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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 ...
f is normally thought of as on the in the function domain by "sending" a point x in its domain to the point f(x). Instead of acting on points, distribution theory reinterprets functions such as f as acting on in a certain way. In applications to physics and engineering, are usually
infinitely differentiable 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 support that are defined on some given non-empty open subset U \subseteq \R^n. (
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 C_c^\infty(U) or \mathcal(U). Most commonly encountered functions, including all continuous maps f : \R \to \R if using U := \R, can be canonically reinterpreted as acting via " integration against a test function." Explicitly, this means that f "acts on" a test function \psi \in \mathcal(\R) 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 ...
\int_\R f \, \psi \, dx, which is often denoted by D_f(\psi). This new action \psi \mapsto D_f(\psi) of f is a scalar-valued map, denoted by D_f, whose domain is the space of test functions \mathcal(\R). This functional D_f turns out to have the two defining properties of what is known as a : it is linear and also continuous when \mathcal(\R) 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 D_f 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 C_c^\infty(U) that is continuous when C_c^\infty(U) is given a topology called the . This leads to space of (all) distributions on U, usually denoted by \mathcal'(U) (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 U (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 C_c^\infty(U)); 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 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: * n is a fixed positive integer and U 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 ...
\R^n. * \N = \ 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. * k will denote a non-negative integer or \infty. * If f 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 \operatorname(f) will denote its domain and the of f, denoted by \operatorname(f), is defined to be the closure of the set \ in \operatorname(f). * For two functions f, g : U \to \Complex, 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 ...
: \langle f, g\rangle := \int_U f(x) g(x) \,dx. * A of size n is an element in \N^n (given that n is fixed, if the size of multi-indices is omitted then the size should be assumed to be n). The of a multi-index \alpha = (\alpha_1, \ldots, \alpha_n) \in \N^n is defined as \alpha_1+\cdots+\alpha_n and denoted by , \alpha, . Multi-indices are particularly useful when dealing with functions of several variables, in particular, we introduce the following notations for a given multi-index \alpha = (\alpha_1, \ldots, \alpha_n) \in \N^n: \begin x^\alpha &= x_1^ \cdots x_n^ \\ \partial^\alpha &= \frac \end We also introduce a partial order of all multi-indices by \beta \ge \alpha if and only if \beta_i \ge \alpha_i for all 1 \le i\le n. When \beta \ge \alpha we define their multi-index binomial coefficient as: \binom := \binom \cdots \binom.


Definitions of test functions and distributions

In this section, some basic notions and definitions needed to define real-valued distributions on are introduced. Further discussion of the topologies on the spaces of test functions and distributions 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 ...
. For all j, k \in \ and any compact subsets K and L of U, we have: \begin C^k(K) &\subseteq C^k_c(U) \subseteq C^k(U) \\ C^k(K) &\subseteq C^k(L) && \text K \subseteq L \\ C^k(K) &\subseteq C^j(K) && \text j \le k \\ C_c^k(U) &\subseteq C^j_c(U) && \text j \le k \\ C^k(U) &\subseteq C^j(U) && \text j \le k \\ \end Distributions on are
continuous linear 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 C_c^\infty(U) 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 C_c^\infty(U) 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 C_c^\infty(U) is continuous, and therefore a distribution, if and only if either of the following equivalent conditions is satisfied: # For every compact subset K\subseteq U there exist constants C>0 and N\in \N (dependent on K) such that for all f \in C_c^\infty(U) with support contained in K, , T(f), \leq C \sup \. # For every compact subset K\subseteq U and every sequence \_^\infty in C_c^\infty(U) whose supports are contained in K, if \_^\infty converges uniformly to zero on U 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. ...
\alpha, then T(f_i) \to 0.


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

We now introduce the seminorms that will define the topology on C^k(U). Different authors sometimes use different families of seminorms so we list the most common families below. However, the resulting topology is the same no matter which family is used. All of the functions above are non-negative \R-valuedThe 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. ...
K under a continuous \R-valued map (for example, x \mapsto \left, \partial^p f(x)\ for x \in U) is itself a compact, and thus bounded, subset of \R. If K \neq \varnothing then this implies that each of the functions defined above is \R-valued (that is, none of the supremums above are ever equal to \infty).
seminorms on C^k(U). As explained in this article, every set of seminorms on a vector space induces a locally convex vector topology. Each of the following sets of seminorms \begin A ~:= \quad &\ \\ B ~:= \quad &\ \\ C ~:= \quad &\ \\ D ~:= \quad &\ \end generate the same locally convex vector topology on C^k(U) (so for example, the topology generated by the seminorms in A is equal to the topology generated by those in C). With this topology, C^k(U) 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 A \cup B \cup C \cup D is a continuous seminorm on C^k(U). 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 ...
(f_i)_ in C^k(U) converges to f \in C^k(U) if and only if for every multi-index p with , p, < k + 1 and every compact K, the net of partial derivatives \left(\partial^p f_i\right)_ converges uniformly to \partial^p f on K. For any k \in \, any (von Neumann) bounded subset of C^(U) 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 C^k(U). In particular, a subset of C^\infty(U) is bounded if and only if it is bounded in C^i(U) for all i \in \N. The space C^k(U) 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 k = \infty. A subset W of C^\infty(U) is open in this topology if and only if there exists i\in \N such that W is open when C^\infty(U) 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 C^i(U).


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

As before, fix k \in \. Recall that if K is any compact subset of U then C^k(K) \subseteq C^k(U). If k is finite then C^k(K) is a Banach space with a topology that can be defined by the norm r_K(f) := \sup_ \left( \sup_ \left, \partial^p f(x_0)\ \right). And when k = 2, then C^k(K) is even a Hilbert space.


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

Suppose U is an open subset of \R^n and K \subseteq U is a compact subset. By definition, elements of C^k(K) are functions with domain U (in symbols, C^k(K) \subseteq C^k(U)), so the space C^k(K) and its topology depend on U; to make this dependence on the open set U clear, temporarily denote C^k(K) by C^k(K;U). Importantly, changing the set U to a different open subset U' (with K \subseteq U') will change the set C^k(K) from C^k(K;U) to C^k(K;U'),Exactly as with C^k(K;U), the space C^k(K; U') is defined to be the vector subspace of C^k(U') consisting of maps with support contained in K endowed with the subspace topology it inherits from C^k(U'). so that elements of C^k(K) will be functions with domain U' instead of U. Despite C^k(K) depending on the open set (U \text U'), the standard notation for C^k(K) makes no mention of it. This is justified because, as this subsection will now explain, the space C^k(K;U) is canonically identified as a subspace of C^k(K;U') (both algebraically and topologically). It is enough to explain how to canonically identify C^k(K; U) with C^k(K; U') when one of U and U' is a subset of the other. The reason is that if V and W are arbitrary open subsets of \R^n containing K then the open set U := V \cap W also contains K, so that each of C^k(K; V) and C^k(K; W) is canonically identified with C^k(K; V \cap W) and now by transitivity, C^k(K; V) is thus identified with C^k(K; W). So assume U \subseteq V are open subsets of \R^n containing K. Given f \in C_c^k(U), its is the function F : V \to \Complex defined by: F(x) = \begin f(x) & x \in U, \\ 0 & \text. \end This trivial extension belongs to C^k(V) (because f \in C_c^k(U) has compact support) and it will be denoted by I(f) (that is, I(f) := F). The assignment f \mapsto I(f) thus induces a map I : C_c^k(U) \to C^k(V) that sends a function in C_c^k(U) to its trivial extension on V. 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 K \subseteq U (where K is also a compact subset of V since K \subseteq U \subseteq V), \begin I\left(C^k(K; U)\right) &~=~ C^k(K; V) \qquad \text \\ I\left(C_c^k(U)\right) &~\subseteq~ C_c^k(V). \end If I is restricted to C^k(K; U) 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 ): \begin \,& C^k(K; U) && \to \,&& C^k(K;V) \\ & f && \mapsto\,&& I(f) \\ \end 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 ...
: \begin \,& C^k(K; U) && \to \,&& C^k(V) \\ & f && \mapsto\,&& I(f). \\ \end Using the injection I : C_c^k(U) \to C^k(V) the vector space C_c^k(U) is canonically identified with its image in C_c^k(V) \subseteq C^k(V). Because C^k(K; U) \subseteq C_c^k(U), through this identification, C^k(K; U) can also be considered as a subset of C^k(V). Thus the topology on C^k(K;U) is independent of the open subset U of \R^n that contains K, which justifies the practice of writing C^k(K) instead of C^k(K; U).


Canonical LF topology

Recall that C_c^k(U) denotes all functions in C^k(U) that have compact support in U, where note that C_c^k(U) is the union of all C^k(K) as K ranges over all compact subsets of U. Moreover, for each k,\, C_c^k(U) is a dense subset of C^k(U). The special case when k = \infty gives us the space of test functions. The canonical LF-topology is metrizable and importantly, it is than the
subspace topology 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 C^\infty(U) induces on C_c^\infty(U). However, the canonical LF-topology does make C_c^\infty(U) into a complete reflexive nuclear Montel bornological barrelled Mackey space; 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 C_c^\infty(U) are known as distributions on U. Other equivalent definitions are described below. There is a canonical duality pairing between a distribution T on U and a test function f \in C_c^\infty(U), which is denoted using angle brackets by \begin \mathcal'(U) \times C_c^\infty(U) \to \R \\ (T, f) \mapsto \langle T, f \rangle := T(f) \end One interprets this notation as the distribution T acting on the test function f to give a scalar, or symmetrically as the test function f acting on the distribution T.


Characterizations of distributions

Proposition. If T is a
linear functional 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 C_c^\infty(U) then the following are equivalent: # is a distribution; # is continuous; # is 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 \left(f_i\right)_^\infty in C_c^\infty(U) that converges in C_c^\infty(U) to some f \in C_c^\infty(U), \lim_ T\left(f_i\right) = T(f);Even though the topology of C_c^\infty(U) is not metrizable, a linear functional on C_c^\infty(U) 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 sequencesA is a sequence that converges to the origin. to null sequences; #* explicitly, for every sequence \left(f_i\right)_^\infty in C_c^\infty(U) that converges in C_c^\infty(U) to the origin (such a sequence is called a ), \lim_ T\left(f_i\right) = 0; #* a is by definition any sequence that converges to the origin; # maps null sequences to bounded subsets; #* explicitly, for every sequence \left(f_i\right)_^\infty in C_c^\infty(U) that converges in C_c^\infty(U) to the origin, the sequence \left(T\left(f_i\right)\right)_^\infty is bounded; # maps Mackey convergent null sequences to bounded subsets; #* explicitly, for every Mackey convergent null sequence \left(f_i\right)_^\infty in C_c^\infty(U), the sequence \left(T\left(f_i\right)\right)_^\infty is bounded; #* a sequence f_ = \left(f_i\right)_^\infty is said to be if there exists a divergent sequence r_ = \left(r_i\right)_^\infty \to \infty of positive real numbers such that the sequence \left(r_i f_i\right)_^\infty 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 C_c^\infty(U); # The graph of is closed; # There exists a continuous seminorm g on C_c^\infty(U) such that , T, \leq g; # There exists a constant C > 0 and a finite subset \ \subseteq \mathcal (where \mathcal is any collection of continuous seminorms that defines the canonical LF topology on C_c^\infty(U)) such that , T, \leq C(g_1 + \cdots + g_m);If \mathcal 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 K\subseteq U there exist constants C>0 and N\in \N such that for all f \in C^\infty(K), , T(f), \leq C \sup \; # For every compact subset K\subseteq U there exist constants C_K>0 and N_K\in \N such that for all f \in C_c^\infty(U) with support contained in K,See for example . , T(f), \leq C_K \sup \; # For any compact subset K\subseteq U and any sequence \_^\infty in C^\infty(K), if \_^\infty converges uniformly to zero for all multi-indices p, then T(f_i) \to 0;


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

The set of all distributions on U is the
continuous dual space 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 C_c^\infty(U), 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 \mathcal'(U). Importantly, unless indicated otherwise, the topology on \mathcal'(U) 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 then this will be indicated. Neither topology is metrizable although unlike the weak-* topology, the strong dual topology makes \mathcal'(U) into a complete nuclear space, to name just a few of its desirable properties. Neither C_c^\infty(U) nor its strong dual \mathcal'(U) 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 \mathcal'(U) converges in the strong dual topology if and only if it converges in the weak-* topology (this leads many authors to use pointwise convergence to 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 \mathcal'(U) 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 \mathcal'(U) 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 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 C_c^\infty(U) (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 \mathcal'(U) at a particular point of . However, as is the case with functions, distributions on restrict to give distributions on open subsets of . Furthermore, distributions are in the sense that a distribution on all of can be assembled from a distribution on an open cover of satisfying some compatibility conditions on the overlaps. Such a structure is known as a
sheaf 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 V \subseteq U be open subsets of \R^n. Every function f \in \mathcal(V) can be from its domain to a function on by setting it equal to 0 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 ...
U \setminus V. This extension is a smooth compactly supported function called the and it will be denoted by E_ (f). This assignment f \mapsto E_ (f) defines the operator E_ : \mathcal(V) \to \mathcal(U), which is a continuous injective linear map. It is used to canonically identify \mathcal(V) 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 \mathcal(U) (although as a topological subspace). Its transpose ( explained here) \rho_ := ^E_ : \mathcal'(U) \to \mathcal'(V), is called the and as the name suggests, the image \rho_(T) of a distribution T \in \mathcal'(U) under this map is a distribution on V called the restriction of T to V. The defining condition of the restriction \rho_(T) is: \langle \rho_ T, \phi \rangle = \langle T, E_ \phi \rangle \quad \text \phi \in \mathcal(V). If V \neq U then the (continuous injective linear) trivial extension map E_ : \mathcal(V) \to \mathcal(U) is a topological embedding (in other words, if this linear injection was used to identify \mathcal(V) as a subset of \mathcal(U) then \mathcal(V)'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 \mathcal(U) induces on it; importantly, it would be a topological subspace since that requires equality of topologies) and its range is also dense in its codomain \mathcal(U). Consequently if V \neq U then the restriction mapping is neither injective nor surjective. A distribution S \in \mathcal'(V) is said to be if it belongs to the range of the transpose of E_ and it is called if it is extendable to \R^n. Unless U = V, the restriction to is neither injective nor surjective. Lack of surjectivity follows since distributions can blow up towards the boundary of . For instance, if U = \R and V = (0, 2), then the distribution T(x) = \sum_^\infty n \, \delta\left(x-\frac\right) is in \mathcal'(V) but admits no extension to \mathcal'(U).


Gluing and distributions that vanish in a set

Let be an open subset of . T \in \mathcal'(U) is said to if for all f \in \mathcal(U) such that \operatorname(f) \subseteq V we have Tf = 0. vanishes in if and only if the restriction of to is equal to 0, or equivalently, if and only if lies in the
kernel 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 \rho_.


Support of a distribution

This last corollary implies that for every distribution on , there exists a unique largest subset of such that vanishes in (and does not vanish in any open subset of that is not contained in ); the complement in of this unique largest open subset is called . Thus \operatorname(T) = U \setminus \bigcup \. If f is a locally integrable function on and if D_f is its associated distribution, then the support of D_f is the smallest closed subset of in the complement of which f 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 f is continuous, then the support of D_f is equal to the closure of the set of points in at which f does not vanish. The support of the distribution associated with the
Dirac measure 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 x_0 is the set \. If the support of a test function f does not intersect the support of a distribution then Tf = 0. A distribution is 0 if and only if its support is empty. If f \in C^\infty(U) is identically 1 on some open set containing the support of a distribution then f T = T. If the support of a distribution is compact then it has finite order and there is a constant C and a non-negative integer N such that: , T \phi, \leq C\, \phi\, _N := C \sup \left\ \quad \text \phi \in \mathcal(U). If has compact support, then it has a unique extension to a continuous linear functional \widehat on C^\infty(U); this function can be defined by \widehat (f) := T(\psi f), where \psi \in \mathcal(U) is any function that is identically 1 on an open set containing the support of . If S, T \in \mathcal'(U) and \lambda \neq 0 then \operatorname(S + T) \subseteq \operatorname(S) \cup \operatorname(T) and \operatorname(\lambda T) = \operatorname(T). Thus, distributions with support in a given subset A \subseteq U form a vector subspace of \mathcal'(U). Furthermore, if P is a differential operator in , then for all distributions on and all f \in C^\infty(U) we have \operatorname (P(x, \partial) T) \subseteq \operatorname(T) and \operatorname(fT) \subseteq \operatorname(f) \cap \operatorname(T).


Distributions with compact support


Support in a point set and Dirac measures

For any x \in U, let \delta_x \in \mathcal'(U) denote the distribution induced by the Dirac measure at x. For any x_0 \in U and distribution T \in \mathcal'(U), the support of is contained in \ if and only if is a finite linear combination of derivatives of the Dirac measure at x_0. If in addition the order of is \leq k then there exist constants \alpha_p such that: T = \sum_ \alpha_p \partial^p \delta_. Said differently, if has support at a single point \, then is in fact a finite linear combination of distributional derivatives of the \delta function at . That is, there exists an integer and complex constants a_\alpha such that T = \sum_ a_\alpha \partial^\alpha(\tau_P\delta) where \tau_P 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 \mathcal(U) (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 ...
\mathcal(\R^n) 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 T \in \mathcal'(U) we can write: T = \sum_^\infty \sum_ \partial^p f_, where P_1, P_2, \ldots are finite sets of multi-indices and the functions f_ are continuous. Note that the infinite sum above is well-defined as a distribution. The value of for a given f \in \mathcal(U) can be computed using the finitely many g_\alpha that intersect the support of f.


Operations on distributions

Many operations which are defined on smooth functions with compact support can also be defined for distributions. In general, if A:\mathcal(U)\to\mathcal(U) 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 A to a map A : \mathcal'(U)\to \mathcal'(U) 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 A : X \to Y is the linear map ^A : Y' \to X' \qquad \text \qquad ^A(y') := y' \circ A, or equivalently, it is the unique map satisfying \langle y', A(x)\rangle = \left\langle ^A (y'), x \right\rangle for all x \in X and all y' \in Y' (the prime symbol in y' does not denote a derivative of any kind; it merely indicates that y' is an element of the continuous dual space Y'). Since A is continuous, the transpose ^A : Y' \to X' 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 A : \mathcal(U) \to \mathcal(U) be a continuous linear map. Then by definition, the transpose of A is the unique linear operator A^t : \mathcal'(U) \to \mathcal'(U) that satisfies: \langle ^A(T), \phi \rangle = \langle T, A(\phi) \rangle \quad \text \phi \in \mathcal(U) \text T \in \mathcal'(U). Since \mathcal(U) is dense in \mathcal'(U) (here, \mathcal(U) actually refers to the set of distributions \left\) it is sufficient that the defining equality hold for all distributions of the form T = D_\psi where \psi \in \mathcal(U). Explicitly, this means that a continuous linear map B : \mathcal'(U) \to \mathcal'(U) is equal to ^A if and only if the condition below holds: \langle B(D_\psi), \phi \rangle = \langle ^A(D_\psi), \phi \rangle \quad \text \phi, \psi \in \mathcal(U) where the right-hand side equals \langle ^A(D_\psi), \phi \rangle = \langle D_\psi, A(\phi) \rangle = \langle \psi, A(\phi) \rangle = \int_U \psi \cdot A(\phi) \,dx.


Differential operators


Differentiation of distributions

Let A : \mathcal(U) \to \mathcal(U) be the partial derivative operator \tfrac. To extend A we compute its transpose: \begin \langle ^A(D_\psi), \phi \rangle &= \int_U \psi (A\phi) \,dx && \text \\ &= \int_U \psi \frac \, dx \\ pt&= -\int_U \phi \frac\, dx && \text \\ pt&= -\left\langle \frac, \phi \right\rangle \\ pt&= -\langle A \psi, \phi \rangle = \langle - A \psi, \phi \rangle \end Therefore ^A = -A. Thus, the partial derivative of T with respect to the coordinate x_k is defined by the formula \left\langle \frac, \phi \right\rangle = - \left\langle T, \frac \right\rangle \qquad \text \phi \in \mathcal(U). With this definition, every distribution is infinitely differentiable, and the derivative in the direction x_k is a linear operator on \mathcal'(U). More generally, if \alpha 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 \partial^\alpha T of the distribution T \in \mathcal'(U) is defined by \langle \partial^\alpha T, \phi \rangle = (-1)^ \langle T, \partial^\alpha \phi \rangle \qquad \text \phi \in \mathcal(U). Differentiation of distributions is a continuous operator on \mathcal'(U); this is an important and desirable property that is not shared by most other notions of differentiation. If T is a distribution in \R then \lim_ \frac = T'\in \mathcal'(\R), where T' is the derivative of T and \tau_x is a translation by x; thus the derivative of T may be viewed as a limit of quotients.


Differential operators acting on smooth functions

A linear differential operator in U with smooth coefficients acts on the space of smooth functions on U. Given such an operator P := \sum_\alpha c_\alpha \partial^\alpha, we would like to define a continuous linear map, D_P that extends the action of P on C^\infty(U) to distributions on U. In other words, we would like to define D_P such that the following diagram commutes: \begin \mathcal'(U) & \stackrel & \mathcal'(U) \\ pt\uparrow & & \uparrow \\ ptC^\infty(U) & \stackrel & C^\infty(U) \end where the vertical maps are given by assigning f \in C^\infty(U) its canonical distribution D_f \in \mathcal'(U), which is defined by: D_f(\phi) = \langle f, \phi \rangle := \int_U f(x) \phi(x) \,dx \quad \text \phi \in \mathcal(U). With this notation, the diagram commuting is equivalent to: D_ = D_PD_f \qquad \text f \in C^\infty(U). To find D_P, the transpose ^ P : \mathcal'(U) \to \mathcal'(U) of the continuous induced map P : \mathcal(U)\to \mathcal(U) defined by \phi \mapsto P(\phi) is considered in the lemma below. This leads to the following definition of the differential operator on U called which will be denoted by P_* to avoid confusion with the transpose map, that is defined by P_* := \sum_\alpha b_\alpha \partial^\alpha \quad \text \quad b_\alpha := \sum_ (-1)^ \binom \partial^ c_\beta. As discussed above, for any \phi \in \mathcal(U), the transpose may be calculated by: \begin \left\langle ^P(D_f), \phi \right\rangle &= \int_U f(x) P(\phi)(x) \,dx \\ &= \int_U f(x) \left sum\nolimits_\alpha c_\alpha(x) (\partial^\alpha \phi)(x) \right\,dx \\ &= \sum\nolimits_\alpha \int_U f(x) c_\alpha(x) (\partial^\alpha \phi)(x) \,dx \\ &= \sum\nolimits_\alpha (-1)^ \int_U \phi(x) (\partial^\alpha(c_\alpha f))(x) \,d x \end For the last line we used integration by parts combined with the fact that \phi and therefore all the functions f (x)c_\alpha (x) \partial^\alpha \phi(x) have compact support.For example, let U = \R and take P to be the ordinary derivative for functions of one real variable and assume the support of \phi to be contained in the finite interval (a,b), then since \operatorname(\phi) \subseteq (a, b) \begin \int_\R \phi'(x)f(x)\,dx &= \int_a^b \phi'(x)f(x) \,dx \\ &= \phi(x)f(x)\big\vert_a^b - \int_a^b f'(x) \phi(x) \,d x \\ &= \phi(b)f(b) - \phi(a)f(a) - \int_a^b f'(x) \phi(x) \,d x \\ &= \int_a^b f'(x) \phi(x) \,d x \end where the last equality is because \phi(a) = \phi(b) = 0. Continuing the calculation above, for all \phi \in \mathcal(U): \begin \left\langle ^P(D_f), \phi \right\rangle &=\sum\nolimits_\alpha (-1)^ \int_U \phi(x) (\partial^\alpha(c_\alpha f))(x) \,dx && \text \\ pt&= \int_U \phi(x) \sum\nolimits_\alpha (-1)^ (\partial^\alpha(c_\alpha f))(x)\,dx \\ pt&= \int_U \phi(x) \sum_\alpha \left sum_ \binom (\partial^c_\alpha)(x) (\partial^f)(x) \right\,dx && \text\\ &= \int_U \phi(x) \left sum_\alpha \sum_ (-1)^ \binom (\partial^c_\alpha)(x) (\partial^f)(x)\right\,dx \\ &= \int_U \phi(x) \left \sum_\alpha_\left[_\sum__(-1)^_\binom_\left(\partial^c_\right)(x)_\right(\partial^\alpha_f)(x)\right.html" ;"title="\sum_ (-1)^ \binom \left(\partial^c_\right)(x) \right">\sum_\alpha \left[ \sum_ (-1)^ \binom \left(\partial^c_\right)(x) \right(\partial^\alpha f)(x)\right">\sum_ (-1)^ \binom \left(\partial^c_\right)(x) \right">\sum_\alpha \left[ \sum_ (-1)^ \binom \left(\partial^c_\right)(x) \right(\partial^\alpha f)(x)\right\,dx && \text f \\ &= \int_U \phi(x) \left[\sum\nolimits_\alpha b_\alpha(x) (\partial^\alpha f)(x) \right] \, dx && b_\alpha:=\sum_ (-1)^ \binom \partial^c_ \\ &= \left\langle \left(\sum\nolimits_\alpha b_\alpha \partial^\alpha \right) (f), \phi \right\rangle \end The Lemma combined with the fact that the formal transpose of the formal transpose is the original differential operator, that is, P_= P, enables us to arrive at the correct definition: the formal transpose induces the (continuous) canonical linear operator P_* : C_c^\infty(U) \to C_c^\infty(U) defined by \phi \mapsto P_*(\phi). We claim that the transpose of this map, ^P_* : \mathcal'(U) \to \mathcal'(U), can be taken as D_P. To see this, for every \phi \in \mathcal(U), compute its action on a distribution of the form D_f with f \in C^\infty(U): \begin \left\langle ^P_*\left(D_f\right),\phi \right\rangle &= \left\langle D_, \phi \right\rangle && \text P_* \text P\\ &= \left\langle D_, \phi \right\rangle && P_ = P \end We call the continuous linear operator D_P := ^P_* : \mathcal'(U) \to \mathcal'(U) the . Its action on an arbitrary distribution S is defined via: D_P(S)(\phi) = S\left(P_*(\phi)\right) \quad \text \phi \in \mathcal(U). If (T_i)_^\infty converges to T \in \mathcal'(U) then for every multi-index \alpha, (\partial^\alpha T_i)_^\infty converges to \partial^\alpha T \in \mathcal'(U).


Multiplication of distributions by smooth functions

A differential operator of order 0 is just multiplication by a smooth function. And conversely, if f is a smooth function then P := f(x) is a differential operator of order 0, whose formal transpose is itself (that is, P_* = P). The induced differential operator D_P : \mathcal'(U) \to \mathcal'(U) maps a distribution T to a distribution denoted by fT := D_P(T). 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 T on U by a smooth function m : U \to \R. The product mT is defined by \langle mT, \phi \rangle = \langle T, m\phi \rangle \qquad \text \phi \in \mathcal(U). This definition coincides with the transpose definition since if M : \mathcal(U) \to \mathcal(U) is the operator of multiplication by the function m (that is, (M\phi)(x) = m(x)\phi(x)), then \int_U (M \phi)(x) \psi(x)\,dx = \int_U m(x) \phi(x) \psi(x)\,d x = \int_U \phi(x) m(x) \psi(x) \,d x = \int_U \phi(x) (M \psi)(x)\,d x, so that ^tM = M. Under multiplication by smooth functions, \mathcal'(U) is a module over the ring C^\infty(U). 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 \delta is the Dirac delta distribution on \R, then m \delta = m(0) \delta, and if \delta^' is the derivative of the delta distribution, then m\delta' = m(0) \delta' - m' \delta = m(0) \delta' - m'(0) \delta. The bilinear multiplication map C^\infty(\R^n) \times \mathcal'(\R^n) \to \mathcal'\left(\R^n\right) given by (f,T) \mapsto fT is continuous; it is however, hypocontinuous. Example. The product of any distribution T with the function that is identically on U is equal to T. Example. Suppose (f_i)_^\infty is a sequence of test functions on U that converges to the constant function 1 \in C^\infty(U). For any distribution T on U, the sequence (f_i T)_^\infty converges to T \in \mathcal'(U). If (T_i)_^\infty converges to T \in \mathcal'(U) and (f_i)_^\infty converges to f \in C^\infty(U) then (f_i T_i)_^\infty converges to fT \in \mathcal'(U).


=Problem of multiplying distributions

= It is easy to define the product of a distribution with a smooth function, or more generally the product of two distributions whose
singular 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 in the 1950s. For example, if \operatorname \frac 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 ...
\left(\operatorname \frac\right)(\phi) = \lim_ \int_ \frac\, dx \quad \text \phi \in \mathcal(\R). If \delta is the Dirac delta distribution then (\delta \times x) \times \operatorname \frac = 0 but, \delta \times \left(x \times \operatorname \frac\right) = \delta so the product of a distribution by a smooth function (which is always well-defined) cannot be extended to an associative product on the space of distributions. Thus, nonlinear problems cannot be posed in general and thus are not solved within distribution theory alone. In the context of quantum field theory, however, solutions can be found. In more than two spacetime dimensions the problem is related to the regularization of divergences. Here
Henri Epstein 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 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 T be a distribution on U. Let V be an open set in \R^n and F : V \to U. If F is a submersion then it is possible to define T \circ F \in \mathcal'(V). This is , and is also called , sometimes written F^\sharp : T \mapsto F^\sharp T = T \circ F. The pullback is often denoted F^*, although this notation should not be confused with the use of '*' to denote the adjoint of a linear mapping. The condition that F be a submersion is equivalent to the requirement that the Jacobian derivative d F(x) of F is a surjective linear map for every x \in V. A necessary (but not sufficient) condition for extending F^ to distributions is that F be an open mapping. 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 F is a submersion, then F^ is defined on distributions by finding the transpose map. The uniqueness of this extension is guaranteed since F^ is a continuous linear operator on \mathcal(U). 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 F 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 V of \R^n onto an open subset U of \R^n change of variables under the integral gives: \int_V \phi\circ F(x) \psi(x)\,dx = \int_U \phi(x) \psi \left(F^(x) \right) \left, \det dF^(x) \\,dx. In this particular case, then, F^ is defined by the transpose formula: \left\langle F^\sharp T, \phi \right\rangle = \left\langle T, \left, \det d(F^) \\phi\circ F^ \right\rangle.


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 f and g are functions on \R^n then we denote by f\ast g defined at x \in \R^n to be the integral (f \ast g)(x) := \int_ f(x-y) g(y) \,dy = \int_ f(y)g(x-y) \,dy provided that the integral exists. If 1 \leq p, q, r \leq \infty are such that \frac = \frac + \frac - 1 then for any functions f \in L^p(\R^n) and g \in L^q(\R^n) we have f \ast g \in L^r(\R^n) and \, f\ast g\, _ \leq \, f\, _ \, g\, _. If f and g are continuous functions on \R^n, at least one of which has compact support, then \operatorname(f \ast g) \subseteq \operatorname (f) + \operatorname (g) and if A\subseteq \R^n then the value of f\ast g on A do depend on the values of f 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 ...
A -\operatorname (g) = \. Importantly, if g \in L^1(\R^n) has compact support then for any 0 \leq k \leq \infty, the convolution map f \mapsto f \ast g is continuous when considered as the map C^k(\R^n) \to C^k(\R^n) or as the map C_c^k(\R^n) \to C_c^k(\R^n).


Translation and symmetry

Given a \in \R^n, the translation operator \tau_a sends f : \R^n \to \Complex to \tau_a f : \R^n \to \Complex, defined by \tau_a f(y) = f(y-a). This can be extended by the transpose to distributions in the following way: given a distribution T, is the distribution \tau_a T : \mathcal(\R^n) \to \Complex defined by \tau_a T(\phi) := \left\langle T, \tau_ \phi \right\rangle. Given f : \R^n \to \Complex, define the function \tilde : \R^n \to \Complex by \tilde(x) := f(-x). Given a distribution T, let \tilde : \mathcal(\R^n) \to \Complex be the distribution defined by \tilde(\phi) := T \left(\tilde\right). The operator T \mapsto \tilde is called .


Convolution of a test function with a distribution

Convolution with f \in \mathcal(\R^n) defines a linear map: \begin C_f : \,& \mathcal(\R^n) && \to \,&& \mathcal(\R^n) \\ & g && \mapsto\,&& f \ast g \\ \end which is 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 \mathcal(\R^n). Convolution of f with a distribution T \in \mathcal'(\R^n) can be defined by taking the transpose of C_f relative to the duality pairing of \mathcal(\R^n) with the space \mathcal'(\R^n) of distributions. If f, g, \phi \in \mathcal(\R^n), 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 ...
\langle C_fg, \phi \rangle = \int_\phi(x)\int_f(x-y) g(y) \,dy \,dx = \left\langle g,C_\phi \right\rangle. Extending by continuity, the convolution of f with a distribution T is defined by \langle f \ast T, \phi \rangle = \left\langle T, \tilde \ast \phi \right\rangle, \quad \text \phi \in \mathcal(\R^n). An alternative way to define the convolution of a test function f and a distribution T is to use the translation operator \tau_a. The convolution of the compactly supported function f and the distribution T is then the function defined for each x \in \R^n by (f \ast T)(x) = \left\langle T, \tau_x \tilde \right\rangle. It can be shown that the convolution of a smooth, compactly supported function and a distribution is a smooth function. If the distribution T has compact support, and if f is a polynomial (resp. an exponential function, an analytic function, the restriction of an entire analytic function on \Complex^n to \R^n, the restriction of an entire function of exponential type in \Complex^n to \R^n), then the same is true of T \ast f. If the distribution T has compact support as well, then f\ast T is a compactly supported function, and the Titchmarsh convolution theorem implies that: \operatorname(\operatorname(f \ast T)) = \operatorname(\operatorname(f)) + \operatorname (\operatorname(T)) where \operatorname denotes the convex hull and \operatorname denotes the support.


Convolution of a smooth function with a distribution

Let f \in C^\infty(\R^n) and T \in \mathcal'(\R^n) and assume that at least one of f and T has compact support. The of f and T, denoted by f \ast T or by T \ast f, is the smooth function: \begin f \ast T : \,& \R^n && \to \,&& \Complex \\ & x && \mapsto\,&& \left\langle T, \tau_x \tilde \right\rangle \\ \end satisfying for all p \in \N^n: \begin &\operatorname(f \ast T) \subseteq \operatorname(f)+ \operatorname(T) \\ pt&\text p \in \N^n: \quad \begin\partial^p \left\langle T, \tau_x \tilde \right\rangle = \left\langle T, \partial^p \tau_x \tilde \right\rangle \\ \partial^p (T \ast f) = (\partial^p T) \ast f = T \ast (\partial^p f). \end \end Let M be the map f \mapsto T \ast f. If T is a distribution, then M is continuous as a map \mathcal(\R^n) \to C^\infty(\R^n). If T also has compact support, then M is also continuous as the map C^\infty(\R^n) \to C^\infty(\R^n) and continuous as the map \mathcal(\R^n) \to \mathcal(\R^n). If L : \mathcal(\R^n) \to C^\infty(\R^n) is a continuous linear map such that L \partial^\alpha \phi = \partial^\alpha L \phi for all \alpha and all \phi \in \mathcal(\R^n) then there exists a distribution T \in \mathcal'(\R^n) such that L \phi = T \circ \phi for all \phi \in \mathcal(\R^n). Example. Let H be the Heaviside function on \R. For any \phi \in \mathcal(\R), (H \ast \phi)(x) = \int_^x \phi(t) \, dt. Let \delta be the Dirac measure at 0 and let \delta' be its derivative as a distribution. Then \delta' \ast H = \delta and 1 \ast \delta' = 0. Importantly, the associative law fails to hold: 1 = 1 \ast \delta = 1 \ast (\delta' \ast H ) \neq (1 \ast \delta') \ast H = 0 \ast H = 0.


Convolution of distributions

It is also possible to define the convolution of two distributions S and T on \R^n, provided one of them has compact support. Informally, to define S \ast T where T has compact support, the idea is to extend the definition of the convolution \,\ast\, to a linear operation on distributions so that the associativity formula S \ast (T \ast \phi) = (S \ast T) \ast \phi continues to hold for all test functions \phi. It is also possible to provide a more explicit characterization of the convolution of distributions. Suppose that S and T are distributions and that S has compact support. Then the linear maps \begin \bullet \ast \tilde : \,& \mathcal(\R^n) && \to \,&& \mathcal(\R^n) && \quad \text \quad && \bullet \ast \tilde : \,&& \mathcal(\R^n) && \to \,&& \mathcal(\R^n) \\ & f && \mapsto\,&& f \ast \tilde && && && f && \mapsto\,&& f \ast \tilde \\ \end are continuous. The transposes of these maps: ^\left(\bullet \ast \tilde\right) : \mathcal'(\R^n) \to \mathcal'(\R^n) \qquad ^\left(\bullet \ast \tilde\right) : \mathcal'(\R^n) \to \mathcal'(\R^n) are consequently continuous and it can also be shown that ^\left(\bullet \ast \tilde\right)(T) = ^\left(\bullet \ast \tilde\right)(S). This common value is called and it is a distribution that is denoted by S \ast T or T \ast S. It satisfies \operatorname (S \ast T) \subseteq \operatorname(S) + \operatorname(T). If S and T are two distributions, at least one of which has compact support, then for any a \in \R^n, \tau_a(S \ast T) = \left(\tau_a S\right) \ast T = S \ast \left(\tau_a T\right). If T is a distribution in \R^n and if \delta 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 T \ast \delta = T = \delta \ast T; thus \delta 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 f is a function then f \ast \delta^ = f^ = \delta^ \ast f where now the associativity of convolution implies that f^ \ast g = g^ \ast f for all functions f and g. Suppose that it is T that has compact support. For \phi \in \mathcal(\R^n) consider the function \psi(x) = \langle T, \tau_ \phi \rangle. It can be readily shown that this defines a smooth function of x, which moreover has compact support. The convolution of S and T is defined by \langle S \ast T, \phi \rangle = \langle S, \psi \rangle. This generalizes the classical notion of
convolution 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 \alpha. \partial^\alpha(S \ast T) = (\partial^\alpha S) \ast T = S \ast (\partial^\alpha T). The convolution of a finite number of distributions, all of which (except possibly one) have compact support, is associative. This definition of convolution remains valid under less restrictive assumptions about S and T. The convolution of distributions with compact support induces a continuous bilinear map \mathcal' \times \mathcal' \to \mathcal' defined by (S,T) \mapsto S * T, where \mathcal' denotes the space of distributions with compact support. However, the convolution map as a function \mathcal' \times \mathcal' \to \mathcal' is continuous although it is separately continuous. The convolution maps \mathcal(\R^n) \times \mathcal' \to \mathcal' and \mathcal(\R^n) \times \mathcal' \to \mathcal(\R^n) given by (f, T) \mapsto f * T both to be continuous. Each of these non-continuous maps is, however, separately continuous and hypocontinuous.


Convolution versus multiplication

In general, regularity is required for multiplication products, and locality is required for convolution products. It is expressed in the following extension of the
Convolution Theorem 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 F(\alpha) = f \in \mathcal'_C be a rapidly decreasing tempered distribution or, equivalently, F(f) = \alpha \in \mathcal_M be an ordinary (slowly growing, smooth) function within the space of tempered distributions and let F be the normalized (unitary, ordinary frequency) Fourier transform. Then, according to , F(f * g) = F(f) \cdot F(g) \qquad \text \qquad F(\alpha \cdot g) = F(\alpha) * F(g) hold within the space of tempered distributions. 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 g \equiv \operatorname 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 \mathcal'_C and the space of all ordinary functions within the space of tempered distributions is also called the space of \mathcal_M. More generally, F(\mathcal'_C) = \mathcal_M and F(\mathcal_M) = \mathcal'_C. A particular case is the Paley-Wiener-Schwartz Theorem which states that F(\mathcal') = \operatorname and F(\operatorname ) = \mathcal'. This is because \mathcal' \subseteq \mathcal'_C and \operatorname \subseteq \mathcal_M. In other words, compactly supported tempered distributions \mathcal' belong to the space of \mathcal'_C and Paley-Wiener functions \operatorname, better known as bandlimited functions, belong to the space of \mathcal_M. For example, let g \equiv \operatorname \in \mathcal' be the Dirac comb and f \equiv \delta \in \mathcal' 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 \alpha \equiv 1 \in \operatorname is the function that is constantly one and both equations yield the Dirac-comb identity. Another example is to let g be the Dirac comb and f \equiv \operatorname \in \mathcal' 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 \alpha \equiv \operatorname \in \operatorname is the sinc function and both equations yield the Classical Sampling Theorem for suitable \operatorname functions. More generally, if g is the Dirac comb and f \in \mathcal \subseteq \mathcal'_C \cap \mathcal_M 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 ( Schwartz function), for example, the Gaussian, then \alpha \in \mathcal 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 U \subseteq \R^m and V \subseteq \R^n be open sets. Assume all vector spaces to be over the field \mathbb, where \mathbb=\R or \Complex. For f \in \mathcal(U \times V) define for every u \in U and every v \in V the following functions: \begin f_u : \,& V && \to \,&& \mathbb && \quad \text \quad && f^v : \,&& U && \to \,&& \mathbb \\ & y && \mapsto\,&& f(u, y) && && && x && \mapsto\,&& f(x, v) \\ \end Given S \in \mathcal^(U) and T \in \mathcal^(V), define the following functions: \begin \langle S, f^\rangle : \,& V && \to \,&& \mathbb && \quad \text \quad && \langle T, f_\rangle : \,&& U && \to \,&& \mathbb \\ & v && \mapsto\,&& \langle S, f^v \rangle && && && u && \mapsto\,&& \langle T, f_u \rangle \\ \end where \langle T, f_\rangle \in \mathcal(U) and \langle S, f^\rangle \in \mathcal(V). These definitions associate every S \in \mathcal'(U) and T \in \mathcal'(V) with the (respective) continuous linear map: \begin \,&& \mathcal(U \times V) & \to \,&& \mathcal(V) && \quad \text \quad && \,& \mathcal(U \times V) && \to \,&& \mathcal(U) \\ && f \ & \mapsto\,&& \langle S, f^ \rangle && && & f \ && \mapsto\,&& \langle T, f_ \rangle \\ \end Moreover, if either S (resp. T) has compact support then it also induces a continuous linear map of C^\infty(U \times V) \to C^\infty(V) (resp. denoted by S \otimes T or T \otimes S, is the distribution in U \times V defined by: (S \otimes T)(f) := \langle S, \langle T, f_ \rangle \rangle = \langle T, \langle S, f^\rangle \rangle.


Spaces of distributions

For all 0 < k < \infty and all 1 < p < \infty, every one of the following canonical injections is continuous and has an image (also called the range) that is a
dense subset 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: \begin C_c^\infty(U) & \to & C_c^k(U) & \to & C_c^0(U) & \to & L_c^\infty(U) & \to & L_c^p(U) & \to & L_c^1(U) \\ \downarrow & &\downarrow && \downarrow \\ C^\infty(U) & \to & C^k(U) & \to & C^0(U) \\ \end where the topologies on L_c^q(U) (1 \leq q \leq \infty) are defined as direct limits of the spaces L_c^q(K) in a manner analogous to how the topologies on C_c^k(U) were defined (so in particular, they are not the usual norm topologies). The range of each of the maps above (and of any composition of the maps above) is dense in its codomain. Suppose that X is one of the spaces C_c^k(U) (for k \in \) or L^p_c(U) (for 1 \leq p \leq \infty) or L^p(U) (for 1 \leq p < \infty). Because the canonical injection \operatorname_X : C_c^\infty(U) \to X 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 ...
^\operatorname_X : X'_b \to \mathcal'(U) = \left(C_c^\infty(U)\right)'_b 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 ...
X' of X to be identified with a certain vector subspace of the space \mathcal'(U) of all distributions (specifically, it is identified with the image of this transpose map). This transpose map is continuous but it is necessarily a
topological embedding 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 \mathcal'(U) 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 \mathcal'(U) = \left(C_c^\infty(U)\right)'_b is called . Almost all of the spaces of distributions mentioned in this article arise in this way (for example, tempered distribution, restrictions, distributions of order \leq some integer, distributions induced by a positive Radon measure, distributions induced by an L^p-function, etc.) and any representation theorem about the continuous dual space of X may, through the transpose ^\operatorname_X : X'_b \to \mathcal'(U), be transferred directly to elements of the space \operatorname \left(^\operatorname_X\right).


Radon measures

The inclusion map \operatorname : C_c^\infty(U) \to C_c^0(U) 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 ...
^\operatorname : (C_c^0(U))'_b \to \mathcal'(U) = (C_c^\infty(U))'_b is also a continuous injection. Note that the continuous dual space (C_c^0(U))'_b 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 T \in (C_c^0(U))'_b and integral with respect to a Radon measure; that is, * if T \in (C_c^0(U))'_b then there exists a Radon measure \mu on such that for all f \in C_c^0(U), T(f) = \int_U f \, d\mu, and * if \mu is a Radon measure on then the linear functional on C_c^0(U) defined by sending f \in C_c^0(U) to \int_U f \, d\mu is continuous. Through the injection ^\operatorname : (C_c^0(U))'_b \to \mathcal'(U), every Radon measure becomes a distribution on . If f 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 \phi \mapsto \int_U f(x) \phi(x) \, dx is a Radon measure; so Radon measures form a large and important space of distributions. The following is the theorem of the structure of distributions of
Radon 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 L^\infty functions on :


Positive Radon measures

A linear function T on a space of functions is called if whenever a function f that belongs to the domain of T is non-negative (that is, f is real-valued and f \geq 0) then T(f) \geq 0. One may show that every positive linear functional on C_c^0(U) is necessarily continuous (that is, necessarily a Radon measure). Lebesgue measure is an example of a positive Radon measure.


Locally integrable functions as distributions

One particularly important class of Radon measures are those that are induced locally integrable functions. The function f : U \to \R is called if it is
Lebesgue integrable 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 L^p functions. The topology on \mathcal(U) is defined in such a fashion that any locally integrable function f yields a continuous linear functional on \mathcal(U) – that is, an element of \mathcal'(U) – denoted here by T_f, whose value on the test function \phi is given by the Lebesgue integral: \langle T_f, \phi \rangle = \int_U f \phi\,dx. Conventionally, one abuses notation by identifying T_f with f, provided no confusion can arise, and thus the pairing between T_f and \phi is often written \langle f, \phi \rangle = \langle T_f, \phi \rangle. If f and g are two locally integrable functions, then the associated distributions T_f and T_g are equal to the same element of \mathcal'(U) if and only if f and g 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 ...
\mu on U defines an element of \mathcal'(U) whose value on the test function \phi is \int\phi \,d\mu. As above, it is conventional to abuse notation and write the pairing between a Radon measure \mu and a test function \phi as \langle \mu, \phi \rangle. Conversely, as shown in a theorem by Schwartz (similar to the
Riesz representation theorem :''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 C_c^\infty(U) 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 \mathcal'(U) with respect to the strong topology on \mathcal'(U). This means that for any T \in \mathcal'(U), there is a sequence of test functions, (\phi_i)_^\infty, that converges to T \in \mathcal'(U) (in its strong dual topology) when considered as a sequence of distributions. Or equivalently, \langle \phi_i, \psi \rangle \to \langle T, \psi \rangle \qquad \text \psi \in \mathcal(U).


Distributions with compact support

The inclusion map \operatorname: C_c^\infty(U) \to C^\infty(U) is a continuous injection whose image is dense in its codomain, so the transpose map ^\operatorname: (C^\infty(U))'_b \to \mathcal'(U) = (C_c^\infty(U))'_b is also a continuous injection. Thus the image of the transpose, denoted by \mathcal'(U), forms a space of distributions. The elements of \mathcal'(U) = (C^\infty(U))'_b can be identified as the space of distributions with compact support. Explicitly, if T is a distribution on then the following are equivalent, * T \in \mathcal'(U). * The support of T is compact. * The restriction of T to C_c^\infty(U), when that space is equipped with the subspace topology inherited from C^\infty(U) (a coarser topology than the canonical LF topology), is continuous. * There is a compact subset of such that for every test function \phi whose support is completely outside of , we have T(\phi) = 0. Compactly supported distributions define continuous linear functionals on the space C^\infty(U); recall that the topology on C^\infty(U) is defined such that a sequence of test functions \phi_k converges to 0 if and only if all derivatives of \phi_k 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 C_c^\infty(U) to C^\infty(U).


Distributions of finite order

Let k \in \N. The inclusion map \operatorname: C_c^\infty(U) \to C_c^k(U) 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 ...
^\operatorname: (C_c^k(U))'_b \to \mathcal'(U) = (C_c^\infty(U))'_b is also a continuous injection. Consequently, the image of ^\operatorname, denoted by \mathcal'^(U), forms a space of distributions. The elements of \mathcal'^k(U) are The distributions of order \,\leq 0, which are also called are exactly the distributions that are Radon measures (described above). For 0 \neq k \in \N, a is a distribution of order \,\leq k that is not a distribution of order \,\leq k - 1. A distribution is said to be of if there is some integer k such that it is a distribution of order \,\leq k, and the set of distributions of finite order is denoted by \mathcal'^(U). Note that if k \leq l then \mathcal'^k(U) \subseteq \mathcal'^l(U) so that \mathcal'^(U) := \bigcup_^\infty \mathcal'^n(U) is a vector subspace of \mathcal'(U), and furthermore, if and only if \mathcal'^(U) = \mathcal'(U).


Structure of distributions of finite order

Every distribution with compact support in is a distribution of finite order. Indeed, every distribution in is a distribution of finite order, in the following sense: If is an open and relatively compact subset of and if \rho_ is the restriction mapping from to , then the image of \mathcal'(U) under \rho_ is contained in \mathcal'^(V). The following is the theorem of the structure of distributions of finite order, which shows that every distribution of finite order can be written as a sum of derivatives of
Radon 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 U := (0, \infty) and for every test function f, let S f := \sum_^\infty (\partial^m f)\left(\frac\right). Then S is a distribution of infinite order on . Moreover, S can not be extended to a distribution on \R; that is, there exists no distribution T on \R such that the restriction of T to is equal to S.


Tempered distributions and Fourier transform

Defined below are the , which form a subspace of \mathcal'(\R^n), the space of distributions on \R^n. This is a proper subspace: while every tempered distribution is a distribution and an element of \mathcal'(\R^n), 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 \mathcal'(\R^n).


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 ...
\mathcal(\R^n) 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 \phi:\R^n\to\R is in the Schwartz space provided that any derivative of \phi, multiplied with any power of , x, , converges to 0 as , x, \to \infty. These functions form a complete TVS with a suitably defined family of seminorms. More precisely, for any multi-indices \alpha and \beta define: p_ (\phi) ~=~ \sup_ \left, x^\alpha \partial^\beta \phi(x) \. Then \phi is in the Schwartz space if all the values satisfy: p_ (\phi) < \infty. The family of seminorms p_ defines a locally convex topology on the Schwartz space. For n = 1, the seminorms are, in fact, norms on the Schwartz space. One can also use the following family of seminorms to define the topology: , f, _ = \sup_ \left(\sup_ \left\\right), \qquad k,m \in \N. Otherwise, one can define a norm on \mathcal(\R^n) via \, \phi\, _ ~=~ \max_ \sup_ \left, x^\alpha \partial^\beta \phi(x)\, \qquad k \ge 1. 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 \partial^\alpha into multiplication by x^\alpha and vice versa, this symmetry implies that the Fourier transform of a Schwartz function is also a Schwartz function. A sequence \ in \mathcal(\R^n) converges to 0 in \mathcal(\R^n) if and only if the functions (1 + , x, )^k (\partial^p f_i)(x) converge to 0 uniformly in the whole of \R^n, which implies that such a sequence must converge to zero in C^\infty(\R^n). \mathcal(\R^n) is dense in \mathcal(\R^n). The subset of all analytic Schwartz functions is dense in \mathcal(\R^n) as well. The Schwartz space is nuclear and the tensor product of two maps induces a canonical surjective TVS-isomorphisms \mathcal(\R^m) \ \widehat\ \mathcal(\R^n) \to \mathcal(\R^), where \widehat represents the completion of the injective tensor product (which in this case is identical to the completion of the projective tensor product).


Tempered distributions

The inclusion map \operatorname: \mathcal(\R^n) \to \mathcal(\R^n) 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 ...
^\operatorname: (\mathcal(\R^n))'_b \to \mathcal'(\R^n) is also a continuous injection. Thus, the image of the transpose map, denoted by \mathcal'(\R^n), forms a space of distributions. The space \mathcal'(\R^n) 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 T is a tempered distribution if and only if \left(\text \alpha, \beta \in \N^n: \lim_ p_ (\phi_m) = 0 \right) \Longrightarrow \lim_ T(\phi_m)=0. 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 L^p(\R^n) for p \geq 1 are tempered distributions. The can also be characterized as , meaning that each derivative of T 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 \phi decays faster than every inverse power of , x, . An example of a rapidly falling function is , x, ^n\exp (-\lambda , x, ^\beta) for any positive n, \lambda, \beta.


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, ...
F : \mathcal(\R^n) \to \mathcal(\R^n) 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 ...
^F : \mathcal'(\R^n) \to \mathcal'(\R^n), which (abusing notation) will again be denoted by F. So the Fourier transform of the tempered distribution T is defined by (FT)(\psi) = T(F \psi) for every Schwartz function \psi. FT is thus again a tempered distribution. The Fourier transform is a TVS isomorphism from the space of tempered distributions onto itself. This operation is compatible with differentiation in the sense that F \dfrac = ixFT and also with convolution: if T is a tempered distribution and \psi is a smooth function on \R^n, \psi T is again a tempered distribution and F(\psi T) = F \psi * FT is the convolution of FT and F \psi. In particular, the Fourier transform of the constant function equal to 1 is the \delta distribution.


Expressing tempered distributions as sums of derivatives

If T \in \mathcal'(\R^n) is a tempered distribution, then there exists a constant C > 0, and positive integers M and N such that for all Schwartz functions \phi \in \mathcal(\R^n) \langle T, \phi \rangle \le C\sum\nolimits_\sup_ \left, x^\alpha \partial^\beta \phi(x) \=C\sum\nolimits_ p_(\phi). 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 F and a multi-index \alpha such that T = \partial^\alpha F.


Restriction of distributions to compact sets

If T \in \mathcal'(\R^n), then for any compact set K \subseteq \R^n, there exists a continuous function Fcompactly supported in \R^n (possibly on a larger set than itself) and a multi-index \alpha such that T = \partial^\alpha F on C_c^\infty(K).


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's algebraic analysis, 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

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Notes


References


Bibliography

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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