transpose

{}t\operatorname{In}:

 k(U))' (C b

\tol{D}'(U)=

 infty(U))' (C b
is also a continuous injection. Consequently, the image of

{}t\operatorname{In},

denoted by

l{D}'k(U),

forms a space of distributions. The elements of

l{D}'k(U)

are The distributions of order

\leq0,

which are also called , are exactly the distributions that are Radon measures (described above).

For

0k\in\N,

a is a distribution of order

\leqk

that is not a distribution of order

\leqk-1

.

A distribution is said to be of if there is some integer

k

such that it is a distribution of order

\leqk,

and the set of distributions of finite order is denoted by

l{D}'F(U).

Note that if

k\leql

then

l{D}'k(U)\subseteql{D}'l(U)

so that

l{D}'F(U):=

 infty cup n=0

l{D}'n(U)

is a vector subspace of

l{D}'(U)

and furthermore, if and only if

l{D}'F(U)=l{D}'(U).

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

\rhoVU

is the restriction mapping from to, then the image of

l{D}'(U)

under

\rhoVU

is contained in

l{D}'F(V).

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. (Distributions of infinite order) Let

U:=(0,infty)

and for every test function

f,

let $S f := \sum_^\infty (\partial^m f)\left(\frac\right).$

Then

S

is a distribution of infinite order on . Moreover,

S

can not be extended to a distribution on

\R

; that is, there exists no distribution

T

on

\R

such that the restriction of

T

to is equal to

S.

Tempered distributions and Fourier transform

Defined below are the , which form a subspace of

l{D}'(\Rn),

the space of distributions on

\Rn.

This is a proper subspace: while every tempered distribution is a distribution and an element of

l{D}'(\Rn),

the converse is not true. 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

l{D}'(\Rn).

Schwartz space

The Schwartz space,

l{S}(\Rn),

is the space of all smooth functions that are rapidly decreasing at infinity along with all partial derivatives. Thus

\phi:\Rn\to\R

is in the Schwartz space provided that any derivative of

\phi,

multiplied with any power of

|x|,

converges to 0 as

|x|\toinfty.

These functions form a complete TVS with a suitably defined family of seminorms. More precisely, for any multi-indices

\alpha

and

\beta

define:$p_ (\phi) ~=~ \sup_ \left|x^\alpha \partial^\beta \phi(x) \right|.$

Then

\phi

is in the Schwartz space if all the values satisfy:$p_ (\phi) < \infty.$

The family of seminorms

p\alpha,\beta

defines a locally convex topology on the Schwartz space. For

n=1,

the seminorms are, in fact, norms on the Schwartz space. One can also use the following family of seminorms to define the topology:$|f|_ = \sup_$
 \le m
\left(\sup_ \left\\right), \qquad k,m \in \N.

Otherwise, one can define a norm on

l{S}(\Rn)

via$\|\phi\|_ ~=~ \max_$
 + \beta \leq k
\sup_ \left| x^\alpha \partial^\beta \phi(x)\right|, \qquad k \ge 1.

The Schwartz space is a Fréchet space (that is, a complete metrizable locally convex space). Because the Fourier transform changes

\partial\alpha

into multiplication by

x\alpha

and vice versa, this symmetry implies that the Fourier transform of a Schwartz function is also a Schwartz function.

A sequence

\{fi\}

in

l{S}(\Rn)

converges to 0 in

l{S}(\Rn)

if and only if the functions

(1+|x|)k(\partialpfi)(x)

converge to 0 uniformly in the whole of

\Rn,

which implies that such a sequence must converge to zero in

Cinfty(\Rn).

l{D}(\Rn)

is dense in

l{S}(\Rn).

The subset of all analytic Schwartz functions is dense in

l{S}(\Rn)

as well.

The Schwartz space is nuclear and the tensor product of two maps induces a canonical surjective TVS-isomorphisms$\mathcal(\R^m) \ \widehat\ \mathcal(\R^n) \to \mathcal(\R^),$where

\widehat{}

represents the completion of the injective tensor product (which in this case is the identical to the completion of the projective tensor product).

Tempered distributions

The inclusion map

\operatorname{In}:l{D}(\Rn)\tol{S}(\Rn)

is a continuous injection whose image is dense in its codomain, so the transpose

{}t\operatorname{In}:

 n))' (l{S}(\R b

\tol{D}'(\Rn)

is also a continuous injection. Thus, the image of the transpose map, denoted by

l{S}'(\Rn),

forms a space of distributions.

The space

l{S}'(\Rn)

is called the space of . It is the continuous dual space of the Schwartz space. Equivalently, a distribution

T

is a tempered distribution if and only if$\left(\text \alpha, \beta \in \N^n: \lim_ p_ (\phi_m) = 0 \right) \Longrightarrow \lim_ T(\phi_m)=0.$

Lp(\Rn)

for

p\geq1

are tempered distributions.

The can also be characterized as, meaning that each derivative of

T

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

\phi

decays faster than every inverse power of

|x|.

An example of a rapidly falling function is

|x|n\exp(|x|\beta)

for any positive

n,λ,\beta.

Fourier transform

F:l{S}(\Rn)\tol{S}(\Rn)

is a TVS-automorphism of the Schwartz space, and the is defined to be its transpose

{}tF:l{S}'(\Rn)\tol{S}'(\Rn),

which (abusing notation) will again be denoted by

F.

So the Fourier transform of the tempered distribution

T

is defined by

(FT)(\psi)=T(F\psi)

for every Schwartz function

\psi.

FT

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$F \dfrac = ixFT$and also with convolution: if

T

is a tempered distribution and

\psi

is a smooth function on

\Rn,

\psiT

is again a tempered distribution and$F(\psi T) = F \psi * FT$is the convolution of

FT

and

F\psi.

In particular, the Fourier transform of the constant function equal to 1 is the

\delta

distribution.

Expressing tempered distributions as sums of derivatives

If

T\inl{S}'(\Rn)

is a tempered distribution, then there exists a constant

C>0,

and positive integers

M

and

N

such that for all Schwartz functions

\phi\inl{S}(\Rn)

$\langle T, \phi \rangle \le C\sum\nolimits_$
 \le N, \beta \le M
\sup_ \left|x^\alpha \partial^\beta \phi(x) \right|=C\sum\nolimits_
 \le N, \beta \le M
p_(\phi).

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

\alpha

such that$T = \partial^\alpha F.$

Restriction of distributions to compact sets

If

T\inl{D}'(\Rn),

then for any compact set

K\subseteq\Rn,

there exists a continuous function

F

compactly supported in

\Rn

(possibly on a larger set than itself) and a multi-index

\alpha

such that

T=\partial\alphaF

on
 infty(K). C c

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.

Bibliography

• Book: Barros-Neto, José. An Introduction to the Theory of Distributions. Dekker. New York, NY. 1973.
• .
• Book: Folland, G.B.. Harmonic Analysis in Phase Space. Princeton University Press. Princeton, NJ. 1989.
• Book: Friedlander. F.G.. Joshi. M.S.. Introduction to the Theory of Distributions. Cambridge University Press. Cambridge, UK. 1998. .
• .
• .
• .
• .
• Book: Petersen, Bent E.. Introduction to the Fourier Transform and Pseudo-Differential Operators. Pitman Publishing. Boston, MA. 1983. .
• .
• .
• .
• .
• .
• Book: Woodward, P.M.. Philip_Woodward. Probability and Information Theory with Applications to Radar. Pergamon Press. Oxford, UK. 1953.

• M. J. Lighthill (1959). Introduction to Fourier Analysis and Generalised Functions. Cambridge University Press. (requires very little knowledge of analysis; defines distributions as limits of sequences of functions under integrals)
• V.S. Vladimirov (2002). Methods of the theory of generalized functions. Taylor & Francis.
• .
• .
• .
• .
• .

Notes and References

1. The image of the compact set

K

under a continuous

\R

-valued map (for example,

x\mapsto\left|\partialpf(x)\right|

for

x\inU

) is itself a compact, and thus bounded, subset of

\R.

If

K\varnothing

then this implies that each of the functions defined above is

\R

-valued (that is, none of the supremums above are ever equal to

infty

).
2. Exactly as with

Ck(K;U),

the space

Ck(K;U')

is defined to be the vector subspace of

Ck(U')

consisting of maps with support contained in

K

endowed with the subspace topology it inherits from

Ck(U')

3. Even though the topology of
 infty(U) C c
is not metrizable, a linear functional on
 infty(U) C c
is continuous if and only if it is sequentially continuous.
4. A is a sequence that converges to the origin.
5. If

l{P}

is also directed under the usual function comparison then we can take the finite collection to consist of a single element.
6. See for example .
7. This approach works for non-linear mappings as well, provided they are assumed to be uniformly continuous.
8. .
9. For example, let

U=\R

and take

P

to be the ordinary derivative for functions of one real variable and assume the support of

\phi

to be contained in the finite interval

(a,b),

then since

\operatorname{supp}(\phi)\subseteq(a,b)

$\begin\int_\R \phi'(x)f(x)\,dx &= \int_a^b \phi'(x)f(x) \,dx \\&= \phi(x)f(x)\big\vert_a^b - \int_a^b f'(x) \phi(x) \,d x \\&= \phi(b)f(b) - \phi(a)f(a) - \int_a^b f'(x) \phi(x) \,d x \\&= \int_a^b f'(x) \phi(x) \,d x\end$where the last equality is because

\phi(a)=\phi(b)=0.

10. Lyons. T.. Differential equations driven by rough signals. 10.4171/RMI/240. Revista Matemática Iberoamericana. 215–310. 1998. free.
11. Hairer. Martin. A theory of regularity structures. Inventiones Mathematicae. 2014. 10.1007/s00222-014-0505-4. 198. 2. 269–504. 2014InMat.198..269H. 1303.5113.
12. See for example .
13. See .
14. See for example .
15. proves the uniqueness of such an extension.
16. See for instance and .
17. Book: Folland, G.B.. Harmonic Analysis in Phase Space. Princeton University Press. Princeton, NJ. 1989.
18. Book: Horváth, John. John Horvath (mathematician). Topological Vector Spaces and Distributions. Addison-Wesley Publishing Company. Reading, MA. 1966.
19. Book: Barros-Neto, José. An Introduction to the Theory of Distributions. Dekker. New York, NY. 1973.
20. Book: Petersen, Bent E.. Introduction to the Fourier Transform and Pseudo-Differential Operators. Pitman Publishing. Boston, MA. 1983.
21. Book: Woodward, P.M.. Probability and Information Theory with Applications to Radar. Pergamon Press. Oxford, UK. 1953.
22. Book: Friedlander. F.G.. Joshi. M.S.. Introduction to the Theory of Distributions. Cambridge University Press. Cambridge, UK. 1998.