Distributions, also known as Schwartz distributions or
generalized function
In mathematics, generalized functions are objects extending the notion of functions. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful in making discontinuous functions ...
s, are objects that generalize the classical notion of functions in
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
. Distributions make it possible to
differentiate functions whose derivatives do not exist in the classical sense. In particular, any
locally integrable In mathematics, a locally integrable function (sometimes also called locally summable function) is a function which is integrable (so its integral is finite) on every compact subset of its domain of definition. The importance of such functions li ...
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
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
and
engineering
Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
where many problems naturally lead to differential equations whose solutions or initial conditions are singular, such as the
Dirac delta
In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution (mathematics), distribution over the real numbers, whose value is zero everywhere except at zero, and who ...
function.
A
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
is normally thought of as on the in the function
domain by "sending" a point
in its 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
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
-valued (or
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
-valued) functions with
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
support that are defined on some given non-empty
open subset . (
Bump function
In mathematics, a bump function (also called a test function) is a function f: \R^n \to \R on a Euclidean space \R^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. The set of all bum ...
s are examples of test functions.) The set of all such test functions forms a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
that is denoted by
or
Most commonly encountered functions, including all
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
maps
if using
can be canonically reinterpreted as acting via "
integration against a test function." Explicitly, this means that
"acts on" a test function
by "sending" it to the
number
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers c ...
which is often denoted by
This new action
of
is a
scalar-valued map, denoted by
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
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
and also
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
when
is given a certain
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
called . The action of this distribution on a test function can be interpreted as a weighted average of the distribution on the
support of the test function, even if the values of the distribution at a single point are not well-defined. Distributions like
that arise from functions in this way are prototypical examples of distributions, but many 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
Measure may refer to:
* Measurement, the assignment of a number to a characteristic of an object or event
Law
* Ballot measure, proposed legislation in the United States
* Church of England Measure, legislation of the Church of England
* Measu ...
. 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
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If is a vector space over a field , the ...
on
that is
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
when
is given a topology called the . This leads to space of (all) distributions on
, usually denoted by
(note the
prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
), which by definition is the
space
Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually cons ...
of all distributions on
(that is, it is the
continuous dual space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
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
In mathematical analysis, the spaces of test functions and distributions are topological vector spaces (TVSs) that are used in the definition and application of distributions.
Test functions are usually infinitely differentiable complex-valued ...
are given in the article on
spaces of test functions and distributions
In mathematical analysis, the spaces of test functions and distributions are topological vector spaces (TVSs) that are used in the definition and application of distributions.
Test functions are usually infinitely differentiable complex-valued ...
. 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
Laurent-Moïse Schwartz (; 5 March 1915 – 4 July 2002) was a French mathematician. He pioneered the theory of distributions, which gives a well-defined meaning to objects such as the Dirac delta function. He was awarded the Fields Medal in ...
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
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
*
denotes the
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal ...
s.
*
will denote a non-negative integer or
* If
is a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
then
will denote its
domain and the of
denoted by
is defined to be the
closure of the set
in
* For two functions
the following notation defines a canonical
pairing
In mathematics, a pairing is an ''R''-bilinear map from the Cartesian product of two ''R''-modules, where the underlying ring ''R'' is commutative.
Definition
Let ''R'' be a commutative ring with unit, and let ''M'', ''N'' and ''L'' be ''R''-mod ...
:
* A of size
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
is defined as
and denoted by
Multi-indices are particularly useful when dealing with functions of several variables, in particular, we introduce the following notations for a given multi-index
:
We also introduce a partial order of all multi-indices by
if and only if
for all
When
we define their multi-index binomial coefficient as:
Definitions of test functions and distributions
In this section, some basic notions and definitions needed to define real-valued distributions on are introduced. Further discussion of the topologies on the spaces of test functions and distributions is given in the article on
spaces of test functions and distributions
In mathematical analysis, the spaces of test functions and distributions are topological vector spaces (TVSs) that are used in the definition and application of distributions.
Test functions are usually infinitely differentiable complex-valued ...
.
![Bump](https://upload.wikimedia.org/wikipedia/commons/0/0a/Bump.png)
For all
and any compact subsets
and
of
, we have:
Distributions on are
continuous linear functional In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces.
An operator between two normed spaces is a bounded linear op ...
s 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
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If is a vector space over a field , the ...
on
is continuous, and therefore a distribution, if and only if either 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
Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an ordered tuple of indices. ...
, then
Topology on ''C''''k''(''U'')
We now introduce the
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk ...
s 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
[The image of the ]compact set
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i. ...
under a continuous -valued map (for example, 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 ). seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk ...
s 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
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces.
They are generalizations of Banach spaces ( normed vector spaces that are complete with respect to th ...
that is
normable. Every element of
is a continuous seminorm on
Under this topology, a
net
Net or net may refer to:
Mathematics and physics
* Net (mathematics), a filter-like topological generalization of a sequence
* Net, a linear system of divisors of dimension 2
* Net (polyhedron), an arrangement of polygons that can be folded up ...
in
converges to
if and only if for every multi-index
with
and every compact
the net of partial derivatives
converges uniformly to
on
For any
any
(von Neumann) bounded subset of
is a
relatively compact
In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact.
Properties
Every subset of a compact topological space is relatively compact (sin ...
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
In functional analysis and related areas of mathematics, a Montel space, named after Paul Montel, is any topological vector space (TVS) in which an analog of Montel's theorem holds. Specifically, a Montel space is a Barrelled space, barrelled topo ...
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
In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
induced on it by
Topology on ''C''''k''(''K'')
As before, fix
Recall that if
is any compact subset of
then
If
is finite then
is a
Banach space with a topology that can be defined by the
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...
And when
then
is even a
Hilbert space.
Trivial extensions and independence of ''C''''k''(''K'')'s topology from ''U''
Suppose
is an open subset of
and
is a compact subset. By definition, elements of
are functions with domain
(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
[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 .] 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
Injection or injected may refer to:
Science and technology
* Injective function, a mathematical function mapping distinct arguments to distinct values
* Injection (medicine), insertion of liquid into the body with a syringe
* Injection, in broadca ...
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
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
(linear homeomorphisms are called ):
and thus the next map is a
topological embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.
When some object X is said to be embedded in another object Y, the embedding is giv ...
:
Using the injection
the vector space
is canonically identified with its image in
Because
through this identification,
can also be considered as a subset of
Thus the topology on
is independent of the open subset
of
that contains
which justifies the practice of writing
instead of
Canonical LF topology
Recall that
denotes all functions in
that have compact
support in
where note that
is the union of all
as
ranges over all compact subsets of
Moreover, for each
is a dense subset of
The special case when
gives us the space of test functions.
The canonical LF-topology is metrizable and importantly, it is
than the
subspace topology
In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
that
induces on
However, the canonical LF-topology does make
into a
complete reflexive nuclear Montel Montel is a given name and surname. Notable people with the name include:
Given name
*Montel Vontavious Porter (born 1973), American professional wrestler
*Montel Williams (born 1956), American television personality and television/radio talk show ...
bornological barrelled Mackey space
In mathematics, particularly in functional analysis, a Mackey space is a locally convex topological vector space ''X'' such that the topology of ''X'' coincides with the Mackey topology τ(''X'',''X′''), the finest topology which still pres ...
; the same is true of its
strong dual space
In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X is the continuous dual space X^ of X equipped with the strong (dual) topology or the topology of uniform convergence on bounded sub ...
(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
Duality may refer to:
Mathematics
* Duality (mathematics), a mathematical concept
** Dual (category theory), a formalization of mathematical duality
** Duality (optimization)
** Duality (order theory), a concept regarding binary relations
** Dual ...
between a distribution
on
and a test function
which is denoted using
angle brackets
A bracket is either of two tall fore- or back-facing punctuation marks commonly used to isolate a segment of text or data from its surroundings. Typically deployed in symmetric pairs, an individual bracket may be identified as a 'left' or 'r ...
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
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If is a vector space over a field , the ...
on
then the following are equivalent:
# is a distribution;
# is
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
;
# is
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
at the origin;
# is
uniformly continuous
In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. In ...
;
# is a
bounded operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y.
If X and Y are normed vector ...
;
# is
sequentially continuous
In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that satisfy a very weak axiom of counta ...
;
#* explicitly, for every sequence
in
that converges in
to some
[Even though the topology of is not metrizable, a linear functional on is continuous if and only if it is sequentially continuous.]
# is
sequentially continuous
In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that satisfy a very weak axiom of counta ...
at the origin; in other words, maps null sequences
[A is a sequence that converges to the origin.] 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
is bounded;
# maps
Mackey convergent null sequences to bounded subsets;
#* explicitly, for every Mackey convergent null sequence
in
the sequence
is bounded;
#* a sequence
is said to be if there exists a divergent sequence
of positive real numbers such that the sequence
is bounded; every sequence that is Mackey convergent to the origin necessarily converges to the origin (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
(where
is any collection of continuous seminorms that defines the canonical LF topology on
) such that
[If is also ]directed
Director may refer to:
Literature
* ''Director'' (magazine), a British magazine
* ''The Director'' (novel), a 1971 novel by Henry Denker
* ''The Director'' (play), a 2000 play by Nancy Hasty
Music
* Director (band), an Irish rock band
* ''D ...
under the usual function comparison then we can take the finite collection to consist of a single element.
# 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
[See for example .]
# 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
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
of
which when endowed with the
strong dual topology
In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X is the continuous dual space X^ of X equipped with the strong (dual) topology or the topology of uniform convergence on bounded sub ...
is denoted by
Importantly, unless indicated otherwise, the topology on
is the
strong dual topology
In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X is the continuous dual space X^ of X equipped with the strong (dual) topology or the topology of uniform convergence on bounded sub ...
; if the topology is instead the
weak-* topology
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
then this will be indicated. Neither topology is metrizable although unlike the weak-* topology, the strong dual topology makes
into a
complete nuclear space
In mathematics, nuclear spaces are topological vector spaces that can be viewed as a generalization of finite dimensional Euclidean spaces and share many of their desirable properties. Nuclear spaces are however quite different from Hilbert spaces ...
, to name just a few of its desirable properties.
Neither
nor its strong dual
is a
sequential space
In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that satisfy a very weak axiom of counta ...
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
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
(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
In mathematical analysis, the spaces of test functions and distributions are topological vector spaces (TVSs) that are used in the definition and application of distributions.
Test functions are usually infinitely differentiable complex-valued ...
and the articles on
polar topologies and
dual system
In mathematics, a dual system, dual pair, or duality over a field \mathbb is a triple (X, Y, b) consisting of two vector spaces X and Y over \mathbb and a non- degenerate bilinear map b : X \times Y \to \mathbb.
Duality theory, the study of dual ...
s.
A
map from
into another
locally convex topological vector space
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ...
(such as any
normed space) is
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
if and only if it is
sequentially continuous
In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that satisfy a very weak axiom of counta ...
at the origin. However, this is no longer guaranteed if the map is not linear or for maps valued in more general
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
s (for example, that are not also locally convex
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
s). The same is true of maps from
(more generally, this is true of maps from any locally convex
bornological space
In mathematics, particularly in functional analysis, a bornological space is a type of space which, in some sense, possesses the minimum amount of structure needed to address questions of boundedness of sets and linear maps, in the same way that ...
).
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
Sheaf may refer to:
* Sheaf (agriculture), a bundle of harvested cereal stems
* Sheaf (mathematics), a mathematical tool
* Sheaf toss, a Scottish sport
* River Sheaf, a tributary of River Don in England
* ''The Sheaf'', a student-run newspaper se ...
.
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
A complement is something that completes something else.
Complement may refer specifically to:
The arts
* Complement (music), an interval that, when added to another, spans an octave
** Aggregate complementation, the separation of pitch-clas ...
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
In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, l ...
of
(although as a
topological subspace
In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
).
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
In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
that
induces on it; importantly, it would be a
topological subspace
In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
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
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
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learn ...
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
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
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
In mathematics, a Dirac measure assigns a size to a set based solely on whether it contains a fixed element ''x'' or not. It is one way of formalizing the idea of the Dirac delta function, an important tool in physics and other technical fields.
...
at a point
is the set
If the support of a test function
does not intersect the support of a distribution then
A distribution is 0 if and only if its support is empty. If
is identically 1 on some open set containing the support of a distribution then
If the support of a distribution is compact then it has finite order and there is a constant
and a non-negative integer
such that:
If has compact support, then it has a unique extension to a continuous linear functional
on
; this function can be defined by
where
is any function that is identically 1 on an open set containing the support of .
If
and
then
and
Thus, distributions with support in a given subset
form a vector subspace of
Furthermore, if
is a differential operator in , then for all distributions on and all
we have
and
Distributions with compact support
Support in a point set and Dirac measures
For any
let
denote the distribution induced by the Dirac measure at
For any
and distribution
the support of is contained in
if and only if is a finite linear combination of derivatives of the Dirac measure at
If in addition the order of is
then there exist constants
such that:
Said differently, if has support at a single point
then is in fact a finite linear combination of distributional derivatives of the
function at . That is, there exists an integer and complex constants
such that
where
is the translation operator.
Distribution with compact support
Distributions of finite order with support in an open subset
Global structure of distributions
The formal definition of distributions exhibits them as a subspace of a very large space, namely the topological dual of
(or the
Schwartz space
In mathematics, Schwartz space \mathcal is the function space of all functions whose derivatives are rapidly decreasing. This space has the important property that the Fourier transform is an automorphism on this space. This property enables on ...
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.
Distributions as sheaves
Decomposition of distributions as sums of derivatives of continuous functions
By combining the above results, one may express any distribution on as the sum of a series of distributions with compact support, where each of these distributions can in turn be written as a finite sum of distributional derivatives of continuous functions on . In other words, for arbitrary
we can write:
where
are finite sets of multi-indices and the functions
are continuous.
Note that the infinite sum above is well-defined as a distribution. The value of for a given
can be computed using the finitely many
that intersect the support of
Operations on distributions
Many operations which are defined on smooth functions with compact support can also be defined for distributions. In general, if
is a linear map that is continuous with respect to the
weak topology
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
, then it is possible to extend
to a map
by passing to the limit.
[This approach works for non-linear mappings as well, provided they are assumed to be ]uniformly continuous
In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. In ...
.
Preliminaries: Transpose of a linear operator
Operations on distributions and spaces of distributions are often defined using the
transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
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
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined o ...
. For instance, the well-known
Hermitian adjoint
In mathematics, specifically in operator theory, each linear operator A on a Euclidean vector space defines a Hermitian adjoint (or adjoint) operator A^* on that space according to the rule
:\langle Ax,y \rangle = \langle x,A^*y \rangle,
where ...
of a linear operator between
Hilbert spaces is just the operator's transpose (but with the
Riesz representation theorem
:''This article describes a theorem concerning the dual of a Hilbert space. For the theorems relating linear functionals to Measure (mathematics), measures, see Riesz–Markov–Kakutani representation theorem.''
The Riesz representation theorem, ...
used to identify each Hilbert space with its
continuous dual space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
). In general, the transpose of a continuous linear map
is the linear map
or equivalently, it is the unique map satisfying
for all
and all
(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
In functional analysis and related areas of mathematics a polar topology, topology of \mathcal-convergence or topology of uniform convergence on the sets of \mathcal is a method to define locally convex topologies on the vector spaces of a pairin ...
and
dual system
In mathematics, a dual system, dual pair, or duality over a field \mathbb is a triple (X, Y, b) consisting of two vector spaces X and Y over \mathbb and a non- degenerate bilinear map b : X \times Y \to \mathbb.
Duality theory, the study of dual ...
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
) it is sufficient that the defining equality hold for all distributions of the form
where
Explicitly, this means that a continuous linear map
is equal to
if and only if the condition below holds:
where the right-hand side equals
Differential operators
Differentiation of distributions
Let
be the partial derivative operator
To extend
we compute its transpose:
Therefore
Thus, the partial derivative of
with respect to the coordinate
is defined by the formula
With this definition, every distribution is infinitely differentiable, and the derivative in the direction
is a
linear operator on
More generally, if
is an arbitrary
multi-index
Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an ordered tuple of indices. ...
, 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
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. ...
combined with the fact that
and therefore all the functions
have compact support.
[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
where the last equality is because ] 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,
can be taken as
To see this, for every
compute its action on a distribution of the form
with
:
We call the continuous linear operator
the . Its action on an arbitrary distribution
is defined via:
If
converges to
then for every multi-index
converges to
Multiplication of distributions by smooth functions
A differential operator of order 0 is just multiplication by a smooth function. And conversely, if
is a smooth function then
is a differential operator of order 0, whose formal transpose is itself (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
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Mo ...
over the
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
With this definition of multiplication by a smooth function, the ordinary
product rule
In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v ...
of calculus remains valid. However, some unusual identities also arise. For example, if
is the Dirac delta distribution on
then
and if
is the derivative of the delta distribution, then
The bilinear multiplication map
given by
is continuous; it is however,
hypocontinuous.
Example. The product of any distribution
with the function that is identically on
is equal to
Example. Suppose
is a sequence of test functions on
that converges to the constant function
For any distribution
on
the sequence
converges to
If
converges to
and
converges to
then
converges to
=Problem of multiplying distributions
=
It is easy to define the product of a distribution with a smooth function, or more generally the product of two distributions whose
singular support
In mathematics, the support of a Real number, real-valued Function (mathematics), function f is the subset of the function Domain of a function, domain containing the elements which are not mapped to zero. If the domain of f is a topological spac ...
s are disjoint. With more effort, it is possible to define a well-behaved product of several distributions provided their
wave front set
In mathematical analysis, more precisely in microlocal analysis, the wave front (set) WF(''f'') characterizes the singularities of a generalized function ''f'', not only in space, but also with respect to its Fourier transform at each point. The t ...
s 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
Laurent-Moïse Schwartz (; 5 March 1915 – 4 July 2002) was a French mathematician. He pioneered the theory of distributions, which gives a well-defined meaning to objects such as the Dirac delta function. He was awarded the Fields Medal in ...
in the 1950s. For example, if
is the distribution obtained by the
Cauchy principal value
In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined.
Formulation
Depending on the type of singularity in the integrand ...
If
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
Regularization may refer to:
* Regularization (linguistics)
* Regularization (mathematics)
* Regularization (physics)
In physics, especially quantum field theory, regularization is a method of modifying observables which have singularities in ...
of
divergences. Here
Henri Epstein
Henri is an Estonian, Finnish, French, German and Luxembourgish form of the masculine given name Henry.
People with this given name
; French noblemen
:'' See the ' List of rulers named Henry' for Kings of France named Henri.''
* Henri I de Mon ...
and
Vladimir Glaser Vladimir Jurko Glaser (21 April 1924 – 22 January 1984) was a Croatian theoretical physicist working on quantum field theory and the canonization of the analytic S-matrix.
Biography
Glaser was born in Gorizia, Italy. His father, 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
In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician Geo ...
of
fluid dynamics.
Several not entirely satisfactory theories of
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
s of
generalized function
In mathematics, generalized functions are objects extending the notion of functions. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful in making discontinuous functions ...
s have been developed, among which
Colombeau's (simplified) algebra is maybe the most popular in use today.
Inspired by Lyons'
rough path In stochastic analysis, a rough path is a generalization of the notion of smooth path allowing to construct a robust solution theory for controlled differential equations driven by classically irregular signals, for example a Wiener process. The the ...
theory,
Martin Hairer
Sir Martin Hairer (born 14 November 1975) is an Austrian-British mathematician working in the field of stochastic analysis, in particular stochastic partial differential equations. He is Professor of Mathematics at EPFL (École Polytechnique F ...
proposed a consistent way of multiplying distributions with certain structures (
regularity structures), 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
In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets.
That is, a function f : X \to Y is open if for any open set U in X, the image f(U) is open in Y.
Likewise, ...
. The
Inverse function theorem
In mathematics, specifically differential calculus, the inverse function theorem gives a sufficient condition for a function to be invertible in a neighborhood of a point in its domain: namely, that its ''derivative is continuous and non-zero at ...
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
Change or Changing may refer to:
Alteration
* Impermanence, a difference in a state of affairs at different points in time
* Menopause, also referred to as "the change", the permanent cessation of the menstrual period
* Metamorphosis, or change, ...
formula, the inverse function theorem (locally), and a
partition of unity
In mathematics, a partition of unity of a topological space is a set of continuous functions from to the unit interval ,1such that for every point x\in X:
* there is a neighbourhood of where all but a finite number of the functions of are 0 ...
argument.
In the special case when
is a
diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.
Definition
Given two ...
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
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
of a function with a distribution, or even the convolution of two distributions.
Recall that if
and
are functions on
then we denote by
defined at
to be the integral
provided that the integral exists. If
are such that
then for any functions
and
we have
and
If
and
are continuous functions on
at least one of which has compact support, then
and if
then the value of
on
do depend on the values of
outside of the
Minkowski sum
In geometry, the Minkowski sum (also known as dilation) of two sets of position vectors ''A'' and ''B'' in Euclidean space is formed by adding each vector in ''A'' to each vector in ''B'', i.e., the set
: A + B = \.
Analogously, the Minkowski ...
Importantly, if
has compact support then for any
the convolution map
is continuous when considered as the map
or as the map
Translation and symmetry
Given
the translation operator
sends
to
defined by
This can be extended by the transpose to distributions in the following way: given a distribution
is the distribution
defined by
Given
define the function
by
Given a distribution
let
be the distribution defined by
The operator
is called .
Convolution of a test function with a distribution
Convolution with
defines a linear map:
which is
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
with respect to the canonical
LF space In mathematics, an ''LF''-space, also written (''LF'')-space, is a topological vector space (TVS) ''X'' that is a locally convex inductive limit of a countable inductive system (X_n, i_) of Fréchet spaces.
This means that ''X'' is a direct limi ...
topology on
Convolution of
with a distribution
can be defined by taking the transpose of
relative to the duality pairing of
with the space
of distributions. If
then by
Fubini's theorem
In mathematical analysis Fubini's theorem is a result that gives conditions under which it is possible to compute a double integral by using an iterated integral, introduced by Guido Fubini in 1907. One may switch the order of integration if th ...
Extending by continuity, the convolution of
with a distribution
is defined by
An alternative way to define the convolution of a test function
and a distribution
is to use the translation operator
The convolution of the compactly supported function
and the distribution
is then the function defined for each
by
It can be shown that the convolution of a smooth, compactly supported function and a distribution is a smooth function. If the distribution
has compact support, and if
is a polynomial (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
and continuous as the map
If
is a continuous linear map such that
for all
and all
then there exists a distribution
such that
for all
Example. Let
be the
Heaviside function
The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argume ...
on
For any
Let
be the Dirac measure at 0 and let
be its derivative as a distribution. Then
and
Importantly, the associative law fails to hold:
Convolution of distributions
It is also possible to define the convolution of two distributions
and
on
provided one of them has compact support. Informally, to define
where
has compact support, the idea is to extend the definition of the convolution
to a linear operation on distributions so that the associativity formula
continues to hold for all test functions
It is also possible to provide a more explicit characterization of the convolution of distributions. Suppose that
and
are distributions and that
has compact support. Then the linear maps
are continuous. The transposes of these maps:
are consequently continuous and it can also be shown that
This common value is called and it is a distribution that is denoted by
or
It satisfies
If
and
are two distributions, at least one of which has compact support, then for any
If
is a distribution in
and if
is a
Dirac measure
In mathematics, a Dirac measure assigns a size to a set based solely on whether it contains a fixed element ''x'' or not. It is one way of formalizing the idea of the Dirac delta function, an important tool in physics and other technical fields.
...
then
; thus
is the
identity element
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
of the convolution operation. Moreover, if
is a function then
where now the associativity of convolution implies that
for all functions
and
Suppose that it is
that has compact support. For
consider the function
It can be readily shown that this defines a smooth function of
which moreover has compact support. The convolution of
and
is defined by
This generalizes the classical notion of
convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''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
The convolution of distributions with compact support induces a continuous bilinear map
defined by
where
denotes the space of distributions with compact support. However, the convolution map as a function
is continuous although it is separately continuous. The convolution maps
and
given by
both to be continuous. Each of these non-continuous maps is, however,
separately continuous
In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example.
Definition
Vector spaces
Let V, W ...
and
hypocontinuous.
Convolution versus multiplication
In general,
regularity is required for multiplication products, and
locality
Locality may refer to:
* Locality (association), an association of community regeneration organizations in England
* Locality (linguistics)
* Locality (settlement)
* Suburbs and localities (Australia), in which a locality is a geographic subdivis ...
is required for convolution products. It is expressed in the following extension of the
Convolution Theorem
In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the pointwise product of their Fourier transforms. More generally, convolution in one domain (e.g ...
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. Then, according to ,
hold within the space of tempered distributions. In particular, these equations become the
Poisson Summation Formula
In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform. Consequently, the periodic summation of ...
if
is the
Dirac Comb
In mathematics, a Dirac comb (also known as shah function, impulse train or sampling function) is a periodic function with the formula
\operatorname_(t) \ := \sum_^ \delta(t - k T)
for some given period T. Here ''t'' is a real variable and th ...
. The space of all rapidly decreasing tempered distributions is also called the space of
and the space of all ordinary functions within the space of tempered distributions is also called the space of
More generally,
and
A particular case is the
Paley-Wiener-Schwartz Theorem which states that
and
This is because
and
In other words, compactly supported tempered distributions
belong to the space of
and
Paley-Wiener functions
better known as
bandlimited functions, belong to the space of
For example, let
be the Dirac comb and
be the
Dirac delta
In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution (mathematics), distribution over the real numbers, whose value is zero everywhere except at zero, and who ...
;then
is the function that is constantly one and both equations yield the
Dirac-comb identity. Another example is to let
be the Dirac comb and
be the
rectangular function
The rectangular function (also known as the rectangle function, rect function, Pi function, Heaviside Pi function, gate function, unit pulse, or the normalized boxcar function) is defined as
\operatorname(t) = \Pi(t) =
\left\{\begin{array}{rl ...
; then
is the
sinc function and both equations yield the
Classical Sampling Theorem for suitable
functions. More generally, if
is the Dirac comb and
is a
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ...
window function
In signal processing and statistics, a window function (also known as an apodization function or tapering function) is a mathematical function that is zero-valued outside of some chosen interval, normally symmetric around the middle of the int ...
(
Schwartz function), for example, the
Gaussian
Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below.
There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
, then
is another smooth window function (Schwartz function). They are known as
mollifier
In mathematics, mollifiers (also known as ''approximations to the identity'') are smooth functions with special properties, used for example in distribution theory to create sequences of smooth functions approximating nonsmooth (generalized) f ...
s, especially in
partial differential equations theory, or as
regularizers in
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
because they allow turning
generalized function
In mathematics, generalized functions are objects extending the notion of functions. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful in making discontinuous functions ...
s into
regular functions.
Tensor products of distributions
Let
and
be open sets. Assume all vector spaces to be over the field
where
or
For
define for every
and every
the following functions:
Given
and
define the following functions:
where
and
These definitions associate every
and
with the (respective) continuous linear map:
Moreover, if either
(resp.
) has compact support then it also induces a continuous linear map of
(resp.
denoted by
or
is the distribution in
defined by:
Spaces of distributions
For all
and all
every one of the following canonical injections is continuous and has an
image (also called the range) that is a
dense subset
In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the ra ...
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
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
is a continuous injection. This injective transpose map thus allows the
continuous dual space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
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
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.
When some object X is said to be embedded in another object Y, the embedding is giv ...
.
A linear subspace of
carrying a
locally convex topology that is finer than the
subspace topology
In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
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
be transferred directly to elements of the space
Radon measures
The inclusion map
is a continuous injection whose image is dense in its codomain, so the
transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
is also a continuous injection.
Note that the continuous dual space
can be identified as the space of
Radon measure
In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space ''X'' that is finite on all compact sets, outer regular on all Borel ...
s, where there is a one-to-one correspondence between the continuous linear functionals
and integral with respect to a Radon measure; that is,
* if
then there exists a Radon measure
on such that for all
and
* if
is a Radon measure on then the linear functional on
defined by sending
to
is continuous.
Through the injection
every Radon measure becomes a distribution on . If
is a
locally integrable In mathematics, a locally integrable function (sometimes also called locally summable function) is a function which is integrable (so its integral is finite) on every compact subset of its domain of definition. The importance of such functions li ...
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 measure
In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space ''X'' that is finite on all compact sets, outer regular on all Borel ...
s, 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
In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Leb ...
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
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
(see, for instance, ). Similarly, every
Radon measure
In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space ''X'' that is finite on all compact sets, outer regular on all Borel ...
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
:''This article describes a theorem concerning the dual of a Hilbert space. For the theorems relating linear functionals to Measure (mathematics), measures, see Riesz–Markov–Kakutani representation theorem.''
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
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
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
is a continuous injection whose image is dense in its codomain, so the
transpose map is also a continuous injection. Thus the image of the transpose, denoted by
forms a space of distributions.
The elements of
can be identified as the space of distributions with compact support. Explicitly, if
is a distribution on then the following are equivalent,
*
* The support of
is compact.
* The restriction of
to
when that space is equipped with the subspace topology inherited from
(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
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
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
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 measure
In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space ''X'' that is finite on all compact sets, outer regular on all Borel ...
s:
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
The
Schwartz space
In mathematics, Schwartz space \mathcal is the function space of all functions whose derivatives are rapidly decreasing. This space has the important property that the Fourier transform is an automorphism on this space. This property enables on ...
is the space of all smooth functions that are
rapidly decreasing In mathematics, a function is said to vanish at infinity if its values approach 0 as the input grows without bounds. There are two different ways to define this with one definition applying to functions defined on normed vector spaces and the other ...
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
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk ...
s. 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
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces.
They are generalizations of Banach spaces ( normed vector spaces that are complete with respect to th ...
(that is, a
complete metrizable
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) s ...
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-isomorphisms
where
represents the completion of the
injective tensor product In mathematics, the injective tensor product of two topological vector spaces (TVSs) was introduced by Alexander Grothendieck and was used by him to define nuclear spaces. An injective tensor product is in general not necessarily complete, so it ...
(which in this case is identical to the completion of the
projective tensor product
The strongest locally convex topological vector space (TVS) topology on X \otimes Y, the tensor product of two locally convex TVSs, making the canonical map \cdot \otimes \cdot : X \times Y \to X \otimes Y (defined by sending (x, y) \in X \times ...
).
Tempered distributions
The inclusion map
is a continuous injection whose image is dense in its codomain, so the
transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
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
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
of the Schwartz space. Equivalently, a distribution
is a tempered distribution if and only if
The derivative of a tempered distribution is again a tempered distribution. Tempered distributions generalize the bounded (or slow-growing) locally integrable functions; all distributions with compact support and all
square-integrable
In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value ...
functions are tempered distributions. More generally, all functions that are products of polynomials with elements of
Lp space for
are tempered distributions.
The can also be characterized as , meaning that each derivative of
grows at most as fast as some
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
. This characterization is dual to the behaviour of the derivatives of a function in the Schwartz space, where each derivative of
decays faster than every inverse power of
An example of a rapidly falling function is
for any positive
Fourier transform
To study the Fourier transform, it is best to consider complex-valued test functions and complex-linear distributions. The ordinary
continuous Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
is a TVS-
automorphism of the Schwartz space, and the is defined to be its
transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
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
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined o ...
, 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 mathematics, hyperfunctions are generalizations of functions, as a 'jump' from one holomorphic function to another at a boundary, and can be thought of informally as distributions of infinite order. Hyperfunctions were introduced by Mikio Sa ...
, in which spaces of
holomorphic functions are used as test functions. A refined theory has been developed, in particular
Mikio Sato
is a Japanese mathematician known for founding the fields of algebraic analysis, hyperfunctions, and holonomic quantum fields. He is a professor at the Research Institute for Mathematical Sciences in Kyoto.
Education
Sato studied at the Univ ...
's
algebraic analysis
Algebraic analysis is an area of mathematics that deals with systems of linear partial differential equations by using sheaf theory and complex analysis to study properties and generalizations of functions such as hyperfunctions and microfunct ...
, using
sheaf theory
In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...
and
several complex variables. This extends the range of symbolic methods that can be made into rigorous mathematics, for example,
Feynman integrals.
See also
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
Notes
References
Bibliography
*
* .
*
*.
* .
* .
* .
* .
*
*
*
* .
*
*
* .
* .
* .
* .
* .
*
*
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.
* .
* .
* .
* .
* .
{{Topological vector spaces
Articles containing proofs
Functional analysis
Generalizations of the derivative
Generalized functions
Smooth functions