Distribution (mathematics)

Distributions (or generalized functions) are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. Distributions are widely used in the theory of partial differential equations, where it may be easier to establish the existence of distributional solutions than classical solutions, or 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 distributions, such as the Dirac delta function.

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 Kolmogorov & Fomin (1957), generalized functions originated in the work of Sergei Sobolev (1936) 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. Gårding (1997) comments that although the ideas in the transformative book by Schwartz (1951) 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.

Distribution theory reinterprets functions as linear functionals acting on a space of test functions. Standard functions act by integration against a test function, but many other linear functionals do not arise in this way, and these are the "generalized functions". There are different possible choices for the space of test functions, leading to different spaces of distributions. The basic space of test function consists of smooth functions with compact support, leading to standard distributions. Use of the space of smooth, rapidly (faster than any polynomial increases) decreasing test functions (these functions are called Schwartz functions) gives instead the tempered distributions, which are important because they have a well-defined distributional Fourier transform. Every tempered distribution is a distribution in the normal sense, but the converse is not true: in general the larger the space of test functions, the more restrictive the notion of distribution. On the other hand, the use of spaces of analytic test functions leads to Sato's theory of hyperfunctions; this theory has a different character from the previous ones because there are no analytic functions with non-empty compact support.

Basic ideaEdit

 
A typical test function, the bump function Ψ(x). It is smooth (infinitely differentiable) and has compact support (is zero outside an interval, in this case the interval [−1, 1]).

Distributions are a class of linear functionals that map a set of test functions (conventional and well-behaved functions) into the set of real numbers. In the simplest case, the set of test functions considered is D(R), which is the set of functions   : RR having two properties:

  •   is smooth (infinitely differentiable);
  •   has compact support (is identically zero outside some bounded interval).

A distribution T is a linear mapping T : D(R) → R. Instead of writing T( ), it is conventional to write   for the value of T acting on a test function  . A simple example of a distribution is the Dirac delta δ, defined by

 

meaning that δ 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 and Radon measures to corresponding distributions, but not all distributions can be formed in this manner.

Functions and measures as distributionsEdit

Suppose that f : RR is a locally integrable function. Then a corresponding distribution Tf may be defined by

 

This integral is a real number which depends linearly and continuously on  . Conversely, the values of the distribution Tf on test functions in D(R) determine the pointwise almost everywhere values of the function f on R. In a conventional abuse of notation, f is often used to represent both the original function f and the corresponding distribution Tf. This example suggests the definition of a distribution as a linear and, in an appropriate sense, continuous functional on the space of test functions D(R).

Similarly, if μ is a Radon measure on R, then a corresponding distribution Rμ may be defined by

 

This integral also depends linearly and continuously on  , so that Rμ is a distribution. If μ is absolutely continuous with respect to Lebesgue measure with density f and dμ = f dx, then this definition for Rμ is the same as the previous one for Tf, but if μ is not absolutely continuous, then Rμ 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 RP = δ is the Dirac delta.

Adding and multiplying distributionsEdit

Distributions may be multiplied by real numbers and added together, so they form a real 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 that extends the usual pointwise product of functions and has the same algebraic properties. This result was shown by Schwartz (1954), and is usually referred to as the Schwartz Impossibility Theorem.

Derivatives of distributionsEdit

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  . If   is a test function, we can use integration by parts to see that

 

where the last equality follows from the fact that   has compact support, so is zero outside of a bounded set. This suggests that if   is a distribution, we should define its derivative   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 (so-called Dirac delta function) is the distribution defined by the equation

 

It is the derivative of the distribution corresponding to the Heaviside step function H: For any test function  ,

 

so H′ = δ. Note,  (∞) = 0 because   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.

Test functions and distributionsEdit

Throughout, n is a fixed positive integer, U is a fixed non-empty open subset of Euclidean space  , and  . In the following, real-valued distributions on an open subset U will be formally defined. With minor modifications, one can also define complex-valued distributions, and one can replace Rn by any (paracompact) smooth manifold.

For any   let   denote the vector space of all k-times continuously differentiable complex-valued functions on  . Recall that the support of a function f, denoted by  , is the closure in   of the set  . For any compact subset  , let   denote all those functions   such that   (note that   depends on both K and   but we will only indicate K, where in particular, if   then the domain of f is   rather than K). Note that if f is a real-valued function on U, then f is an element of   if and only if it is a   bump function.

Distributions on U are defined to be the continuous linear functionals on the space   of test functions on U when this space is endowed with a particular topology (called the canonical LF topology). The space of test functions on U is also denoted is also D(U) or  . So to define the space of distributions, which is denoted by  , we must first define the topology on the space  , which requires that several other topological vector spaces (TVSs) be defined first.

To define the various special subspaces of   (such as distributions of order   some integer, etc.), we construct the spaces   and   for arbitrary  , which adds almost no additional complexity to the presentation and allows us to easily define additional spaces of distributions later. Once the topology on   is defined, we can then immediately place a topology on the space of distributions.

Throughout, we will let   be any collection of compact subsets of   such that (1)  , and (2) for any compact   there exists some   such that  . The most common choices for   are:

  • the set of all compact subsets of  , or
  • a set   where  , and for all i,   and   is a relatively compact non-empty open subset of   (i.e. "relatively compact" means that the closure of  , in either U or  , is compact).

We make   into a directed set by defining   if and only if  . Note that although the definitions of the subsequently defined topologies explicitly reference  , in reality they do not depend on the choice of  ; that is, if   and   are any two such collections of compact subsets of  , then the topologies defined on   and   by using   in place of   are the same as those defined by using   in place of  .

Topology on the space of Ck mapsEdit

We give   the Fréchet space topology defined by the family of seminorms

 

as i ranges over   and K ranges over  , where   is a multi-index of non-negative integers and   denotes its length. Under this topology, a net of functions   in   converges to a function   if and only if for every multi-index p with   and every  ,   converges to   uniformly on K.[1] Note that   is a Montel space if and only if  . [2]

Topology on the space of Ck maps with support in a compact setEdit

For any compact  ,   is a closed subspace of   and we give   the subspace topology induced by  , thereby making it into a Fréchet space. For all compact   with  , denote the natural inclusion by   where note that this map is a linear embedding of TVSs whose range is closed in its codomain (said differently, the topology on   is identical to the subspace topology it inherits from  , and   is a closed subspace of  ). When   then   is a Banach space and when  , it is even a Hilbert space.[3] If K is non-trivial then   is not a Banach space but   is a Fréchet space for all  .

Topology on the space of test functionsEdit

Let   denote all those functions in   that have compact support[disambiguation needed] in  , where note that   is the union of all   as K ranges over  . Moreover, for every k,   is a dense subset of  . If   then   is called the space of test functions and it may also be denoted by  . For all compact   with  , there are natural inclusions  , which form a direct system in the category of locally convex TVSs that is directed by   (under subset inclusion). This system's direct limit (in the category of locally convex TVSs) is the locally convex TVS   together with  , where   is the natural inclusion and where the topology on   is the strongest locally convex topology making all of these inclusion maps continuous. This topology, called the canonical LF topology, makes   into a complete Hausdorff locally convex LF-space (and also an LB-space when  ) and this topology is finer than the subspace topology that   inherits from   (thus the natural inclusion   is continuous but in general not a topological embedding).[3] From basic category theory, we know that this topology is independent of the particular choice of the directed collection of compact subsets   (as mentioned earlier). The spaces  ,  , and the strong duals   and   are also barreled nuclear Montel bornological Mackey spaces.

It may be shown that for any compact subset  , the natural inclusion   is an embedding of TVSs. Furthermore, a sequence   in   converges in   if and only if there exists some   such that   contains this sequence and this sequence converges in  . From the universal property of direct limits, we know that if   is a linear map into a locally convex space Y (not necessarily Hausdorff), then u is continuous   u is bounded   for every  ,  's restriction to  ,  , is continuous (or bounded).[4][5] A subset B of   is bounded in   if and only if there exists some   such that   and B is a bounded subset of  .[5] Moreover, if   is compact and   then S is bounded in   if and only if it is bounded in  . For any  , any bounded subset of   (resp.  ) is a relatively compact subset of   (resp.  ), where  .[5]

For all compact KU, the interior of   in   is empty so   is of the first category in itself. By Baire's theorem,   is not metrizable.

The bilinear multiplication map   given by   is not continuous; it is however, hypocontinuous.[6]

Topology defined via neighborhoodsEdit

The canonical LF topology may also be defined by defining the neighborhoods of the origin as follows: if U is a convex subset of  , then U is a neighborhood of the origin in the canonical LF topology if and only if for all compact  ,   is a neighborhood of the origin in  .

Topology defined via differential operatorsEdit

A linear differential operator in U with C coefficients is a sum   where all but finitely many of the C(U) functions   are identically 0. The integer   is called the order of the differential operator  , where   when  . If   is a linear differential operator of order k then it induces a canonical linear map   defined by  , where we shall reuse notation and also denote this map by  .[7]

For any  , the canonical LF topology on   is the weakest locally convex TVS topology making all linear differential operators in U of order < k + 1 into continuous maps from   into  .[7]

Sequentional definition of the topology on D(U)Edit

The space D(U) of test functions on U, which is a real vector space, can be given a topology by defining the limit of a sequence of elements of D(U). A sequence ( k) in D(U) is said to converge to   ∈ D(U) if the following two conditions hold:[8]

  • There is a compact set K ⊂ U containing the supports of all  k:
 
  • For each multi-index α, the sequence of partial derivatives   tends uniformly to  .

With this definition, D(U) becomes a complete locally convex topological vector space satisfying the Heine–Borel property.[9]

This topology can be placed in the context of the following general construction: let

 

be a countable increasing union of locally convex topological vector spaces and ιi : XiX be the inclusion maps. In this context, the inductive limit topology, or final topology, τ on X is the finest locally convex vector space topology making all the inclusion maps   continuous. The topology τ can be explicitly described as follows: let β be the collection of convex balanced subsets W of X such that WXi is open for all i. A base for the inductive limit topology τ then consists of the sets of the form x + W, where x in X and W in β.

The proof that τ is a vector space topology makes use of the assumption that each Xi is locally convex. By construction, β is a local base for τ. That any locally convex vector space topology on X must necessarily contain τ means it is the weakest one. One can also show that, for each i, the subspace topology Xi inherits from τ coincides with its original topology. When each Xi is a Fréchet space, (X, τ) is called an LF space.

Now let U be the union of Ui where {Ui} is a countable nested family of open subsets of U with compact closures Ki = Ui. Then we have the countable increasing union

 

where   is the set of all smooth functions on U with support lying in Ki. On each  , consider the topology given by the seminorms

 

i.e. the topology of uniform convergence of derivatives of arbitrary order. This makes each   a Fréchet space. The resulting LF space structure on D(U) is the topology described in the beginning of the section.

On D(U), one can also consider the topology given by the seminorms

 

However, this topology has the disadvantage of not being complete. On the other hand, because of the particular features of the  's, a set is bounded with respect to τ if and only if it lies in some  's. The completeness of (D(U), τ) then follow from that of DKi's.

The topology τ is not metrizable by the Baire category theorem, since D(U) is the union of subspaces of the first category in D(U).[10]

DistributionsEdit

A distribution is a continuous linear functional on  . If T is a linear functional on   then the following are equivalent:

  • T is a distribution;
  • T is continuous[disambiguation needed];
  • T is continuous[disambiguation needed] at the origin;
  • T is bounded;
  • T is sequentially continuous; i.e. for every sequence   of test functions in   that converges to f in  ,  ;
  • Even though the topology of   is not metrizable, a linear functional on   is continuous if and only if it is sequentially continuous.
  • T is sequentially continuous at the origin; i.e. for every sequence   of test functions in   that converges to 0 in  ,  ;
  • the kernel of T is a closed subspace of  ;
  • to every compact subset K of U, there is an integer   and a constant   such that for all  ,[11]
 ;
  • the statement above but with the compact set K restricted to lie in  ;
  • for every compact subset K of U there exists a positive constant CK and a non-negative integer NK such that
 

for all test functions f with support contained in K.[12]

  • Note that if N can be chosen to be independent of K then the distribution is said to be of finite order and the smallest such N is called the order of the distribution. A distribution is said to have infinite order if it does not have finite order.
  • the statement above but with the compact set K restricted to lie in  ;
  • for any compact subset K of U and any sequence of test functions   belonging to  , if   converges uniformly to zero for all multi-indices p, then  ;
  • the statement above but with the compact set K restricted to lie in  ;

We have the canonical duality pairing between a distribution   and a test function  ), which is denoted using angle brackets by

 

so that ⟨T, ⟩ = T( ). One interprets this notation as the distribution T acting on the test function   to give a scalar, or symmetrically as the test function   acting on the distribution T.

Topology on the space of distributionsEdit

The space of distributions, denoted by  , is the continuous dual space of   with the topology of uniform convergence on bounded subsets of  [3] (this is written as  ), where this topology is also called the strong dual topology (in functional analysis, the strong dual topology is the "standard" or "default" topology placed on the continuous dual space  , where if X is a normed space then this strong dual topology is the same as the usual norm-induced topology on  ). This topology is chosen because it is with this topology that D′(U) becomes a nuclear Montel space and it is with this topology that the kernels theorem of Schwartz holds.[13] No matter what dual topology is placed on D′(U), a sequence of distributions converges in this topology if and only if it converges pointwise (although this need not be true of a net), which is why the topology is sometimes defined to be the weak-* topology. No matter which topology is chosen, D′(U) will be a non-metrizable, locally convex topological vector space.

Each of  ,  ,  , and   are nuclear Montel spaces.[14] One reason for giving   the canonical LF topology is because it is with this topology that   and its continuous dual space both become nuclear spaces, which have many nice properties and which may be viewed as a generalization of finite-dimensional spaces (for comparison, normed spaces are another generalization of finite-dimensional spaces that have many "nice" properties). It is precisely because   is a nuclear space that the Schwartz kernel theorem holds, as Alexander Grothendieck discovered when he investigated why the theorem works for the space of distributions but not for other "nice" spaces like the Hilbert space   (this led him to discover nuclear maps and nuclear spaces, among other things). One of the primary results of the Schwartz kernel theorem is that for any open subsets   and  , the canonical map   is an isomorphism of TVSs (where   has the topology of uniform convergence on bounded subsets);[15] this result is false if one replaces the space   with   (which is a reflexive space that is even isomorphic to its own strong dual space) and replaces   with the dual of this   space.[16]

Each of   and   is a nuclear[17] Montel space,[18] which implies that they are each reflexive, barreled, Mackey, and have the Heine-Borel property. Other consequences are that:

  • the topology on   (the strong dual topology) is identical to the topology of uniform convergence on compact subsets of  ;
  • the strong dual space of   is TVS isomorphic to   via the canonical TVS-isomorphism   defined by sending   to value at   (i.e. to the linear functional on   defined by sending   to  );
  • on any bounded subset of  , the weak and strong subspace topologies coincide; the same is true for  ;
  • every weakly convergent sequence in   is strongly convergent (although this does not necessarily extend to nets).

Sequences of distributionsEdit

A sequence of distributions (Tk) converges with respect to the weak-* topology on D′(U) to a distribution T if and only if

 

for every test function   in D(U). For example, if fk : RR is the function

 

and Tk is the distribution corresponding to fk, then

 

as k → ∞, so Tkδ in D′(R). Thus, for large k, the function fk can be regarded as an approximation of the Dirac delta distribution.

Localization of distributionsEdit

There is no way to define the value of a distribution in D′(U) 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 locally determined 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 overlap. Such a structure is known as a sheaf.

Restrictions to an open subsetEdit

Let U and V be open subsets of Rn with V ⊂ U. Let InVU : D(V) → D(U) be the operator which extends by zero a given smooth function compactly supported in V to a smooth function compactly supported in the larger set U. The transpose of InVU is called the restriction mapping and is denoted by  .

The map InVU : D(V) → D(U) is a continuous injection where if   then it is not a topological embedding and its range is not dense in D(U), which implies that this map's transpose is neither injective nor surjective and that the topology that InVU transfers from D(V) onto its image is strictly finer than the subspace topology that D(U) induces on this same set.[19] A distribution S ∈ D′(V) is said to be extendible to U if it belongs to the range of the transpose of InVU and it is called extendible if it is extendable to  .[19]

For any distribution T ∈ D′(U), the restriction ρVU(T) is a distribution in the dual space D′(V) defined by

 

for all test functions   ∈ D(V).

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 D′(V) but admits no extension to D′(U).

Gluing and distributions that vanish in a setEdit

Theorem[20] Let I be a non-empty set and let   be a collection of open subsets of  . For each  , let   and suppose that for all  , the restriction of   to   is equal to the restriction of   to   (note that both restrictions are elements of  ). Then there exists a unique distribution   such that for all  , the restriction of T to   is equal to  .

If V is an open subset of U then we say that a distribution   vanishes in V if for each test function  , if   then  . 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 ρVU.

Corollary: Let   be a collection of open subsets of   and let  . Then T = 0 if and only if for each  , the restriction of T to   is equal to 0.[20]
Corollary:[20] The union of all open subsets of U in which a distribution T vanishes is an open subset of U in which T vanishes.

Support of a distributionEdit

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 doesn't 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 the support of T.[20] Thus

 

If f is a locally integrable function on U and if   is its associated distribution, then the support of   is the smallest closed subset of U in the complement of which f is almost everywhere equal to 0.[20] If f is continuous, then the support of   is equal to the closure of the set of points in U at which f doesn't vanish.[20] The support of the distribution associated with the Dirac measure at a point   is the set  .[20] 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 the product f T = T. If the support of a distribution T is compact then it has finite order and furthermore, there is a constant C and a non-negative integer N such that

 

for all  .[21] If T has compact support then it has a unique extension to a continuous linear functional   on  ; this functional can be defined by  , where   is any function that is identically 1 on an open set containing the support of T.[21]

If   and   then   and  . Thus, distributions with support in a given subset   form a vector subspace of  ; such a subspace is weakly closed in   if and only if A is closed in U.[22] Furthermore, if   is a differential operator in U, then for all distributions T on U and all   we have   and  .[22]

Support in a point set and Dirac measuresEdit

For any  , let   denote the distribution induced by the Dirac measure at x. For any   and distribution  , the support of T is contained in   if and only if T is a finite linear combination of derivatives of the Dirac measure at x[23], where if in addition the order of T is   (k a non-negative integer) then   so some scalars   (  ranges over all multi-indices such that  ).[24]

Decomposition of distributionsEdit

Theorem[25] Let  . There exists a sequence distributions   in   such that each   has compact support, every compact subset   intersects the support of only finitely many  's, and the partial sums   converge in   to T (i.e.  ).

Operations on distributionsEdit

Many operations which are defined on smooth functions with compact support can also be defined for distributions. In general, if A : D(U) → D(U) is a linear mapping of vector spaces which is continuous with respect to the weak-* topology, then it is possible to extend A to a mapping A : D′(U) → D′(U) by passing to the limit. (This approach works for non-linear mappings as well, provided they are assumed to be uniformly continuous.)

In practice, however, it is more convenient to define operations on distributions by means of the transpose.[26] If A : D(U) → D(U) is a continuous linear operator, then the transpose is an operator At : D(U) → D(U) such that for  ,   is the map   satisfying

 

(For operators acting on spaces of complex-valued test functions, the transpose At differs from the adjoint A* in that it does not include a complex conjugate.)

If such an operator At exists and is continuous on D(U), then the original operator A may be extended to D′(U) by defining AT for a distribution T as

 

Differential operatorsEdit

DifferentiationEdit

Suppose A : D(U) → D(U) is the partial derivative operator

 

If   and ψ are in D(U), then an integration by parts gives

 

so that At = −A. This operator is a continuous linear transformation on D(U). So, if T ∈ D′(U) is a distribution, then the partial derivative of T with respect to the coordinate xk is defined by the formula

 

With this definition, every distribution is infinitely differentiable, and the derivative in the direction xk is a linear operator on D′(U).

More generally, if α = (α1, ..., αn) is an arbitrary multi-index and ∂α is the associated partial derivative operator, then the partial derivative ∂αT of the distribution T ∈ D′(U) is defined by

 

Differentiation of distributions is a continuous operator on D′(U); this is an important and desirable property that is not shared by most other notions of differentiation.

If T is a distribution in   then   in   where D T is the derivative of T and   is translation by x; thus the derivative of T may be viewed as a limit of quotients.[21]

Differential operatorsEdit

A linear differential operator in U with C coefficients is a sum   where all but finitely many of the C(U) functions   are identically 0. The integer   is called the order of the differential operator  , where   when  . If   is a linear differential operator of order k then it deduces a canonical linear map   defined by  , where we shall reuse notation and again denote this map by  . The restriction of the canonical map to   induces a continuous linear map   as well as another continuous linear map  ;[7] the transposes of these two map are consequently continuous linear maps   and  , respectively, where   denotes the continuous dual space of  .[7] Recall that the transpose of a continuous linear map   is the map   defined by  , or equivalently, it is the unique map satisfying   for all   and all  .

Differential operators acting on distributions and formal transposesEdit

For any two complex-valued functions f and g on U, define   (if the integral exists) and if   and   then define  . Given any  , let   be the canonical distribution defined by  .

We want to extend the action of a differential linear operator   to distributions, where this extension (however it may be defined) will be the map denoted by  . One property that this extension should reasonably be required to have is that   for any  , where   is the smooth function that results from applying the differential operator   to  . It is natural to consider the transpose map   that   canonically induces but this is not the extension of   to distributions that we want because it doesn't have the aforementioned property. But by investigating this map's action on  , we will be led to the appropriate definitions.

Note that for all  

 

Since   has compact support, so too does every function   so when we integrate by parts we get

 

(for instance, if  ,  , and the support of   is contained in the interval [a, b], then  , where  ).

We now define the formal transpose of  , denoted by  , to be the differential operator in U defined by   where

 

with   and with   if and only if   for all i. This definition stems from the fact that by using the Leibniz rule, we obtain

 .

We've thus shown that   for all   from which it follows that  . Note that the formal transpose of the formal transpose is the original differential operator, i.e.  .[7]

The formal transpose of   induces a continuous linear map   from   into  , whose transpose will be denoted by   or   or simply   and called a differential operator, where this map is a linear map  .[7] Explicitly, for any   and  ,  . For any  , we have  ,[7] which justifies our definition.

If   converges to T in   then for every multi-index  ,   converges to   in  .

Multiplication of a distribution by smooth functionEdit

Observe that a differential operator of degree 0 is just multiplication by a C function. And conversely, if   then   is a differential operator of degree 0, whose formal transpose is itself (i.e.  ). The induced differential operator   maps a distribution T to a distribution denoted by  . We have thus defined the multiplication of a distribution by a   function. If   converges to T in   and if   converges to f in   then   converges to f T in  .

We now give an alternative presentation of multiplication by a smooth function.

Multiplication by a smooth function

If m : UR is an infinitely differentiable function and T is a distribution on U, then the product mT is defined by

 

This definition coincides with the transpose definition since if M : D(U) → D(U) is the operator of multiplication by the function m (i.e., M  = m  ), then

 

so that Mt = M.

Under multiplication by smooth functions, D′(U) is a module over the ring C(U). With this definition of multiplication by a smooth function, the ordinary product rule of calculus remains valid. However, a number of unusual identities also arise. For example, if δ is the Dirac delta distribution on R, then  = m(0)δ, and if δ′ is the derivative of the delta distribution, then

 

These definitions of differentiation and multiplication also make it possible to define the operation of a linear differential operator with smooth coefficients on a distribution. A linear differential operator P takes a distribution T ∈ D′(U) to another distribution PT given by a sum of the form

 

where the coefficients pα are smooth functions on U. The action of the distribution PT on a test function   is given by

 

The minimum integer k for which such an expansion holds for every distribution T is called the order of P. The space D′(U) is a D-module with respect to the action of the ring of linear differential operators.

The bilinear multiplication map   given by   is not continuous; it is however, hypocontinuous.[6]

Composition with a smooth functionEdit

Let T be a distribution on an open set U ⊂ Rn. Let V be an open set in Rn, and F : V → U. Then provided F is a submersion, it is possible to define

 

This is the composition of the distribution T with F, and is also called the pullback of T along F, 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 dF(x) of F is a surjective linear map for every x ∈ V. A necessary (but not sufficient) condition for extending F# to distributions is that F be an open mapping.[27] The inverse function theorem ensures that a submersion satisfies this condition.

If F is a submersion, then F# is defined on distributions by finding the transpose map. Uniqueness of this extension is guaranteed since F# is a continuous linear operator on D(U). Existence, however, requires using the change of variables formula, the inverse function theorem (locally) and a partition of unity argument.[28]

In the special case when F is a diffeomorphism from an open subset V of Rn onto an open subset U of Rn change of variables under the integral gives

 

In this particular case, then, F# is defined by the transpose formula:

 

ConvolutionEdit

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   then we denote by fg the convolution of f and g, defined at   to be the integral

 

provided that the integral exists. If   are such that 1/r = (1/p) + (1/q) - 1 then for any functions   and   we have   and  .[29] If f and g are continuous functions on  , at least one of which has compact support, then   and if   then the value of fg in the set A do not depend on the values of f outside of the Minkowski sum  .[29]

Importantly, if   has compact support then for any  , the convolution map   is continuous when considered as the map   or as the map  .[29]

Translation and symmetry

Given  , the translation operator τa is sends a function   to the function   defined by  . This can be extended by the transpose to distributions in the following way: given a distribution T, the translation of T by a is the distribution   defined by  .[30][31]

Given a function  , define the function   by  . Given a distribution T, let   be the distribution defined by  . The operator   is called the symmetry with respect to the origin.[30]

Convolution of a smooth function with a distributionEdit

Let   and   and assume that at least one of f and T has compact support. For any  , note that   is the function   given by  . The convolution of f and T, denoted by   or by  , is the smooth function   defined by  .[30] We have  . For all multi-indices p, it satisfies   and  .[30]

If T is a distribution then the map   is continuous as a map   where if in addition T has compact support then it is also continuous as the map   and continuous as the map  .[30]

If   is a continuous linear map such that   for all   and all   then there exists a distribution   such that   for all  .[21]

Example

Let H be the Heavyside function on   and   be the Dirac measure at 0. Then  , where   is the derivative of  , and  . Moreover, for any  ,  .[21] Importantly, the associative law fails to hold:[21]

  but  .

Convolution of a test function with a distributionEdit

If   is a compactly supported smooth test function, then convolution with f defines a linear map   given by  , 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 Cf relative to the duality pairing of   with the space   of distributions (Trèves 1967, Chapter 27). If fg , then by Fubini's theorem

 

where  . Extending by continuity, the convolution of f with a distribution T is defined by

 

for all test functions  .

An alternative way to define the convolution of a function f and a distribution T is to use the translation operator τa. The convolution of the compactly supported function f and the distribution T is then the function defined for each   by

 

It can be shown that the convolution of a smooth, compactly supported function and a distribution is a smooth function. If the distribution T has compact support then if f is a polynomial (resp. an exponential function, an analytic function, the restriction to   of an entire analytic function on  , the restriction to   of an entire function of exponential type in  ) then the same is true of  .[30] If the distribution T has compact support as well, then fT is a compactly supported function, and the Titchmarsh convolution theorem (Hörmander 1983, Theorem 4.3.3) implies that

 

where ch denotes the convex hull and supp denotes the support.

Convolution of distributionsEdit

It is also possible to define the convolution of two distributions S and T on  , provided one of them has compact support. Informally, in order to define ST where T has compact support, the idea is to extend the definition of the convolution ∗ to a linear operation on distributions so that the associativity formula

 

continues to hold for all test functions  .[32]

It is also possible to provide a more explicit characterization of the convolution of distributions (Trèves 1967, Chapter 27). Suppose that S and T are distributions and that S has compact support. Then the linear map   defined by   is continuous as is the linear map   defined by  . The transposes of these maps,   and  , are consequently continuous and one may show that  .[30] This common value is called the convolution of S and T and it is a distribution that is denoted by   or  . It satisfies  .[30] If S and T are two distributions, at least one of which has compact support, then for any  ,