Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. Distributions are widely used in the theory of partial differential equations, where it may be easier to establish the existence of distributional solutions (weak solutions) than classical solutions, or where appropriate classical solutions may not exist. Distributions are also important in physics and engineering where many problems naturally lead to differential equations whose solutions or initial conditions are singular, such as the Dirac delta function. A function f is normally thought of as on the in the function domain by "sending" a point x in the domain to the point f(x). Instead of acting on points, distribution theory reinterprets functions such as f as acting on in a certain way. In**** applications**** to**** physics and engineering,**** **** are usually infinitely**** differentiable**** complex-valued**** (or real****-valued****)**** functions with**** compact support that**** are defined on**** some**** given non-empty open**** subset****.**** (Bump functions are examples of test functions.) The set of all such test functions forms a vector space that is denoted by or Most commonly encountered functions, including all continuous maps if using U := \R, can be canonically reinterpreted as acting via "integration against a test function." Explicitly, this means that such a function f "acts on" a test function by "sending" it to the number which is often denoted by D_f(\psi). This new action of f defines a scalar-valued map whose domain is the space of test functions This functional D_f turns out to have the two defining properties of what is known as a : it is linear, and it is also continuous when is given a certain topology called. The action (the integration ) of this distribution D_f on a test function \psi 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 there exist many distributions that cannot be defined by integration against any function. Examples of the latter include the Dirac delta function and distributions defined to act by integration of test functions against certain measures \mu on U. Nonetheless, it is still always possible to reduce any arbitrary distribution down to a simpler of related distributions that do arise via such actions of integration. More**** generally,**** a is**** by**** definition**** a linear**** functional**** on**** **** that**** is**** continuous**** when**** **** is**** given a topology**** called**** the . This leads to space of (all) distributions on U, usually denoted by (note the prime), which by definition is the space of all distributions on U (that is, it is the continuous dual space of ); it is these distributions that are the main focus of this article. Definitions of the appropriate topologies on spaces of test functions and distributions are given in the article on spaces of test functions and distributions. This article is primarily concerned with the definition of distributions, together with their properties and some important examples.

History

The practical use of distributions can be traced back to the use of Green's functions in the 1830s to solve ordinary differential equations, but was not formalized until much later. According to, generalized functions originated in the work of on second-order hyperbolic partial differential equations, and the ideas were developed in somewhat extended form by Laurent Schwartz in the late 1940s. According to his autobiography, Schwartz introduced the term "distribution" by analogy with a distribution of electrical charge, possibly including not only point charges but also dipoles and so on. comments that although the ideas in the transformative book by were not entirely new, it was Schwartz's broad attack and conviction that distributions would be useful almost everywhere in analysis that made the difference. A detailed history of the theory of distributions was given by.

Notation

The following notation will be used throughout this article:

<!-- # Basic idea Distributions are a class of [linear functionals](https://bliptext.com/articles/linear-functional) that map a set of (conventional and [well-behaved](https://bliptext.com/articles/well-behaved) functions) into the set of real or complex numbers. In the simplest case, the set of test functions considered is which is the set of functions having two properties: A distribution T is a continuous linear mapping Instead of writing T(\varphi), it is conventional to write for the value of T acting on a test function \varphi. A simple example of a distribution is the [Dirac delta](https://bliptext.com/articles/dirac-delta), \delta, defined by meaning that \delta evaluates a test function at 0 . Its physical interpretation is as the density of a point source. As described next, there are straightforward mappings from both [locally integrable functions](https://bliptext.com/articles/locally-integrable-function) and [Radon measures](https://bliptext.com/articles/radon-measure) to corresponding distributions, but not all distributions can be formed in this manner. # [A typical test function, the bump function \Psi(x). It is smooth (infinitely differentiable) and has compact support (is zero outside an interval, in this case the interval [-1, 1]). | upload.wikimedia.org/wikipedia/commons/3/37/Mollifier///Illustration.svg] ## Functions and measures as distributions Suppose is a [locally integrable function](https://bliptext.com/articles/locally-integrable-function). Then a corresponding distribution, denoted by T_f, may be defined by This integral is a [real number](https://bliptext.com/articles/real-number) which depends [linearly](https://bliptext.com/articles/linear-operator) and [continuously](https://bliptext.com/articles/continuous-function) on \varphi. Conversely, the values of the distribution T_f on test functions in determine the pointwise almost everywhere values of the function f on \R. In a conventional [abuse of notation](https://bliptext.com/articles/abuse-of-notation), f is often used to represent both the original function f and the corresponding distribution T_f.. This example suggests the definition of a distribution as a linear and, in an appropriate sense, continuous [functional](https://bliptext.com/articles/functional-mathematics) on the space of test functions Similarly, if \mu is a [Radon measure](https://bliptext.com/articles/radon-measure) on \R, then a corresponding distribution, denoted by R_{\mu}, may be defined by This integral also depends linearly and continuously on \varphi, so that R_{\mu} is a distribution. If \mu is [absolutely continuous](https://bliptext.com/articles/absolute-continuity) with respect to Lebesgue measure with density f and then this definition for R_{\mu} is the same as the previous one for T_f, but if \mu is not [absolutely continuous](https://bliptext.com/articles/absolute-continuity), then R_{\mu} is a distribution that is not associated with a function. For example, if P is the point-mass measure on \R that assigns measure one to the singleton set \{0\} and measure zero to sets that do not contain zero, then so that is the Dirac delta. ## Adding and multiplying distributions Distributions may be multiplied by real numbers and added together, so they form a real [vector space](https://bliptext.com/articles/vector-space). A distribution may also be multiplied by a rapidly decreasing infinitely differentiable function to get another distribution, but [it is not possible to define a product of general distributions](https://bliptext.com/articles/distribution-mathematics) that extends the usual pointwise product of functions and has the same algebraic properties. This result was shown by, and is usually referred to as the. ## Derivatives of distributions It is desirable to choose a definition for the derivative of a distribution which, at least for distributions derived from smooth functions, has the property that (i.e. where f' is the usual derivative of f and (T_f)' denotes the derivative of the distribution T_f, which we wish to define). If \phi is a test function, we can use [integration by parts](https://bliptext.com/articles/integration-by-parts) to see that where the last equality follows from the fact that \phi has compact support, so is zero outside of a bounded set. This suggests that if T is a, we should define its derivative T' by It turns out that this is the proper definition; it extends the ordinary definition of derivative, every distribution becomes infinitely differentiable and the usual properties of derivatives hold. **Example**: Recall that the [Dirac delta](https://bliptext.com/articles/dirac-delta) (i.e. the so-called [Dirac delta](https://bliptext.com/articles/dirac-delta) "function") is the distribution defined by the equation It is the derivative of the distribution corresponding to the [Heaviside step function](https://bliptext.com/articles/heaviside-step-function) H: For any test function \phi so H = \delta. Note, because \phi has compact support by our definition of a test function. Similarly, the derivative of the Dirac delta is the distribution defined by the equation This latter distribution is an example of a distribution that is not derived from a function or a measure. Its physical interpretation is the density of a dipole source. Just as the Dirac impulse can be realized in the weak limit as a sequence of various kinds of constant norm bump functions of ever increasing amplitude and narrowing support, its derivative can by definition be realized as the weak limit of the negative derivatives of said functions, which are now antisymmetric about the eventual distribution's point of singular support.-->

Definitions of test functions and distributions

In this section, some basic notions and definitions needed to define real-valued distributions on U 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. Bump.png where and This function is a test function on \R^2 and is an element of The support of this function is the closed unit disk in \R^2. It is non-zero on the open unit disk and it is equal to 0 everywhere outside of it.]] For all and any compact subsets K and L of U, we have: Distributions on**** U are continuous**** linear**** functionals on**** when**** this**** vector**** space is**** endowed with**** a particular**** topology**** called**** the . The following proposition states two necessary and sufficient conditions for the continuity of a linear function on that are often straightforward to verify. Proposition:**** A linear**** functional**** T on**** is**** continuous****,**** and therefore a , if**** and only**** if**** any of**** the following equivalent**** conditions**** is**** satisfied:****

Topology on Ck(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-valued 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 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 that is normable. Every element of is a continuous seminorm on C^k(U). Under this topology, a net in C^k(U) converges to if and only if for every multi-index p with |p|< k + 1 and every compact K, the net of partial derivatives converges uniformly to on K. For any any (von Neumann) bounded subset of C^{k+1}(U) is a relatively compact 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 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 induced on it by C^i(U).

Topology on Ck(K)

As before, fix Recall that if K is any compact subset of U then If k is finite then C^k(K) is a Banach space with a topology that can be defined by the norm And when k = 2, then C^k(K) is even a Hilbert space.

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

Suppose U is an open subset of \R^n and is a compact subset. By definition, elements of C^k(K) are functions with domain U (in symbols, ), 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 ) will change the set C^k(K) from C^k(K;U) to C^k(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, 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 also contains K, so that each of C^k(K; V) and C^k(K; W) is canonically identified with and now by transitivity, C^k(K; V) is thus identified with C^k(K; W). So assume are open subsets of \R^n containing K. Given its is the function defined by: This trivial extension belongs to C^k(V) (because has compact support) and it will be denoted by I(f) (that is, I(f) := F). The assignment thus induces a map that sends a function in C_c^k(U) to its trivial extension on V. This map is a linear injection and for every compact subset (where K is also a compact subset of V since ), If I is restricted to C^k(K; U) then the following induced linear map is a homeomorphism (linear homeomorphisms are called ): and thus the next map is a topological embedding: Using the injection the vector space C_c^k(U) is canonically identified with its image in Because 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 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 Comparison of topologies than the subspace topology that C^\infty(U) induces on However, the canonical LF-topology does make into a complete reflexive nuclear Montel bornological barrelled Mackey space; the same is true of its strong dual space (that is, the space of all distributions with its usual topology). The canonical LF-topology can be defined in various ways.

Distributions

As discussed earlier, continuous linear functionals on a are known as distributions on U. Other equivalent definitions are described below. There is a canonical duality pairing between a distribution T on U and a test function which is denoted using angle brackets by One interprets this notation as the distribution T acting on the test function f to give a scalar, or symmetrically as the test function f acting on the distribution T.

Characterizations of distributions

Proposition. If T is a linear functional on then the following are equivalent:

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

The set of all distributions on U is the continuous dual space of which when endowed with the strong dual topology is denoted by Importantly, unless indicated otherwise, the topology on is the strong dual topology; if the topology is instead the weak-* topology then this will be indicated. Neither topology is metrizable although unlike the weak-* topology, the strong dual topology makes into a complete nuclear space, to name just a few of its desirable properties. Neither nor its strong dual is a sequential space and so neither of their topologies can be fully described by sequences (in other words, defining only what sequences converge in these spaces is enough to fully/correctly define their topologies). However, a in converges in the strong dual topology if and only if it converges in the weak-* topology (this leads many authors to use pointwise convergence to the convergence of a sequence of distributions; this is fine for sequences but this is guaranteed to extend to the convergence of nets of distributions because a net may converge pointwise but fail to converge in the strong dual topology). More information about the topology that is endowed with can be found in the article on spaces of test functions and distributions and the articles on polar topologies and dual systems. A map from into another locally convex topological vector space (such as any normed space) is continuous if and only if it is sequentially continuous at the origin. However, this is no longer guaranteed if the map is not linear or for maps valued in more general topological spaces (for example, that are not also locally convex topological vector spaces). The same is true of maps from (more generally, this is true of maps from any locally convex bornological space).

Localization of distributions

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

Extensions and restrictions to an open subset

Let be open subsets of \R^n. Every function can be from its domain V to a function on U by setting it equal to 0 on the complement This extension is a smooth compactly supported function called the and it will be denoted by E_{VU} (f). This assignment defines the operator which is a continuous injective linear map. It is used to canonically identify as a vector subspace of (although as a topological subspace). Its transpose (explained here) is**** called**** the **** and as**** the name**** suggests****,**** the image of**** a distribution**** **** under this**** map is**** a distribution**** on**** V called**** the restriction of**** T to**** V.**** The defining**** condition of**** the restriction **** is****:**** If V \neq U then the (continuous injective linear) trivial extension map is a topological embedding (in other words, if this linear injection was used to identify as a subset of then 's topology would strictly finer than the subspace topology that induces on it; importantly, it would be a topological subspace since that requires equality of topologies) and its range is also dense in its codomain Consequently if V \neq U then the restriction mapping is neither injective nor surjective. A distribution is said to be **' if it belongs to the range of the transpose of E_{VU} and it is called **' if it is extendable to \R^n. Unless U = V, the restriction to V is neither injective nor surjective. Lack of surjectivity follows since distributions can blow up towards the boundary of V. For instance, if U = \R and V = (0, 2), then the distribution is in but admits no extension to

Gluing and distributions that vanish in a set

Let V be**** an**** open**** subset**** of**** U.**** is**** said**** to**** **** if**** for all **** such**** that**** **** we**** have**** Tf**** = 0.**** T vanishes in V if and only if the restriction of T to V is equal to 0, or equivalently, if and only if T lies in the kernel of the restriction map \rho_{VU}.

Support of a distribution

This last corollary implies that for every distribution T on U, there exists a unique largest subset V of U such that T vanishes in V (and does not vanish in any open subset of U that is not contained in V); the complement in U of this unique largest open subset is called. Thus If f is a locally integrable function on U and if D_f is its associated distribution, then the support of D_f is the smallest closed subset of U in the complement of which f is almost everywhere equal to 0. If f is continuous, then the support of D_f is equal to the closure of the set of points in U at which f does not vanish. The support of the distribution associated with the Dirac measure at a point x_0 is the set {x_0}. If the support of a test function f does not intersect the support of a distribution T then Tf = 0. A distribution T is 0 if and only if its support is empty. If is identically 1 on some open set containing the support of a distribution T then f T = T. If the support of a distribution T is compact then it has finite order and there is a constant C and a non-negative integer N such that: If T has compact support, then it has a unique extension to a continuous linear functional \widehat{T} on C^\infty(U); this function can be defined by where is any function that is identically 1 on an open set containing the support of T. If and then and Thus, distributions with support in a given subset form a vector subspace of Furthermore, if P is a differential operator in U, then for all distributions T on U and all we have and

Distributions with compact support

Support in a point set and Dirac measures

For any x \in U, let denote the distribution induced by the Dirac measure at x. For any x_0 \in U and distribution the support of T is contained in {x_0} if and only if T is a finite linear combination of derivatives of the Dirac measure at x_0. If in addition the order of T is \leq k then there exist constants \alpha_p such that: Said differently, if T has support at a single point {P}, then T is in fact a finite linear combination of distributional derivatives of the \delta function at P. That is, there exists an integer m and complex constants a_\alpha such that 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 (or the Schwartz space for tempered distributions). It is not immediately clear from the definition how exotic a distribution might be. To answer this question, it is instructive to see distributions built up from a smaller space, namely the space of continuous functions. Roughly, any distribution is locally a (multiple) derivative of a continuous function. A precise version of this result, given below, holds for distributions of compact support, tempered distributions, and general distributions. Generally speaking, no proper subset of the space of distributions contains all continuous functions and is closed under differentiation. This says that distributions are not particularly exotic objects; they are only as complicated as necessary.

Distributions as sheaves

Decomposition of distributions as sums of derivatives of continuous functions

By combining the above results, one may express any distribution on U 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 U. In other words, for arbitrary we can write: where are finite sets of multi-indices and the functions f_{ip} are continuous. Note that the infinite sum above is well-defined as a distribution. The value of T for a given 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 is a linear map that is continuous with respect to the weak topology, then it is not always possible to extend A to a map by classic extension theorems of topology or linear functional analysis. The “distributional” extension of the above linear continuous operator A is possible if and only if A admits a Schwartz adjoint, that is another linear continuous operator B of the same type such that , for every pair of test functions. In that condition, B is unique and the extension A’ is the transpose of the Schwartz adjoint B.

Preliminaries: Transpose of a linear operator

Operations on distributions and spaces of distributions are often defined using the transpose of a linear operator. This is because the transpose allows for a unified presentation of the many definitions in the theory of distributions and also because its properties are well-known in functional analysis. For instance, the well-known Hermitian adjoint of a linear operator between Hilbert spaces is just the operator's transpose (but with the Riesz representation theorem used to identify each Hilbert space with its continuous dual space). In general, the transpose of a continuous linear map A : X \to Y is the linear map or equivalently, it is the unique map satisfying 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 is also continuous when both duals are endowed with their respective strong dual topologies; it is also continuous when both duals are endowed with their respective weak* topologies (see the articles polar topology and dual system for more details). In the context of distributions, the characterization of the transpose can be refined slightly. Let be a continuous linear map. Then by definition, the transpose of A is the unique linear operator that satisfies: Since is dense in (here, actually refers to the set of distributions ) it is sufficient that the defining equality hold for all distributions of the form T = D_\psi where Explicitly, this means that a continuous linear map is equal to {}^{t}A if and only if the condition below holds: where the right-hand side equals

Differential operators

Differentiation of distributions

Let be the partial derivative operator To extend A we compute its transpose: Therefore Thus, the partial derivative of T with respect to the coordinate x_k is defined by the formula With this definition, every distribution is infinitely differentiable, and the derivative in the direction x_k is a linear operator on More generally, if \alpha is an arbitrary multi-index, then the partial derivative of the distribution is defined by Differentiation of distributions is a continuous operator on this is an important and desirable property that is not shared by most other notions of differentiation. If T is a distribution in \R then 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 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: where the vertical maps are given by assigning its canonical distribution which is defined by: With this notation, the diagram commuting is equivalent to: To find D_P, the transpose of the continuous induced map defined by is considered in the lemma below. This leads to the following definition of the differential operator on U called which will be denoted by P_* to avoid confusion with the transpose map, that is defined by As discussed above, for any the transpose may be calculated by: For the last line we used integration by parts combined with the fact that \phi and therefore all the functions have compact support. Continuing the calculation above, for all The Lemma combined with the fact that the formal transpose of the formal transpose is the original differential operator, that is, P_{**}= P, enables us to arrive at the correct definition: the formal transpose induces the (continuous) canonical linear operator defined by We claim that the transpose of this map, can be taken as D_P. To see this, for every compute its action on a distribution of the form D_f with : We**** call**** the continuous**** linear**** operator**** the . Its action on an arbitrary distribution S is defined via: If converges to then for every multi-index converges to

Multiplication of distributions by smooth functions

A differential operator of order 0 is just multiplication by a smooth function. And conversely, if 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 maps a distribution T to a distribution denoted by We have thus defined the multiplication of a distribution by a smooth function. We now give an alternative presentation of the multiplication of a distribution T on U by a smooth function The product mT is defined by This definition coincides with the transpose definition since if is the operator of multiplication by the function m (that is, ), then so that {}^tM = M. Under multiplication by smooth functions, is a module over the ring With this definition of multiplication by a smooth function, the ordinary product rule of calculus remains valid. However, some unusual identities also arise. For example, if \delta is the Dirac delta distribution on \R, then and if \delta^' is the derivative of the delta distribution, then The bilinear multiplication map given by is continuous; it is however, hypocontinuous. Example. The product of any distribution T with the function that is identically 1 on U is equal to T. Example. Suppose is a sequence of test functions on U that converges to the constant function For any distribution T on U, the sequence converges to If converges to and converges to then converges to

Problem of multiplying distributions

It is easy to define the product of a distribution with a smooth function, or more generally the product of two distributions whose singular supports are disjoint. With more effort, it is possible to define a well-behaved product of several distributions provided their wave front sets at each point are compatible. A limitation of the theory of distributions (and hyperfunctions) is that there is no associative product of two distributions extending the product of a distribution by a smooth function, as has been proved by Laurent Schwartz in the 1950s. For example, if is the distribution obtained by the Cauchy principal value If \delta is the Dirac delta distribution then but, so the product of a distribution by a smooth function (which is always well-defined) cannot be extended to an associative product on the space of distributions. Thus, nonlinear problems cannot be posed in general and thus are not solved within distribution theory alone. In the context of quantum field theory, however, solutions can be found. In more than two spacetime dimensions the problem is related to the regularization of divergences. Here Henri Epstein and Vladimir Glaser developed the mathematically rigorous (but extremely technical). This does not solve the problem in other situations. Many other interesting theories are non-linear, like for example the Navier–Stokes equations of fluid dynamics. Several not entirely satisfactory theories of algebras of generalized functions have been developed, among which Colombeau's (simplified) algebra is maybe the most popular in use today. Inspired by Lyons' rough path theory, 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 If F is a submersion then it is possible to define This is, and is also called , sometimes written The pullback is often denoted F^, although this notation should not be confused with the use of '' to denote the adjoint of a linear mapping. The condition that F be a submersion is equivalent to the requirement that the Jacobian derivative 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 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 Existence, however, requires using the change of variables formula, the inverse function theorem (locally), and a partition of unity argument. In the special case when F is a diffeomorphism from an open subset V of \R^n onto an open subset U of \R^n change of variables under the integral gives: In this particular case, then, F^{#} is defined by the transpose formula:

Convolution

Under some circumstances, it is possible to define the convolution of a function with a distribution, or even the convolution of two distributions. Recall that if f and g are functions on \R^n then we denote by f\ast g defined at x \in \R^n to be the integral provided that the integral exists. If are such that then for any functions and we have and If f and g are continuous functions on \R^n, at least one of which has compact support, then and if then the value of f\ast g on A do depend on the values of f outside of the Minkowski sum Importantly, if has compact support then for any the convolution map is continuous when considered as the map or as the map

Translation and symmetry

Given a \in \R^n, the translation operator \tau_a sends to defined by This can be extended by the transpose to distributions in the following way: given a distribution T, is the distribution defined by Given define**** the function**** **** by**** **** Given a distribution**** T,**** let **** be**** the distribution**** defined by**** **** The operator**** **** is**** called**** .

Convolution of a test function with a distribution

Convolution with defines a linear map: which is continuous with respect to the canonical LF space topology on Convolution of f with a distribution can be defined by taking the transpose of C_f relative to the duality pairing of with the space of distributions. If then by Fubini's theorem Extending by continuity, the convolution of f with a distribution T is defined by 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 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: where denotes the convex hull and denotes the support.

Convolution of a smooth function with a distribution

Let and and assume that at least one of 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****:**** satisfying for all p \in \N^n: Let M be the map. If T is a distribution, then M is continuous as a map. If T also has compact support, then M is also continuous as the map and continuous as the map If is a continuous linear map such that for all \alpha and all then there exists a distribution such that for all Example. Let H be the Heaviside function on \R. For any Let \delta be the Dirac measure at 0 and let \delta' be its derivative as a distribution. Then and Importantly, the associative law fails to hold:

Convolution of distributions

It is also possible to define the convolution of two distributions 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 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 are continuous. The transposes of these maps: are consequently continuous and it can also be shown that This common value is called and it is a distribution that is denoted by S \ast T or T \ast S. It satisfies If S and T are two distributions, at least one of which has compact support, then for any a \in \R^n, If T is a distribution in \R^n and if \delta is a Dirac measure then ; thus \delta is the identity element of the convolution operation. Moreover, if f is a function then where now the associativity of convolution implies that for all functions f and g. Suppose that it is T that has compact support. For consider the function It can be readily shown that this defines a smooth function of x, which moreover has compact support. The convolution of S and T is defined by This generalizes the classical notion of convolution of functions and is compatible with differentiation in the following sense: for every multi-index \alpha. 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 defined by where denotes the space of distributions with compact support. However, the convolution map as a function is continuous although it is separately continuous. The convolution maps and given by both to be continuous. Each of these non-continuous maps is, however, separately continuous and hypocontinuous.

Convolution versus multiplication

In general, regularity is required for multiplication products, and locality is required for convolution products. It is expressed in the following extension of the Convolution Theorem which guarantees the existence of both convolution and multiplication products. Let be a rapidly decreasing tempered distribution or, equivalently, be an ordinary (slowly growing, smooth) function within the space of tempered distributions and let F be the normalized (unitary, ordinary frequency) Fourier transform. Then, according to , hold within the space of tempered distributions. In particular, these equations become the Poisson Summation Formula if is the Dirac Comb. The space of all rapidly decreasing tempered distributions is also called the space of and the space of all ordinary functions within the space of tempered distributions is also called the space of More generally, and A particular case is the Paley-Wiener-Schwartz Theorem which states that and This is because and In other words, compactly supported tempered distributions belong to the space of and Paley-Wiener functions better known as bandlimited functions, belong to the space of For example, let be the Dirac comb and be the Dirac delta;then 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 be the rectangular function; then is the sinc function and both equations yield the Classical Sampling Theorem for suitable functions. More generally, if g is the Dirac comb and is a smooth window function (Schwartz function), for example, the Gaussian, then is another smooth window function (Schwartz function). They are known as mollifiers, especially in partial differential equations theory, or as regularizers in physics because they allow turning generalized functions into regular functions.

Tensor products of distributions

Let and be open sets. Assume all vector spaces to be over the field \mathbb{F}, where or \Complex. For define for every u \in U and every v \in V the following functions: Given and define the following functions: where and These definitions associate every and with the (respective) continuous linear map: Moreover, if either S (resp. T) has compact support then it also induces a continuous linear map of (resp. ). denoted by S \otimes T or is the distribution in U \times V defined by:

Spaces of distributions

For all and all every one of the following canonical injections is continuous and has an image (also called the range) that is a dense subset of its codomain: where the topologies on L_c^q(U) 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 ) or L^p_c(U) (for ) or L^p(U) (for ). Because the canonical injection is a continuous injection whose image is dense in the codomain, this map's transpose is a continuous injection. This injective transpose map thus allows the continuous dual space X' of X to be identified with a certain vector subspace of the space of all distributions (specifically, it is identified with the image of this transpose map). This transpose map is continuous but it is necessarily a topological embedding. A linear**** subspace**** of**** carrying**** a locally convex**** topology**** that**** is**** finer than**** the subspace**** topology**** induced on**** it**** by**** **** is**** called**** . Almost all of the spaces of distributions mentioned in this article arise in this way (for example, tempered distribution, restrictions, distributions of order \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 be transferred directly to elements of the space

Radon measures

The inclusion map is a continuous injection whose image is dense in its codomain, so the transpose is also a continuous injection. Note that the continuous dual space can be identified as the space of Radon measures, where there is a one-to-one correspondence between the continuous linear functionals and integral with respect to a Radon measure; that is, Through the injection every Radon measure becomes a distribution on U. If f is a locally integrable function on U then the distribution is a Radon measure; so Radon measures form a large and important space of distributions. The following is the theorem of the structure of distributions of Radon measures, which shows that every Radon measure can be written as a sum of derivatives of locally L^\infty functions on U:

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**** 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**** is**** called**** **** if**** it**** is**** Lebesgue**** integrable**** over**** every compact subset**** K of**** U.**** This**** is**** a large class of**** functions that**** includes**** all continuous**** functions and all Lp**** space L^p functions.**** The topology on is defined in such a fashion that any locally integrable function f yields a continuous linear functional on – that is, an element of – denoted here by T_f, whose value on the test function \phi is given by the Lebesgue integral: 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 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 if and only if f and g are equal almost everywhere (see, for instance, ). Similarly, every Radon measure \mu on U defines an element of whose value on the test function \phi is As above, it is conventional to abuse notation and write the pairing between a Radon measure \mu and a test function \phi as Conversely, as shown in a theorem by Schwartz (similar to the Riesz representation theorem), every distribution which is non-negative on non-negative functions is of this form for some (positive) Radon measure.

Test functions as distributions

The test functions are themselves locally integrable, and so define distributions. The space of test functions is sequentially dense in with respect to the strong topology on This means that for any there is a sequence of test functions, that converges to (in its strong dual topology) when considered as a sequence of distributions. Or equivalently,

Distributions with compact support

The inclusion map is a continuous injection whose image is dense in its codomain, so the transpose map is also a continuous injection. Thus the image of the transpose, denoted by forms a space of distributions. The elements of can be identified as the space of distributions with compact support. Explicitly, if T is a distribution on U then the following are equivalent, 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 U. Conversely, it can be shown that every continuous linear functional on this space defines a distribution of compact support. Thus compactly supported distributions can be identified with those distributions that can be extended from to

Distributions of finite order

Let k \in \N. The inclusion map is a continuous injection whose image is dense in its codomain, so the transpose is also a continuous injection. Consequently, the image of denoted by forms a space of distributions. The elements of are ' The distributions of order ,\leq 0, which are also called ' are exactly the distributions that are Radon measures (described above). For a **** is a distribution**** of**** order *,*leq**** k that**** is**** not a distribution**** of**** order.**** 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**** Note**** that**** if**** k *leq** l the**n*** **** so**** that**** **** is**** a vector**** subspace**** of****,**** and furthermore,**** if**** and only**** if****

Structure of distributions of finite order

Every distribution with compact support in U is a distribution of finite order. Indeed, every distribution in U is a distribution of finite order, in the following sense: If V is an open and relatively compact subset of U and if \rho_{VU} is the restriction mapping from U to V, then the image of under \rho_{VU} is contained in The following is the theorem of the structure of distributions of finite order, which shows that every distribution of finite order can be written as a sum of derivatives of Radon measures: Example. (Distributions of infinite order) Let and for every test function f, let Then S is a distribution of infinite order on U. 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 U is equal to S.

Tempered distributions and Fourier transform

Defined below are the , which form**** a subspace**** of**** the space of**** distributions on**** *R*^n.**** This is a proper subspace: while every tempered distribution is a distribution and an element of the converse is not true. Tempered distributions are useful if one studies the Fourier transform since all tempered distributions have a Fourier transform, which is not true for an arbitrary distribution in

Schwartz space

The Schwartz space is the space of all smooth functions that are rapidly decreasing at infinity along with all partial derivatives. Thus is in the Schwartz space provided that any derivative of \phi, multiplied with any power of |x|, converges to 0 as These functions form a complete TVS with a suitably defined family of seminorms. More precisely, for any multi-indices \alpha and \beta define Then \phi is in the Schwartz space if all the values satisfy The family of seminorms 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: Otherwise, one can define a norm on via The Schwartz space is a Fréchet space (that is, a complete metrizable locally convex space). Because the Fourier transform changes into multiplication by x^\alpha and vice versa, this symmetry implies that the Fourier transform of a Schwartz function is also a Schwartz function. A sequence {f_i} in converges to 0 in if and only if the functions converge to 0 uniformly in the whole of \R^n, which implies that such a sequence must converge to zero in is dense in The subset of all analytic Schwartz functions is dense in as well. The Schwartz space is nuclear, and the tensor product of two maps induces a canonical surjective TVS-isomorphisms where represents the completion of the injective tensor product (which in this case is identical to the completion of the projective tensor product).

Tempered distributions

The inclusion map is a continuous injection whose image is dense in its codomain, so the transpose is also a continuous injection. Thus, the image of the transpose map, denoted by forms a space of distributions. The space is called the space of. It is the continuous dual space of the Schwartz space. Equivalently, a distribution T is a tempered distribution if and only if The derivative of a tempered distribution is again a tempered distribution. Tempered distributions generalize the bounded (or slow-growing) locally integrable functions; all distributions with compact support and all square-integrable 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. 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 for any positive

Fourier transform

To study the Fourier transform, it is best to consider complex-valued test functions and complex-linear distributions. The ordinary**** continuous**** Fourier transform is**** a TVS-automorphism**** of**** the Schwartz**** space,**** and the **** is**** defined to**** be**** its transpose **** which (abusing**** notation****)**** will**** again be**** denoted by**** F.**** So**** the Fourier transform of**** the tempered**** distribution**** T is**** defined by**** **** 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 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 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 is a tempered distribution, then there exists a constant C > 0, and positive integers M and N such that for all Schwartz functions This estimate, along with some techniques from functional analysis, can be used to show that there is a continuous slowly increasing function F and a multi-index \alpha such that

Restriction of distributions to compact sets

If then for any compact set there exists a continuous function Fcompactly supported in \R^n (possibly on a larger set than K itself) and a multi-index \alpha such that on

Using holomorphic functions as test functions

The success of the theory led to an investigation of the idea of hyperfunction, in which spaces of holomorphic functions are used as test functions. A refined theory has been developed, in particular Mikio Sato's algebraic analysis, using sheaf theory and several complex variables. This extends the range of symbolic methods that can be made into rigorous mathematics, for example, Feynman integrals.