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 (weak 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.
is normally thought of as on the in the function
domain by "sending" a point
in the domain to the point
Instead of acting on points, distribution theory reinterprets functions such as
as acting on in a certain way. In applications to physics and engineering,
are usually
infinitely differentiable complex-valued (or
real-valued) functions with
compact support that are defined on some given non-empty
open subset
. (
Bump functions are examples of test functions.) The set of all such test functions forms a
vector space that is denoted by
or
Most commonly encountered functions, including all continuous maps
if using
can be canonically reinterpreted as acting via "
integration against a test function." Explicitly, this means that such a function
"acts on" a test function
by "sending" it to the
number which is often denoted by
This new action
of
defines a
scalar-valued map
whose domain is the space of test functions
This
functional
turns out to have the two defining properties of what is known as a : it is
linear, and it is also
continuous when
is given a certain
topology called . The action (the integration
) of this distribution
on a test function
can be interpreted as a weighted average of the distribution on the
support of the test function, even if the values of the distribution at a single point are not well-defined. Distributions like
that arise from functions in this way are prototypical examples of distributions, but there exist many distributions that cannot be defined by integration against any function. Examples of the latter include the
Dirac delta function and distributions defined to act by integration of test functions
against certain
measures
on
Nonetheless, it is still always possible to reduce any arbitrary distribution down to a simpler of related distributions that do arise via such actions of integration.
More generally, a is by definition a linear functional on
that is continuous when
is given a topology called the
. This leads to space of (all) distributions on
, usually denoted by
(note the
prime), which by definition is the
space of all distributions on
(that is, it is the continuous dual space of
); 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's functions in the 1830s to solve ordinary differential equations, but was not formalized until much later. According to, generalized functions originated in the work of on second-order hyperbolic partial differential equations, and the ideas were developed in somewhat extended form by Laurent Schwartz in the late 1940s. According to his autobiography, Schwartz introduced the term "distribution" by analogy with a distribution of electrical charge, possibly including not only point charges but also dipoles and so on. comments that although the ideas in the transformative book by were not entirely new, it was Schwartz's broad attack and conviction that distributions would be useful almost everywhere in analysis that made the difference.
Notation
The following notation will be used throughout this article:
is a fixed positive integer and
is a fixed non-empty
open subset of
Euclidean space
denotes the
natural numbers.
will denote a non-negative integer or
is a
function then
will denote its
domain and the
of
denoted by
is defined to be the
closure of the set
\{x\in\operatorname{Dom}(f):f(x) ≠ 0\}
in
the following notation defines a canonical
pairing:
is an element in
(given that
is fixed, if the size of multi-indices is omitted then the size should be assumed to be
). The
of a multi-index
\alpha=(\alpha1,\ldots,\alphan)\in\Nn
is defined as
and denoted by
Multi-indices are particularly useful when dealing with functions of several variables, in particular, we introduce the following notations for a given multi-index
\alpha=(\alpha1,\ldots,\alphan)\in\Nn
:
We also introduce a partial order of all multi-indices by
if and only if
for all
When
we define their multi-index binomial coefficient as:
Definitions of test functions and distributions
In this section, some basic notions and definitions needed to define real-valued distributions on are introduced. Further discussion of the topologies on the spaces of test functions and distributions is given in the article on spaces of test functions and distributions.
For all
j,k\in\{0,1,2,\ldots,infty\}
and any compact subsets
and
of
, we have:
Distributions on are continuous linear functionals on
when this vector space is endowed with a particular topology called the
. The following proposition states two necessary and sufficient conditions for the continuity of a linear function on
that are often straightforward to verify.
Proposition: A linear functional on
is continuous, and therefore a
, if and only if any of the following equivalent conditions is satisfied:
- For every compact subset
there exist constants
and
(dependent on
) such that for all
with support contained in
,
- For every compact subset
and every sequence
in
whose supports are contained in
, if
converges uniformly to zero on
for every
multi-index
, then
Topology on Ck(U)
We now introduce the seminorms that will define the topology on
Different authors sometimes use different families of seminorms so we list the most common families below. However, the resulting topology is the same no matter which family is used.
All of the functions above are non-negative
-valued
[1] seminorms on
As explained in this article, every set of seminorms on a vector space induces a
locally convex vector topology.
Each of the following sets of seminorms generate the same locally convex vector topology on
(so for example, the topology generated by the seminorms in
is equal to the topology generated by those in
).
With this topology,
becomes a locally convex
Fréchet space that is normable. Every element of
is a continuous seminorm on
Under this topology, a
net
in
converges to
if and only if for every multi-index
with
and every compact
the net of partial derivatives
\left(\partialpfi\right)i
converges uniformly to
on
For any
k\in\{0,1,2,\ldots,infty\},
any
(von Neumann) bounded subset of
is a
relatively compact subset of
In particular, a subset of
is bounded if and only if it is bounded in
for all
The space
is a
Montel space if and only if
A subset
of
is open in this topology if and only if there exists
such that
is open when
is endowed with the
subspace topology induced on it by
Topology on Ck(K)
As before, fix
k\in\{0,1,2,\ldots,infty\}.
Recall that if
is any compact subset of
then
If
is finite then
is a
Banach space with a topology that can be defined by the
normAnd when
then
is even a
Hilbert space.
Trivial extensions and independence of Ck(K)'s topology from U
Suppose
is an open subset of
and
is a compact subset. By definition, elements of
are functions with domain
(in symbols,
), so the space
and its topology depend on
to make this dependence on the open set
clear, temporarily denote
by
Importantly, changing the set
to a different open subset
(with
) will change the set
from
to
[2] so that elements of
will be functions with domain
instead of
Despite
depending on the open set (
), the standard notation for
makes no mention of it. This is justified because, as this subsection will now explain, the space
is canonically identified as a subspace of
(both algebraically and topologically).
It is enough to explain how to canonically identify
with
when one of
and
is a subset of the other. The reason is that if
and
are arbitrary open subsets of
containing
then the open set
also contains
so that each of
and
is canonically identified with
and now by transitivity,
is thus identified with
So assume
are open subsets of
containing
Given
its is the function
defined by:
This trivial extension belongs to
(because
has compact support) and it will be denoted by
(that is,
). The assignment
thus induces a map
that sends a function in
to its trivial extension on
This map is a linear
injection and for every compact subset
(where
is also a compact subset of
since
),
If
is restricted to
then the following induced linear map is a
homeomorphism (linear homeomorphisms are called):
and thus the next map is a topological embedding:
Using the injection
the vector space
is canonically identified with its image in
Because
through this identification,
can also be considered as a subset of
Thus the topology on
is independent of the open subset
of
that contains
which justifies the practice of writing
instead of
Canonical LF topology
See main article: Spaces of test functions and distributions.
See also: LF-space and Topology of uniform convergence.
Recall that
denotes all functions in
that have compact support in
where note that
is the union of all
as
ranges over all compact subsets of
Moreover, for each
is a dense subset of
The special case when
gives us the space of test functions.
The canonical LF-topology is metrizable and importantly, it is than the subspace topology that
induces on
However, the canonical LF-topology does make
into a
complete reflexive nuclear Montel bornological barrelled Mackey space; the same is true of its
strong dual space (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
are known as distributions on
Other equivalent definitions are described below.
There is a canonical duality pairing between a distribution
on
and a test function
which is denoted using angle brackets by
One interprets this notation as the distribution
acting on the test function
to give a scalar, or symmetrically as the test function
acting on the distribution
Characterizations of distributions
Proposition. If
is a
linear functional on
then the following are equivalent:
- is a distribution;
- is continuous;
- is continuous at the origin;
- is uniformly continuous;
- is a bounded operator;
- is sequentially continuous;
- explicitly, for every sequence
in
that converges in
to some
[3] - is sequentially continuous at the origin; in other words, maps null sequences[4] to null sequences;
- explicitly, for every sequence
in
that converges in
to the origin (such a sequence is called a),
- a is by definition any sequence that converges to the origin;
- maps null sequences to bounded subsets;
- explicitly, for every sequence
in
that converges in
to the origin, the sequence
\left(T\left(fi\right)\right)
is bounded;
- maps Mackey convergent null sequences to bounded subsets;
- explicitly, for every Mackey convergent null sequence
in
the sequence
\left(T\left(fi\right)\right)
is bounded;
-
is said to be if there exists a divergent sequence
r\bull=\left(ri\right)
\toinfty
of positive real numbers such that the sequence
is bounded; every sequence that is Mackey convergent to the origin necessarily converges to the origin (in the usual sense);
- The kernel of is a closed subspace of
- The graph of is closed;
- There exists a continuous seminorm
on
such that
- There exists a constant
and a finite subset
\{g1,\ldots,gm\}\subseteql{P}
(where
is any collection of continuous seminorms that defines the canonical LF topology on
) such that
[5] - For every compact subset
there exist constants
and
such that for all
- For every compact subset
there exist constants
and
such that for all
with support contained in
[6] - For any compact subset
and any sequence
in
if
converges uniformly to zero for all
multi-indices
then
Topology on the space of distributions and its relation to the weak-* topology
The set of all distributions on
is the continuous dual space of
which when endowed with the
strong dual topology is denoted by
Importantly, unless indicated otherwise, the topology on
is the
strong dual topology; if the topology is instead the weak-* topology then this will be indicated. Neither topology is metrizable although unlike the weak-* topology, the strong dual topology makes
into a
complete nuclear space, to name just a few of its desirable properties.
Neither
nor its strong dual
is a
sequential space and so neither of their topologies can be fully described by sequences (in other words, defining only what sequences converge in these spaces is enough to fully/correctly define their topologies).However, a in
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 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
is endowed with can be found in the article on
spaces of test functions and distributions and the articles on
polar topologies and
dual systems.
A map from
into another
locally convex topological vector space (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
(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
at a particular point of . However, as is the case with functions, distributions on restrict to give distributions on open subsets of . Furthermore, distributions are in the sense that a distribution on all of can be assembled from a distribution on an open cover of satisfying some compatibility conditions on the overlaps. Such a structure is known as a
sheaf.
Extensions and restrictions to an open subset
Let
be open subsets of
Every function
can be from its domain to a function on by setting it equal to
on the
complement
This extension is a smooth compactly supported function called the and it will be denoted by
This assignment
defines the operator
which is a continuous injective linear map. It is used to canonically identify
as a vector subspace of
(although as a
topological subspace). Its transpose (explained here)
is called the
and as the name suggests, the image
of a distribution
under this map is a distribution on
called the
restriction of
to
The defining condition of the restriction
is:
If
then the (continuous injective linear) trivial extension map
is a topological embedding (in other words, if this linear injection was used to identify
as a subset of
then
's topology would
strictly finer than the
subspace topology that
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
Consequently if
then the restriction mapping is neither injective nor surjective. A distribution
is said to be
if it belongs to the range of the transpose of
and it is called
if it is extendable to
Unless
the restriction to is neither
injective nor
surjective. Lack of surjectivity follows since distributions can blow up towards the boundary of . For instance, if
and
then the distribution
is in
but admits no extension to
Gluing and distributions that vanish in a set
Let be an open subset of .
is said to
if for all
such that
\operatorname{supp}(f)\subseteqV
we have
vanishes in if and only if the restriction of to is equal to 0, or equivalently, if and only if lies in the
kernel of the restriction map
Support of a distribution
This last corollary implies that for every distribution on, there exists a unique largest subset of such that vanishes in (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
If
is a locally integrable function on and if
is its associated distribution, then the support of
is the smallest closed subset of in the complement of which
is
almost everywhere equal to 0. If
is continuous, then the support of
is equal to the closure of the set of points in at which
does not vanish. The support of the distribution associated with the
Dirac measure at a point
is the set
If the support of a test function
does not intersect the support of a distribution then
A distribution is 0 if and only if its support is empty. If
is identically 1 on some open set containing the support of a distribution then
If the support of a distribution is compact then it has finite order and there is a constant
and a non-negative integer
such that:
If has compact support, then it has a unique extension to a continuous linear functional
on
; this function can be defined by
\widehat{T}(f):=T(\psif),
where
is any function that is identically 1 on an open set containing the support of .
If
and
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
form a vector subspace of
Furthermore, if
is a differential operator in, then for all distributions on and all
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
let
denote the distribution induced by the Dirac measure at
For any
and distribution
the support of is contained in
if and only if is a finite linear combination of derivatives of the Dirac measure at
If in addition the order of is
then there exist constants
such that:
Said differently, if has support at a single point
then is in fact a finite linear combination of distributional derivatives of the
function at . That is, there exists an integer and complex constants
such that
where
is the translation operator.
Distributions of finite order with support in an open subset
Global structure of distributions
The formal definition of distributions exhibits them as a subspace of a very large space, namely the topological dual of
(or the
Schwartz space
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
we can write:
where
are finite sets of multi-indices and the functions
are continuous.
Note that the infinite sum above is well-defined as a distribution. The value of for a given
can be computed using the finitely many
that intersect the support of
Operations on distributions
Many operations which are defined on smooth functions with compact support can also be defined for distributions. In general, if
is a linear map that is continuous with respect to the
weak topology, then it is not always possible to extend
to a map
by classic extension theorems of topology or linear functional analysis.
[7] The “distributional” extension of the above linear continuous operator A is possible if and only if A admits a Schwartz adjoint, that is another linear continuous operator B of the same type such that
\langleAf,g\rangle=\langlef,Bg\rangle
,for every pair of test functions. In that condition, B is unique and the extension A’ is the transpose of the Schwartz adjoint B.
[8] Preliminaries: Transpose of a linear operator
See main article: Transpose of a linear map.
Operations on distributions and spaces of distributions are often defined using the transpose of a linear operator. This is because the transpose allows for a unified presentation of the many definitions in the theory of distributions and also because its properties are well-known in functional analysis.[9] 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
is the linear map
or equivalently, it is the unique map satisfying
\langley',A(x)\rangle=\left\langle{}tA(y'),x\right\rangle
for all
and all
(the prime symbol in
does not denote a derivative of any kind; it merely indicates that
is an element of the continuous dual space
). Since
is continuous, the transpose
is also continuous when both duals are endowed with their respective
strong dual topologies; it is also continuous when both duals are endowed with their respective weak* topologies (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
be a continuous linear map. Then by definition, the transpose of
is the unique linear operator
that satisfies:
Since
is dense in
(here,
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
where
Explicitly, this means that a continuous linear map
is equal to
if and only if the condition below holds:
where the right-hand side equals
\langle{}tA(D\psi),\phi\rangle=\langleD\psi,A(\phi)\rangle=\langle\psi,A(\phi)\rangle=\intU\psi ⋅ A(\phi)dx.
Differential operators
Differentiation of distributions
Let
be the partial derivative operator
\tfrac{\partial}{\partialxk}.
To extend
we compute its transpose:
Therefore
Thus, the partial derivative of
with respect to the coordinate
is defined by the formula
With this definition, every distribution is infinitely differentiable, and the derivative in the direction
is a
linear operator on
More generally, if
is an arbitrary
multi-index, then the partial derivative
of the distribution
is defined by
Differentiation of distributions is a continuous operator on
this is an important and desirable property that is not shared by most other notions of differentiation.
If
is a distribution in
then
where
is the derivative of
and
is a translation by
thus the derivative of
may be viewed as a limit of quotients.
Differential operators acting on smooth functions
A linear differential operator in
with smooth coefficients acts on the space of smooth functions on
Given such an operator
we would like to define a continuous linear map,
that extends the action of
on
to distributions on
In other words, we would like to define
such that the following diagram
commutes:
where the vertical maps are given by assigning
its canonical distribution
which is defined by:
With this notation, the diagram commuting is equivalent to:
To find
the transpose
of the continuous induced map
defined by
is considered in the lemma below. This leads to the following definition of the differential operator on
called which will be denoted by
to avoid confusion with the transpose map, that is defined by
As discussed above, for any
the transpose may be calculated by:
For the last line we used integration by parts combined with the fact that
and therefore all the functions
f(x)c\alpha(x)\partial\alpha\phi(x)
have compact support.
[10] Continuing the calculation above, for all
The Lemma combined with the fact that the formal transpose of the formal transpose is the original differential operator, that is,
enables us to arrive at the correct definition: the formal transpose induces the (continuous) canonical linear operator
defined by
We claim that the transpose of this map,
{}tP*:l{D}'(U)\tol{D}'(U),
can be taken as
To see this, for every
compute its action on a distribution of the form
with
:
We call the continuous linear operator
DP:={}tP*:l{D}'(U)\tol{D}'(U)
the
. Its action on an arbitrary distribution
is defined via:
If
converges to
then for every multi-index
\alpha,(\partial\alphaTi)
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
is a smooth function then
is a differential operator of order 0, whose formal transpose is itself (that is,
). The induced differential operator
maps a distribution
to a distribution denoted by
We have thus defined the multiplication of a distribution by a smooth function.
We now give an alternative presentation of the multiplication of a distribution
on
by a smooth function
The product
is defined by
This definition coincides with the transpose definition since if
is the operator of multiplication by the function
(that is,
), then
so that
Under multiplication by smooth functions,
is a
module over the
ring
With this definition of multiplication by a smooth function, the ordinary
product rule of calculus remains valid. However, some unusual identities also arise. For example, if
is the Dirac delta distribution on
then
and if
is the derivative of the delta distribution, then
The bilinear multiplication map
Cinfty(\Rn) x l{D}'(\Rn)\tol{D}'\left(\Rn\right)
given by
is continuous; it is however,
hypocontinuous.
Example. The product of any distribution
with the function that is identically on
is equal to
Example. Suppose
is a sequence of test functions on
that converges to the constant function
For any distribution
on
the sequence
converges to
If
converges to
and
converges to
then
converges to
Problem of multiplying distributions
It is easy to define the product of a distribution with a smooth function, or more generally the product of two distributions whose singular supports are disjoint.[11] 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
is the distribution obtained by the
Cauchy principal valueIf
is the Dirac delta distribution then
but,
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 are not solved within distribution theory alone. In the context of quantum field theory, however, solutions can be found. In more than two spacetime dimensions the problem is related to the regularization of divergences. Here Henri Epstein and Vladimir Glaser developed the mathematically rigorous (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,[12] Martin Hairer proposed a consistent way of multiplying distributions with certain structures (regularity structures[13]), 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
be a distribution on
Let
be an open set in
and
If
is a
submersion then it is possible to define
This is, and is also called, sometimes written
The pullback is often denoted
although this notation should not be confused with the use of '*' to denote the adjoint of a linear mapping.
The condition that
be a submersion is equivalent to the requirement that the
Jacobian derivative
of
is a
surjective linear map for every
A necessary (but not sufficient) condition for extending
to distributions is that
be an
open mapping.
[14] The
Inverse function theorem ensures that a submersion satisfies this condition.
If
is a submersion, then
is defined on distributions by finding the transpose map. The uniqueness of this extension is guaranteed since
is a continuous linear operator on
Existence, however, requires using the
change of variables formula, the inverse function theorem (locally), and a
partition of unity argument.
[15] In the special case when
is a
diffeomorphism from an open subset
of
onto an open subset
of
change of variables under the integral gives:
In this particular case, then,
is defined by the transpose formula:
Convolution
Under some circumstances, it is possible to define the convolution of a function with a distribution, or even the convolution of two distributions.Recall that if
and
are functions on
then we denote by
defined at
to be the integral
provided that the integral exists. If
are such that
then for any functions
and
we have
and
If
and
are continuous functions on
at least one of which has compact support, then
\operatorname{supp}(f\astg)\subseteq\operatorname{supp}(f)+\operatorname{supp}(g)
and if
then the value of
on
do depend on the values of
outside of the
Minkowski sum A-\operatorname{supp}(g)=\{a-s:a\inA,s\in\operatorname{supp}(g)\}.
Importantly, if
has compact support then for any
the convolution map
is continuous when considered as the map
or as the map
Translation and symmetry
Given
the translation operator
sends
to
defined by
This can be extended by the transpose to distributions in the following way: given a distribution
is the distribution
\tauaT:l{D}(\Rn)\to\Complex
defined by
\tauaT(\phi):=\left\langleT,\tau-a\phi\right\rangle.
[16] Given
define the function
by
Given a distribution
let
\tilde{T}:l{D}(\Rn)\to\Complex
be the distribution defined by
\tilde{T}(\phi):=T\left(\tilde{\phi}\right).
The operator
is called
.
Convolution of a test function with a distribution
Convolution with
defines a linear map:
which is
continuous with respect to the canonical
LF space topology on
Convolution of
with a distribution
can be defined by taking the transpose of
relative to the duality pairing of
with the space
of distributions. If
then by
Fubini's theoremExtending by continuity, the convolution of
with a distribution
is defined by
An alternative way to define the convolution of a test function
and a distribution
is to use the translation operator
The convolution of the compactly supported function
and the distribution
is then the function defined for each
by
It can be shown that the convolution of a smooth, compactly supported function and a distribution is a smooth function. If the distribution
has compact support, and if
is a polynomial (resp. an exponential function, an analytic function, the restriction of an entire analytic function on
to
the restriction of an entire function of exponential type in
to
), then the same is true of
If the distribution
has compact support as well, then
is a compactly supported function, and the
Titchmarsh convolution theorem implies that:
where
denotes the
convex hull and
denotes the support.
Convolution of a smooth function with a distribution
Let
and
and assume that at least one of
and
has compact support. The
of
and
denoted by
or by
is the smooth function:
satisfying for all
:
Let
be the map
. If
is a distribution, then
is continuous as a map
. If
also has compact support, then
is also continuous as the map
Cinfty(\Rn)\toCinfty(\Rn)
and continuous as the map
If
L:l{D}(\Rn)\toCinfty(\Rn)
is a continuous linear map such that
L\partial\alpha\phi=\partial\alphaL\phi
for all
and all
then there exists a distribution
such that
for all
Example. Let
be the
Heaviside function on
For any
Let
be the Dirac measure at 0 and let
be its derivative as a distribution. Then
and
Importantly, the associative law fails to hold:
Convolution of distributions
It is also possible to define the convolution of two distributions
and
on
provided one of them has compact support. Informally, to define
where
has compact support, the idea is to extend the definition of the convolution
to a linear operation on distributions so that the associativity formula
continues to hold for all test functions
[17] It is also possible to provide a more explicit characterization of the convolution of distributions. Suppose that
and
are distributions and that
has compact support. Then the linear maps
are continuous. The transposes of these maps:
are consequently continuous and it can also be shown that
This common value is called and it is a distribution that is denoted by
or
It satisfies
\operatorname{supp}(S\astT)\subseteq\operatorname{supp}(S)+\operatorname{supp}(T).
If
and
are two distributions, at least one of which has compact support, then for any
\taua(S\astT)=\left(\tauaS\right)\astT=S\ast\left(\tauaT\right).
If
is a distribution in
and if
is a
Dirac measure then
T\ast\delta=T=\delta\astT
; thus
is the
identity element of the convolution operation. Moreover, if
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
and
Suppose that it is
that has compact support. For
consider the function
It can be readily shown that this defines a smooth function of
which moreover has compact support. The convolution of
and
is defined by
This generalizes the classical notion of convolution of functions and is compatible with differentiation in the following sense: for every multi-index
The convolution of a finite number of distributions, all of which (except possibly one) have compact support, is associative.
This definition of convolution remains valid under less restrictive assumptions about
and
[18] The convolution of distributions with compact support induces a continuous bilinear map
defined by
where
denotes the space of distributions with compact support. However, the convolution map as a function
is continuous although it is separately continuous. The convolution maps
l{D}(\Rn) x l{D}'\tol{D}'
and
l{D}(\Rn) x l{D}'\tol{D}(\Rn)
given by
both to be continuous. Each of these non-continuous maps is, however, separately continuous and
hypocontinuous.
Convolution versus multiplication
In general, regularity is required for multiplication products, and locality is required for convolution products. It is expressed in the following extension of the Convolution Theorem which guarantees the existence of both convolution and multiplication products. Let
be a rapidly decreasing tempered distribution or, equivalently,
be an ordinary (slowly growing, smooth) function within the space of tempered distributions and let
be the normalized (unitary, ordinary frequency)
Fourier transform.
[19] Then, according to,
hold within the space of tempered distributions.
[20] [21] [22] In particular, these equations become the
Poisson Summation Formula if
} is the
Dirac Comb.
[23] The space of all rapidly decreasing tempered distributions is also called the space of
and the space of all ordinary functions within the space of tempered distributions is also called the space of
More generally,
and
[24] 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
and
\operatorname{PW}\subseteql{O}M.
In other words, compactly supported tempered distributions
belong to the space of
andPaley-Wiener functions
better known as
bandlimited functions, belong to the space of
For example, let
} \in \mathcal' be the Dirac comb and
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
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
functions. More generally, if
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
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
and
be open sets. Assume all vector spaces to be over the field
where
or
For
define for every
and every
the following functions:
Given
and
define the following functions:
where
\langleT,f\bullet\rangle\inl{D}(U)
and
\langleS,f\bullet\rangle\inl{D}(V).
These definitions associate every
and
with the (respective) continuous linear map:
Moreover, if either
(resp.
) has compact support then it also induces a continuous linear map of
Cinfty(U x V)\toCinfty(V)
(resp.
denoted by
or
is the distribution in
defined by:
Spaces of distributions
See also: Spaces of test functions and distributions.
For all
and all
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:
where the topologies on
(
) are defined as direct limits of the spaces
in a manner analogous to how the topologies on
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
is one of the spaces
(for
) or
(for
) or
(for
). Because the canonical injection
is a continuous injection whose image is dense in the codomain, this map's
transpose {}t\operatorname{In}X:X'b\tol{D}'(U)=
is a continuous injection. This injective transpose map thus allows the continuous dual space
of
to be identified with a certain vector subspace of the space
of all distributions (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
carrying a
locally convex topology that is finer than the
subspace topology induced on it by
is called
.Almost all of the spaces of distributions mentioned in this article arise in this way (for example, tempered distribution, restrictions, distributions of order
some integer, distributions induced by a positive Radon measure, distributions induced by an
-function, etc.) and any representation theorem about the continuous dual space of
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).
Radon measures
The inclusion map
is a continuous injection whose image is dense in its codomain, so the
transpose {}t\operatorname{In}:
\tol{D}'(U)=
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,
then there exists a Radon measure
on such that for all
and
is a Radon measure on then the linear functional on
defined by sending
to
is continuous.
Through the injection
{}t\operatorname{In}:
\tol{D}'(U),
every Radon measure becomes a distribution on . If
is a
locally integrable function on then the distribution
is a Radon measure; so Radon measures form a large and important space of distributions.
The following is the theorem of the structure of distributions of Radon measures, which shows that every Radon measure can be written as a sum of derivatives of locally
functions on :
Positive Radon measures
A linear function
on a space of functions is called
if whenever a function
that belongs to the domain of
is non-negative (that is,
is real-valued and
) then
One may show that every positive linear functional on
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
is called
if it is
Lebesgue integrable over every compact subset of . This is a large class of functions that includes all continuous functions and all
Lp space
functions. The topology on
is defined in such a fashion that any locally integrable function
yields a continuous linear functional on
– that is, an element of
– denoted here by
whose value on the test function
is given by the Lebesgue integral:
Conventionally, one abuses notation by identifying
with
provided no confusion can arise, and thus the pairing between
and
is often written
If
and
are two locally integrable functions, then the associated distributions
and
are equal to the same element of
if and only if
and
are equal
almost everywhere (see, for instance,). Similarly, every
Radon measure
on
defines an element of
whose value on the test function
is
As above, it is conventional to abuse notation and write the pairing between a Radon measure
and a test function
as
Conversely, as shown in a theorem by Schwartz (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
is sequentially
dense in
with respect to the strong topology on
This means that for any
there is a sequence of test functions,
that converges to
(in its strong dual topology) when considered as a sequence of distributions. Or equivalently,
Distributions with compact support
The inclusion map
\operatorname{In}:
\toCinfty(U)
is a continuous injection whose image is dense in its codomain, so the
transpose map {}t\operatorname{In}:
\tol{D}'(U)=
is also a continuous injection. Thus the image of the transpose, denoted by
forms a space of distributions.
The elements of
can be identified as the space of distributions with compact support. Explicitly, if
is a distribution on then the following are equivalent,
is compact.
to
when that space is equipped with the subspace topology inherited from
(a coarser topology than the canonical LF topology), is continuous.
- There is a compact subset of such that for every test function
whose support is completely outside of, we have
Compactly supported distributions define continuous linear functionals on the space
; recall that the topology on
is defined such that a sequence of test functions
converges to 0 if and only if all derivatives of
converge uniformly to 0 on every compact subset of . Conversely, it can be shown that every continuous linear functional on this space defines a distribution of compact support. Thus compactly supported distributions can be identified with those distributions that can be extended from
to
Distributions of finite order
Let
The inclusion map
is a continuous injection whose image is dense in its codomain, so the
transpose {}t\operatorname{In}:
\tol{D}'(U)=
is also a continuous injection. Consequently, the image of
denoted by
forms a space of distributions. The elements of
are
The distributions of order
which are also called
are exactly the distributions that are Radon measures (described above).
For
a
is a distribution of order
that is not a distribution of order
.
A distribution is said to be of if there is some integer
such that it is a distribution of order
and the set of distributions of finite order is denoted by
Note that if
then
l{D}'k(U)\subseteql{D}'l(U)
so that
is a vector subspace of
, and furthermore, if and only if
Structure of distributions of finite order
Every distribution with compact support in is a distribution of finite order. Indeed, every distribution in is a distribution of finite order, in the following sense: If is an open and relatively compact subset of and if
is the restriction mapping from to, then the image of
under
is contained in
The following is the theorem of the structure of distributions of finite order, which shows that every distribution of finite order can be written as a sum of derivatives of Radon measures:
Example. (Distributions of infinite order) Let
and for every test function
let
Then
is a distribution of infinite order on . Moreover,
can not be extended to a distribution on
; that is, there exists no distribution
on
such that the restriction of
to is equal to
Tempered distributions and Fourier transform
Defined below are the , which form a subspace of
the space of distributions on
This is a proper subspace: while every tempered distribution is a distribution and an element of
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
Schwartz space
is the space of all smooth functions that are rapidly decreasing at infinity along with all partial derivatives. Thus
is in the Schwartz space provided that any derivative of
multiplied with any power of
converges to 0 as
These functions form a complete TVS with a suitably defined family of
seminorms. More precisely, for any
multi-indices
and
define
Then
is in the Schwartz space if all the values satisfy
The family of seminorms
defines a
locally convex topology on the Schwartz space. For
the seminorms are, in fact,
norms on the Schwartz space. One can also use the following family of seminorms to define the topology:
Otherwise, one can define a norm on
via
The Schwartz space is a Fréchet space (that is, a complete metrizable locally convex space). Because the Fourier transform changes
into multiplication by
and vice versa, this symmetry implies that the Fourier transform of a Schwartz function is also a Schwartz function.
A sequence
in
converges to 0 in
if and only if the functions
converge to 0 uniformly in the whole of
which implies that such a sequence must converge to zero in
is dense in
The subset of all analytic Schwartz functions is dense in
as well.
The Schwartz space is nuclear, and the tensor product of two maps induces a canonical surjective TVS-isomorphismswhere
represents the completion of the
injective tensor product (which in this case is 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}:
\tol{D}'(\Rn)
is also a continuous injection. Thus, the image of the transpose map, denoted by
forms a space of distributions.
The space
is called the space of . It is the continuous dual space of the Schwartz space. Equivalently, a distribution
is a tempered distribution if and only if
for
are tempered distributions.
The can also be characterized as, meaning that each derivative of
grows at most as fast as some
polynomial. This characterization is dual to the behaviour of the derivatives of a function in the Schwartz space, where each derivative of
decays faster than every inverse power of
An example of a rapidly falling function is
for any positive
Fourier transform
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
So the Fourier transform of the tempered distribution
is defined by
for every Schwartz function
is thus again a tempered distribution. The Fourier transform is a TVS isomorphism from the space of tempered distributions onto itself. This operation is compatible with differentiation in the sense that
and also with convolution: if
is a tempered distribution and
is a smooth function on
is again a tempered distribution and
is the convolution of
and
In particular, the Fourier transform of the constant function equal to 1 is the
distribution.
Expressing tempered distributions as sums of derivatives
If
is a tempered distribution, then there exists a constant
and positive integers
and
such that for all
Schwartz functions
This estimate, along with some techniques from functional analysis, can be used to show that there is a continuous slowly increasing function
and a multi-index
such that
Restriction of distributions to compact sets
If
then for any compact set
there exists a continuous function
compactly supported in
(possibly on a larger set than itself) and a multi-index
such that
on
Using holomorphic functions as test functions
The success of the theory led to an investigation of the idea of hyperfunction, in which spaces of holomorphic functions are used as test functions. A refined theory has been developed, in particular Mikio Sato's algebraic analysis, using sheaf theory and several complex variables. This extends the range of symbolic methods that can be made into rigorous mathematics, for example, Feynman integrals.
See also
Differential equations related
Generalizations of distributions
Bibliography
- Book: Barros-Neto, José. An Introduction to the Theory of Distributions. Dekker. New York, NY. 1973.
- .
- Book: Folland, G.B.. Gerald Folland. 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.
Further reading
- 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
- The image of the compact set
under a continuous
-valued map (for example,
x\mapsto\left|\partialpf(x)\right|
for
) is itself a compact, and thus bounded, subset of
If
then this implies that each of the functions defined above is
-valued (that is, none of the supremums above are ever equal to
).
- Exactly as with
the space
is defined to be the vector subspace of
consisting of maps with support contained in
endowed with the subspace topology it inherits from
.
- Even though the topology of
is not metrizable, a linear functional on
is continuous if and only if it is sequentially continuous.
- A is a sequence that converges to the origin.
- If
is also directed under the usual function comparison then we can take the finite collection to consist of a single element.
- See for example .
- The extension theorem for mappings defined from a subspace S of a topological vector space E to the topological space E itself works for non-linear mappings as well, provided they are assumed to be uniformly continuous. But, unfortunately, this is not our case, we would desire to “extend” a linear continuous mapping A from a tvs E into another tvs F, in order to obtain a linear continuous mapping from the dual E’ to the dual F’ (note the order of spaces). In general, this is not even an extension problem, because (in general) E is not necessarily a subset of its own dual E’. Moreover, It is not a classic topological transpose problem, because the transpose of A goes from F’ to E’ and not from E’ to F’. Our case needs, indeed, a new order of ideas, involving the specific topological properties of the Laurent Schwartz spaces D(U) and D’(U), together with the fundamental concept of weak (or Schwartz) adjoint of the linear continuous operator A.
- Book: Strichartz, Robert . A Guide to Distribution Theory and Fourier Transforms . 1993 . USA . 17 . English.
- .
- For example, let
and take
to be the ordinary derivative for functions of one real variable and assume the support of
to be contained in the finite interval
then since
\operatorname{supp}(\phi)\subseteq(a,b)
where the last equality is because
- Web site: Multiplication of two distributions whose singular supports are disjoint. Jun 27, 2017. Stack Exchange Network. Per Persson (username: md2perpe).
- Lyons. T.. Differential equations driven by rough signals. 10.4171/RMI/240. Revista Matemática Iberoamericana. 215–310. 1998. 14 . 2 . free.
- 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. 119138901 .
- See for example .
- See .
- See for example .
- proves the uniqueness of such an extension.
- See for instance and .
- Book: Folland, G.B.. Harmonic Analysis in Phase Space. Princeton University Press. Princeton, NJ. 1989.
- Book: Horváth, John. John Horvath (mathematician). Topological Vector Spaces and Distributions. Addison-Wesley Publishing Company. Reading, MA. 1966.
- Book: Barros-Neto, José. An Introduction to the Theory of Distributions. Dekker. New York, NY. 1973.
- Book: Petersen, Bent E.. Introduction to the Fourier Transform and Pseudo-Differential Operators. Pitman Publishing. Boston, MA. 1983.
- Book: Woodward, P.M.. Probability and Information Theory with Applications to Radar. Pergamon Press. Oxford, UK. 1953.
- Book: Friedlander. F.G.. Joshi. M.S.. Introduction to the Theory of Distributions. Cambridge University Press. Cambridge, UK. 1998.