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 acting on the points 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 test functions in a certain way. In applications to physics and engineering, test functions are usually infinitely differentiablecomplex-valued (or real-valued) functions with compactsupport that are defined on some given non-empty open subset$U\subseteq \mathbb {R} ^{n}$. (Bump functions are examples of test functions.) The set of all such test functions forms a vector space that is denoted by $C_{c}^{\infty }(U)$ or ${\mathcal {D}}(U).$
Most commonly encountered functions, including all continuous maps $f:\mathbb {R} \to \mathbb {R}$ if using $U:=\mathbb {R} ,$ 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 $\psi \in {\mathcal {D}}(\mathbb {R} )$ by "sending" it to the number${\textstyle \int _{\mathbb {R} }f\,\psi \,dx,}$ which is often denoted by $D_{f}(\psi ).$ This new action ${\textstyle \psi \mapsto D_{f}(\psi )}$ of $f$ defines a scalar-valued map$D_{f}:{\mathcal {D}}(\mathbb {R} )\to \mathbb {C} ,$ whose domain is the space of test functions ${\mathcal {D}}(\mathbb {R} ).$ This functional$D_{f}$ turns out to have the two defining properties of what is known as a distribution on $U=\mathbb {R}$: it is linear, and it is also continuous when ${\mathcal {D}}(\mathbb {R} )$ is given a certain topology called the canonical LF topology. The action (the integration ${\textstyle \psi \mapsto \int _{\mathbb {R} }f\,\psi \,dx}$) 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 ${\textstyle \psi \mapsto \int _{U}\psi d\mu }$ against certain measures$\mu$ on $U.$ Nonetheless, it is still always possible to reduce any arbitrary distribution down to a simpler family of related distributions that do arise via such actions of integration.
More generally, a distribution on $U$ is by definition a linear functional on $C_{c}^{\infty }(U)$ that is continuous when $C_{c}^{\infty }(U)$ is given a topology called the canonical LF topology. This leads to the space of (all) distributions on $U$, usually denoted by ${\mathcal {D}}'(U)$ (note the prime), which by definition is the space of all distributions on $U$ (that is, it is the continuous dual space of $C_{c}^{\infty }(U)$); it is these distributions that are the main focus of this article.
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 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.
Notationedit
The following notation will be used throughout this article:
$n$ is a fixed positive integer and $U$ is a fixed non-empty open subset of Euclidean space$\mathbb {R} ^{n}.$
$\mathbb {N} =\{0,1,2,\ldots \}$ denotes the natural numbers.
$k$ will denote a non-negative integer or $\infty .$
If $f$ is a function then $\operatorname {Dom} (f)$ will denote its domain and the support of $f,$ denoted by $\operatorname {supp} (f),$ is defined to be the closure of the set $\{x\in \operatorname {Dom} (f):f(x)\neq 0\}$ in $\operatorname {Dom} (f).$
For two functions $f,g:U\to \mathbb {C} ,$ the following notation defines a canonical pairing:
$\langle f,g\rangle :=\int _{U}f(x)g(x)\,dx.$
A multi-index of size$n$ is an element in $\mathbb {N} ^{n}$ (given that $n$ is fixed, if the size of multi-indices is omitted then the size should be assumed to be $n$). The length of a multi-index $\alpha =(\alpha _{1},\ldots ,\alpha _{n})\in \mathbb {N} ^{n}$ is defined as $\alpha _{1}+\cdots +\alpha _{n}$ and denoted by $|\alpha |.$ Multi-indices are particularly useful when dealing with functions of several variables, in particular, we introduce the following notations for a given multi-index $\alpha =(\alpha _{1},\ldots ,\alpha _{n})\in \mathbb {N} ^{n}$:
We also introduce a partial order of all multi-indices by $\beta \geq \alpha$ if and only if $\beta _{i}\geq \alpha _{i}$ for all $1\leq i\leq n.$ When $\beta \geq \alpha$ we define their multi-index binomial coefficient as:
Definitions of test functions and distributionsedit
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.
For any compact subset $K\subseteq U,$ let $C^{k}(K)$ and $C^{k}(K;U)$ both denote the vector space of all those functions $f\in C^{k}(U)$ such that $\operatorname {supp} (f)\subseteq K.$
If $f\in C^{k}(K)$ then the domain of $f$ is U and not K. So although $C^{k}(K)$ depends on both K and U, only K is typically indicated. The justification for this common practice is detailed below. The notation $C^{k}(K;U)$ will only be used when the notation $C^{k}(K)$ risks being ambiguous.
Every $C^{k}(K)$ contains the constant 0 map, even if $K=\varnothing .$
Let $C_{c}^{k}(U)$ denote the set of all $f\in C^{k}(U)$ such that $f\in C^{k}(K)$ for some compact subset K of U.
Equivalently, $C_{c}^{k}(U)$ is the set of all $f\in C^{k}(U)$ such that $f$ has compact support.
$C_{c}^{k}(U)$ is equal to the union of all $C^{k}(K)$ as $K\subseteq U$ ranges over all compact subsets of $U.$
If $f$ is a real-valued function on $U$, then $f$ is an element of $C_{c}^{k}(U)$ if and only if $f$ is a $C^{k}$bump function. Every real-valued test function on $U$ is also a complex-valued test function on $U.$
For all $j,k\in \{0,1,2,\ldots ,\infty \}$ and any compact subsets $K$ and $L$ of $U$, we have:
Definition: Elements of $C_{c}^{\infty }(U)$ are called test functions on U and $C_{c}^{\infty }(U)$ is called the space of test functions on U. We will use both ${\mathcal {D}}(U)$ and $C_{c}^{\infty }(U)$ to denote this space.
Distributions on U are continuous linear functionals on $C_{c}^{\infty }(U)$ when this vector space is endowed with a particular topology called the canonical LF-topology. The following proposition states two necessary and sufficient conditions for the continuity of a linear function on $C_{c}^{\infty }(U)$ that are often straightforward to verify.
Proposition: A linear functionalT on $C_{c}^{\infty }(U)$ is continuous, and therefore a distribution, if and only if any of the following equivalent conditions is satisfied:
For every compact subset $K\subseteq U$ there exist constants $C>0$ and $N\in \mathbb {N}$ (dependent on $K$) such that for all $f\in C_{c}^{\infty }(U)$ with support contained in $K$,^{[1]}^{[2]}
For every compact subset $K\subseteq U$ and every sequence $\{f_{i}\}_{i=1}^{\infty }$ in $C_{c}^{\infty }(U)$ whose supports are contained in $K$, if $\{\partial ^{\alpha }f_{i}\}_{i=1}^{\infty }$ converges uniformly to zero on $U$ for every multi-index$\alpha$, then $T(f_{i})\to 0.$
Topology on C^{k}(U)edit
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.
Suppose $k\in \{0,1,2,\ldots ,\infty \}$ and $K$ is an arbitrary compact subset of $U.$ Suppose $i$ an integer such that $0\leq i\leq k$^{[note 1]} and $p$ is a multi-index with length $|p|\leq k.$ For $K\neq \varnothing ,$ define:
while for $K=\varnothing ,$ define all the functions above to be the constant 0 map.
All of the functions above are non-negative $\mathbb {R}$-valued^{[note 2]}seminorms on $C^{k}(U).$ As explained in this article, every set of seminorms on a vector space induces a locally convexvector topology.
generate the same locally convexvector 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 $C$).
The vector space $C^{k}(U)$ is endowed with the locally convex topology induced by any one of the four families $A,B,C,D$ of seminorms described above. This topology is also equal to the vector topology induced by all of the seminorms in $A\cup B\cup C\cup D.$
With this topology, $C^{k}(U)$ becomes a locally convex Fréchet space that is notnormable. Every element of $A\cup B\cup C\cup D$ is a continuous seminorm on $C^{k}(U).$
Under this topology, a net$(f_{i})_{i\in I}$ in $C^{k}(U)$ converges to $f\in C^{k}(U)$ if and only if for every multi-index $p$ with $|p|<k+1$ and every compact $K,$ the net of partial derivatives $\left(\partial ^{p}f_{i}\right)_{i\in I}$converges uniformly to $\partial ^{p}f$ on $K.$^{[3]} For any $k\in \{0,1,2,\ldots ,\infty \},$ any (von Neumann) bounded subset of $C^{k+1}(U)$ is a relatively compact subset of $C^{k}(U).$^{[4]} In particular, a subset of $C^{\infty }(U)$ is bounded if and only if it is bounded in $C^{i}(U)$ for all $i\in \mathbb {N} .$^{[4]} The space $C^{k}(U)$ is a Montel space if and only if $k=\infty .$^{[5]}
A subset $W$ of $C^{\infty }(U)$ is open in this topology if and only if there exists $i\in \mathbb {N}$ such that $W$ is open when $C^{\infty }(U)$ is endowed with the subspace topology induced on it by $C^{i}(U).$
Topology on C^{k}(K)edit
As before, fix $k\in \{0,1,2,\ldots ,\infty \}.$ Recall that if $K$ is any compact subset of $U$ then $C^{k}(K)\subseteq C^{k}(U).$
Assumption: For any compact subset $K\subseteq U,$ we will henceforth assume that $C^{k}(K)$ is endowed with the subspace topology it inherits from the Fréchet space$C^{k}(U).$
If $k$ is finite then $C^{k}(K)$ is a Banach space^{[6]} with a topology that can be defined by the norm
And when $k=2,$ then $C^{k}(K)$ is even a Hilbert space.^{[6]}
Trivial extensions and independence of C^{k}(K)'s topology from Uedit
Suppose $U$ is an open subset of $\mathbb {R} ^{n}$ and $K\subseteq U$ is a compact subset. By definition, elements of $C^{k}(K)$ are functions with domain $U$ (in symbols, $C^{k}(K)\subseteq C^{k}(U)$), so the space $C^{k}(K)$ and its topology depend on $U;$ to make this dependence on the open set $U$ clear, temporarily denote $C^{k}(K)$ by $C^{k}(K;U).$
Importantly, changing the set $U$ to a different open subset $U'$ (with $K\subseteq U'$) will change the set $C^{k}(K)$ from $C^{k}(K;U)$ to $C^{k}(K;U'),$^{[note 3]} so that elements of $C^{k}(K)$ will be functions with domain $U'$ instead of $U.$
Despite $C^{k}(K)$ depending on the open set ($U{\text{ or }}U'$), the standard notation for $C^{k}(K)$ makes no mention of it.
This is justified because, as this subsection will now explain, the space $C^{k}(K;U)$ is canonically identified as a subspace of $C^{k}(K;U')$ (both algebraically and topologically).
It is enough to explain how to canonically identify $C^{k}(K;U)$ with $C^{k}(K;U')$ when one of $U$ and $U'$ is a subset of the other. The reason is that if $V$ and $W$ are arbitrary open subsets of $\mathbb {R} ^{n}$ containing $K$ then the open set $U:=V\cap W$ also contains $K,$ so that each of $C^{k}(K;V)$ and $C^{k}(K;W)$ is canonically identified with $C^{k}(K;V\cap W)$ and now by transitivity, $C^{k}(K;V)$ is thus identified with $C^{k}(K;W).$
So assume $U\subseteq V$ are open subsets of $\mathbb {R} ^{n}$ containing $K.$
Given $f\in C_{c}^{k}(U),$ its trivial extension to $V$ is the function $F:V\to \mathbb {C}$ defined by:
This trivial extension belongs to $C^{k}(V)$ (because $f\in C_{c}^{k}(U)$ has compact support) and it will be denoted by $I(f)$ (that is, $I(f):=F$). The assignment $f\mapsto I(f)$ thus induces a map $I:C_{c}^{k}(U)\to C^{k}(V)$ that sends a function in $C_{c}^{k}(U)$ to its trivial extension on $V.$ This map is a linear injection and for every compact subset $K\subseteq U$ (where $K$ is also a compact subset of $V$ since $K\subseteq U\subseteq V$),
${\begin{alignedat}{4}I\left(C^{k}(K;U)\right)&~=~C^{k}(K;V)\qquad {\text{ and thus }}\\I\left(C_{c}^{k}(U)\right)&~\subseteq ~C_{c}^{k}(V).\end{alignedat}}$
If $I$ is restricted to $C^{k}(K;U)$ then the following induced linear map is a homeomorphism (linear homeomorphisms are called TVS-isomorphisms):
the vector space $C_{c}^{k}(U)$ is canonically identified with its image in $C_{c}^{k}(V)\subseteq C^{k}(V).$ Because $C^{k}(K;U)\subseteq C_{c}^{k}(U),$ through this identification, $C^{k}(K;U)$ can also be considered as a subset of $C^{k}(V).$
Thus the topology on $C^{k}(K;U)$ is independent of the open subset $U$ of $\mathbb {R} ^{n}$ that contains $K,$^{[7]} which justifies the practice of writing $C^{k}(K)$ instead of $C^{k}(K;U).$
Canonical LF topologyedit
Recall that $C_{c}^{k}(U)$ denotes all functions in $C^{k}(U)$ that have compact support in $U,$ where note that $C_{c}^{k}(U)$ is the union of all $C^{k}(K)$ as $K$ ranges over all compact subsets of $U.$ Moreover, for each $k,\,C_{c}^{k}(U)$ is a dense subset of $C^{k}(U).$ The special case when $k=\infty$ gives us the space of test functions.
$C_{c}^{\infty }(U)$ is called the space of test functions on $U$ and it may also be denoted by ${\mathcal {D}}(U).$ Unless indicated otherwise, it is endowed with a topology called the canonical LF topology, whose definition is given in the article: Spaces of test functions and distributions.
As discussed earlier, continuous linear functionals on a $C_{c}^{\infty }(U)$ are known as distributions on $U.$ Other equivalent definitions are described below.
By definition, a distribution on $U$ is a continuouslinear functional on $C_{c}^{\infty }(U).$ Said differently, a distribution on $U$ is an element of the continuous dual space of $C_{c}^{\infty }(U)$ when $C_{c}^{\infty }(U)$ is endowed with its canonical LF topology.
There is a canonical duality pairing between a distribution $T$ on $U$ and a test function $f\in C_{c}^{\infty }(U),$ which is denoted using angle brackets by
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 distributionsedit
Proposition. If $T$ is a linear functional on $C_{c}^{\infty }(U)$ then the following are equivalent:
explicitly, for every sequence $\left(f_{i}\right)_{i=1}^{\infty }$ in $C_{c}^{\infty }(U)$ that converges in $C_{c}^{\infty }(U)$ to some $f\in C_{c}^{\infty }(U),$${\textstyle \lim _{i\to \infty }T\left(f_{i}\right)=T(f);}$^{[note 4]}
T is sequentially continuous at the origin; in other words, T maps null sequences^{[note 5]} to null sequences;
explicitly, for every sequence $\left(f_{i}\right)_{i=1}^{\infty }$ in $C_{c}^{\infty }(U)$ that converges in $C_{c}^{\infty }(U)$ to the origin (such a sequence is called a null sequence), ${\textstyle \lim _{i\to \infty }T\left(f_{i}\right)=0;}$
a null sequence is by definition any sequence that converges to the origin;
T maps null sequences to bounded subsets;
explicitly, for every sequence $\left(f_{i}\right)_{i=1}^{\infty }$ in $C_{c}^{\infty }(U)$ that converges in $C_{c}^{\infty }(U)$ to the origin, the sequence $\left(T\left(f_{i}\right)\right)_{i=1}^{\infty }$ is bounded;
explicitly, for every Mackey convergent null sequence $\left(f_{i}\right)_{i=1}^{\infty }$ in $C_{c}^{\infty }(U),$ the sequence $\left(T\left(f_{i}\right)\right)_{i=1}^{\infty }$ is bounded;
a sequence $f_{\bullet }=\left(f_{i}\right)_{i=1}^{\infty }$ is said to be Mackey convergent to the origin if there exists a divergent sequence $r_{\bullet }=\left(r_{i}\right)_{i=1}^{\infty }\to \infty$ of positive real numbers such that the sequence $\left(r_{i}f_{i}\right)_{i=1}^{\infty }$ is bounded; every sequence that is Mackey convergent to the origin necessarily converges to the origin (in the usual sense);
The kernel of T is a closed subspace of $C_{c}^{\infty }(U);$
The graph of T is closed;
There exists a continuous seminorm $g$ on $C_{c}^{\infty }(U)$ such that $|T|\leq g;$
There exists a constant $C>0$ and a finite subset $\{g_{1},\ldots ,g_{m}\}\subseteq {\mathcal {P}}$ (where ${\mathcal {P}}$ is any collection of continuous seminorms that defines the canonical LF topology on $C_{c}^{\infty }(U)$) such that $|T|\leq C(g_{1}+\cdots +g_{m});$^{[note 6]}
For every compact subset $K\subseteq U$ there exist constants $C>0$ and $N\in \mathbb {N}$ such that for all $f\in C^{\infty }(K),$^{[1]}
For every compact subset $K\subseteq U$ there exist constants $C_{K}>0$ and $N_{K}\in \mathbb {N}$ such that for all $f\in C_{c}^{\infty }(U)$ with support contained in $K,$^{[10]}
For any compact subset $K\subseteq U$ and any sequence $\{f_{i}\}_{i=1}^{\infty }$ in $C^{\infty }(K),$ if $\{\partial ^{p}f_{i}\}_{i=1}^{\infty }$ converges uniformly to zero for all multi-indices$p,$ then $T(f_{i})\to 0;$
Topology on the space of distributions and its relation to the weak-* topologyedit
The set of all distributions on $U$ is the continuous dual space of $C_{c}^{\infty }(U),$ which when endowed with the strong dual topology is denoted by ${\mathcal {D}}'(U).$ Importantly, unless indicated otherwise, the topology on ${\mathcal {D}}'(U)$ 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 ${\mathcal {D}}'(U)$ into a completenuclear space, to name just a few of its desirable properties.
Neither $C_{c}^{\infty }(U)$ nor its strong dual ${\mathcal {D}}'(U)$ 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 not enough to fully/correctly define their topologies).
However, a sequence in ${\mathcal {D}}'(U)$ converges in the strong dual topology if and only if it converges in the weak-* topology (this leads many authors to use pointwise convergence to define the convergence of a sequence of distributions; this is fine for sequences but this is not guaranteed to extend to the convergence of nets of distributions because a net may converge pointwise but fail to converge in the strong dual topology).
More information about the topology that ${\mathcal {D}}'(U)$ 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.
There is no way to define the value of a distribution in ${\mathcal {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 overlaps. Such a structure is known as a sheaf.
Extensions and restrictions to an open subsetedit
Let $V\subseteq U$ be open subsets of $\mathbb {R} ^{n}.$
Every function $f\in {\mathcal {D}}(V)$ can be extended by zero from its domain V to a function on U by setting it equal to $0$ on the complement$U\setminus V.$ This extension is a smooth compactly supported function called the trivial extension of $f$ to $U$ and it will be denoted by $E_{VU}(f).$
This assignment $f\mapsto E_{VU}(f)$ defines the trivial extension operator
$E_{VU}:{\mathcal {D}}(V)\to {\mathcal {D}}(U),$
which is a continuous injective linear map. It is used to canonically identify ${\mathcal {D}}(V)$ as a vector subspace of ${\mathcal {D}}(U)$ (although not as a topological subspace).
Its transpose (explained here)
is called the restriction to $V$ of distributions in $U$^{[11]} and as the name suggests, the image $\rho _{VU}(T)$ of a distribution $T\in {\mathcal {D}}'(U)$ under this map is a distribution on $V$ called the restriction of $T$ to $V.$ The defining condition of the restriction $\rho _{VU}(T)$ is:
$\langle \rho _{VU}T,\phi \rangle =\langle T,E_{VU}\phi \rangle \quad {\text{ for all }}\phi \in {\mathcal {D}}(V).$
If $V\neq U$ then the (continuous injective linear) trivial extension map $E_{VU}:{\mathcal {D}}(V)\to {\mathcal {D}}(U)$ is not a topological embedding (in other words, if this linear injection was used to identify ${\mathcal {D}}(V)$ as a subset of ${\mathcal {D}}(U)$ then ${\mathcal {D}}(V)$'s topology would strictly finer than the subspace topology that ${\mathcal {D}}(U)$ induces on it; importantly, it would not be a topological subspace since that requires equality of topologies) and its range is also not dense in its codomain${\mathcal {D}}(U).$^{[11]} Consequently if $V\neq U$ then the restriction mapping is neither injective nor surjective.^{[11]} A distribution $S\in {\mathcal {D}}'(V)$ is said to be extendible to U if it belongs to the range of the transpose of $E_{VU}$ and it is called extendible if it is extendable to $\mathbb {R} ^{n}.$^{[11]}
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=\mathbb {R}$ and $V=(0,2),$ then the distribution
is in ${\mathcal {D}}'(V)$ but admits no extension to ${\mathcal {D}}'(U).$
Gluing and distributions that vanish in a setedit
Theorem^{[12]} — Let $(U_{i})_{i\in I}$ be a collection of open subsets of $\mathbb {R} ^{n}.$ For each $i\in I,$ let $T_{i}\in {\mathcal {D}}'(U_{i})$ and suppose that for all $i,j\in I,$ the restriction of $T_{i}$ to $U_{i}\cap U_{j}$ is equal to the restriction of $T_{j}$ to $U_{i}\cap U_{j}$ (note that both restrictions are elements of ${\mathcal {D}}'(U_{i}\cap U_{j})$). Then there exists a unique ${\textstyle T\in {\mathcal {D}}'(\bigcup _{i\in I}U_{i})}$ such that for all $i\in I,$ the restriction of T to $U_{i}$ is equal to $T_{i}.$
Let V be an open subset of U. $T\in {\mathcal {D}}'(U)$ is said to vanish in V if for all $f\in {\mathcal {D}}(U)$ such that $\operatorname {supp} (f)\subseteq V$ 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}.$
Corollary^{[12]} — Let $(U_{i})_{i\in I}$ be a collection of open subsets of $\mathbb {R} ^{n}$ and let ${\textstyle T\in {\mathcal {D}}'(\bigcup _{i\in I}U_{i}).}$$T=0$ if and only if for each $i\in I,$ the restriction of T to $U_{i}$ is equal to 0.
Corollary^{[12]} — 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 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 the support of T.^{[12]} 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.^{[12]} 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.^{[12]} The support of the distribution associated with the Dirac measure at a point $x_{0}$ is the set $\{x_{0}\}.$^{[12]} 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 $f\in C^{\infty }(U)$ is identically 1 on some open set containing the support of a distribution T then $fT=T.$ 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:^{[7]}
$|T\phi |\leq C\|\phi \|_{N}:=C\sup \left\{\left|\partial ^{\alpha }\phi (x)\right|:x\in U,|\alpha |\leq N\right\}\quad {\text{ for all }}\phi \in {\mathcal {D}}(U).$
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 ${\widehat {T}}(f):=T(\psi f),$ where $\psi \in {\mathcal {D}}(U)$ is any function that is identically 1 on an open set containing the support of T.^{[7]}
If $S,T\in {\mathcal {D}}'(U)$ and $\lambda \neq 0$ then $\operatorname {supp} (S+T)\subseteq \operatorname {supp} (S)\cup \operatorname {supp} (T)$ and $\operatorname {supp} (\lambda T)=\operatorname {supp} (T).$ Thus, distributions with support in a given subset $A\subseteq U$ form a vector subspace of ${\mathcal {D}}'(U).$^{[13]} Furthermore, if $P$ is a differential operator in U, then for all distributions T on U and all $f\in C^{\infty }(U)$ we have $\operatorname {supp} (P(x,\partial )T)\subseteq \operatorname {supp} (T)$ and $\operatorname {supp} (fT)\subseteq \operatorname {supp} (f)\cap \operatorname {supp} (T).$^{[13]}
Distributions with compact supportedit
Support in a point set and Dirac measuresedit
For any $x\in U,$ let $\delta _{x}\in {\mathcal {D}}'(U)$ denote the distribution induced by the Dirac measure at $x.$ For any $x_{0}\in U$ and distribution $T\in {\mathcal {D}}'(U),$ 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 $x_{0}.$^{[14]} If in addition the order of T is $\leq k$ then there exist constants $\alpha _{p}$ such that:^{[15]}
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
Theorem^{[7]} — Suppose T is a distribution on U with compact support K. There exists a continuous function $f$ defined on U and a multi-index p such that
$T=\partial ^{p}f,$
where the derivatives are understood in the sense of distributions. That is, for all test functions $\phi$ on U,
Distributions of finite order with support in an open subsetedit
Theorem^{[7]} — Suppose T is a distribution on U with compact support K and let V be an open subset of U containing K. Since every distribution with compact support has finite order, take N to be the order of T and define $P:=\{0,1,\ldots ,N+2\}^{n}.$ There exists a family of continuous functions $(f_{p})_{p\in P}$ defined on Uwith support in V such that
$T=\sum _{p\in P}\partial ^{p}f_{p},$
where the derivatives are understood in the sense of distributions. That is, for all test functions $\phi$ on U,
The formal definition of distributions exhibits them as a subspace of a very large space, namely the topological dual of ${\mathcal {D}}(U)$ (or the Schwartz space${\mathcal {S}}(\mathbb {R} ^{n})$ for tempered distributions). It is not immediately clear from the definition how exotic a distribution might be. To answer this question, it is instructive to see distributions built up from a smaller space, namely the space of continuous functions. Roughly, any distribution is locally a (multiple) derivative of a continuous function. A precise version of this result, given below, holds for distributions of compact support, tempered distributions, and general distributions. Generally speaking, no proper subset of the space of distributions contains all continuous functions and is closed under differentiation. This says that distributions are not particularly exotic objects; they are only as complicated as necessary.
Theorem^{[16]} — Let T be a distribution on U.
There exists a sequence $(T_{i})_{i=1}^{\infty }$ in ${\mathcal {D}}'(U)$ such that each T_{i} has compact support and every compact subset $K\subseteq U$ intersects the support of only finitely many $T_{i},$ and the sequence of partial sums $(S_{j})_{j=1}^{\infty },$ defined by $S_{j}:=T_{1}+\cdots +T_{j},$ converges in ${\mathcal {D}}'(U)$ to T; in other words we have:
$T=\sum _{i=1}^{\infty }T_{i}.$
Recall that a sequence converges in ${\mathcal {D}}'(U)$ (with its strong dual topology) if and only if it converges pointwise.
Decomposition of distributions as sums of derivatives of continuous functionsedit
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 $T\in {\mathcal {D}}'(U)$ we can write:
where $P_{1},P_{2},\ldots$ are finite sets of multi-indices and the functions $f_{ip}$ are continuous.
Theorem^{[17]} — Let T be a distribution on U. For every multi-index p there exists a continuous function $g_{p}$ on U such that
any compact subset K of U intersects the support of only finitely many $g_{p},$ and
$T=\sum \nolimits _{p}\partial ^{p}g_{p}.$
Moreover, if T has finite order, then one can choose $g_{p}$ in such a way that only finitely many of them are non-zero.
Note that the infinite sum above is well-defined as a distribution. The value of T for a given $f\in {\mathcal {D}}(U)$ can be computed using the finitely many $g_{\alpha }$ that intersect the support of $f.$
Operations on distributionsedit
Many operations which are defined on smooth functions with compact support can also be defined for distributions. In general, if $A:{\mathcal {D}}(U)\to {\mathcal {D}}(U)$ 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 $A':{\mathcal {D}}'(U)\to {\mathcal {D}}'(U)$ by classic extension theorems of topology or linear functional analysis.^{[note 7]} The “distributional” extension of the above linear continuous operator A is possible if and only if A admits a Schwartz adjoint, that is another linear continuous operator B of the same type such that
$\langle Af,g\rangle =\langle f,Bg\rangle$,
for every pair of test functions. In that condition, B is unique and the extension A’ is the transpose of the Schwartz adjoint B.^{[citation needed]}^{[18]}^{[clarification needed]}
Preliminaries: Transpose of a linear operatoredit
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.^{[19]} 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
${}^{t}A:Y'\to X'\qquad {\text{ defined by }}\qquad {}^{t}A(y'):=y'\circ A,$
or equivalently, it is the unique map satisfying $\langle y',A(x)\rangle =\left\langle {}^{t}A(y'),x\right\rangle$ for all $x\in X$ and all $y'\in Y'$ (the prime symbol in $y'$ does not denote a derivative of any kind; it merely indicates that $y'$ is an element of the continuous dual space $Y'$). Since $A$ is continuous, the transpose ${}^{t}A:Y'\to X'$ is also continuous when both duals are endowed with their respective strong dual topologies; it is also continuous when both duals are endowed with their respective weak* topologies (see the articles polar topology and dual system for more details).
In the context of distributions, the characterization of the transpose can be refined slightly. Let $A:{\mathcal {D}}(U)\to {\mathcal {D}}(U)$ be a continuous linear map. Then by definition, the transpose of $A$ is the unique linear operator ${}^{t}A:{\mathcal {D}}'(U)\to {\mathcal {D}}'(U)$ that satisfies:
$\langle {}^{t}A(T),\phi \rangle =\langle T,A(\phi )\rangle \quad {\text{ for all }}\phi \in {\mathcal {D}}(U){\text{ and all }}T\in {\mathcal {D}}'(U).$
Since ${\mathcal {D}}(U)$ is dense in ${\mathcal {D}}'(U)$ (here, ${\mathcal {D}}(U)$ actually refers to the set of distributions $\left\{D_{\psi }:\psi \in {\mathcal {D}}(U)\right\}$) it is sufficient that the defining equality hold for all distributions of the form $T=D_{\psi }$ where $\psi \in {\mathcal {D}}(U).$ Explicitly, this means that a continuous linear map $B:{\mathcal {D}}'(U)\to {\mathcal {D}}'(U)$ is equal to ${}^{t}A$ if and only if the condition below holds:
$\langle B(D_{\psi }),\phi \rangle =\langle {}^{t}A(D_{\psi }),\phi \rangle \quad {\text{ for all }}\phi ,\psi \in {\mathcal {D}}(U)$
where the right-hand side equals $\langle {}^{t}A(D_{\psi }),\phi \rangle =\langle D_{\psi },A(\phi )\rangle =\langle \psi ,A(\phi )\rangle =\int _{U}\psi \cdot A(\phi )\,dx.$
Differential operatorsedit
Differentiation of distributionsedit
Let $A:{\mathcal {D}}(U)\to {\mathcal {D}}(U)$ be the partial derivative operator ${\tfrac {\partial }{\partial x_{k}}}.$ To extend $A$ we compute its transpose:
Therefore ${}^{t}A=-A.$ Thus, the partial derivative of $T$ with respect to the coordinate $x_{k}$ is defined by the formula
$\left\langle {\frac {\partial T}{\partial x_{k}}},\phi \right\rangle =-\left\langle T,{\frac {\partial \phi }{\partial x_{k}}}\right\rangle \qquad {\text{ for all }}\phi \in {\mathcal {D}}(U).$
With this definition, every distribution is infinitely differentiable, and the derivative in the direction $x_{k}$ is a linear operator on ${\mathcal {D}}'(U).$
More generally, if $\alpha$ is an arbitrary multi-index, then the partial derivative $\partial ^{\alpha }T$ of the distribution $T\in {\mathcal {D}}'(U)$ is defined by
$\langle \partial ^{\alpha }T,\phi \rangle =(-1)^{|\alpha |}\langle T,\partial ^{\alpha }\phi \rangle \qquad {\text{ for all }}\phi \in {\mathcal {D}}(U).$
Differentiation of distributions is a continuous operator on ${\mathcal {D}}'(U);$ this is an important and desirable property that is not shared by most other notions of differentiation.
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.^{[20]}
Differential operators acting on smooth functionsedit
A linear differential operator in $U$ with smooth coefficients acts on the space of smooth functions on $U.$ Given such an operator
${\textstyle P:=\sum _{\alpha }c_{\alpha }\partial ^{\alpha },}$
we would like to define a continuous linear map, $D_{P}$ that extends the action of $P$ on $C^{\infty }(U)$ to distributions on $U.$ In other words, we would like to define $D_{P}$ such that the following diagram commutes:
where the vertical maps are given by assigning $f\in C^{\infty }(U)$ its canonical distribution $D_{f}\in {\mathcal {D}}'(U),$ which is defined by:
$D_{f}(\phi )=\langle f,\phi \rangle :=\int _{U}f(x)\phi (x)\,dx\quad {\text{ for all }}\phi \in {\mathcal {D}}(U).$
With this notation, the diagram commuting is equivalent to:
$D_{P(f)}=D_{P}D_{f}\qquad {\text{ for all }}f\in C^{\infty }(U).$
To find $D_{P},$ the transpose ${}^{t}P:{\mathcal {D}}'(U)\to {\mathcal {D}}'(U)$ of the continuous induced map $P:{\mathcal {D}}(U)\to {\mathcal {D}}(U)$ defined by $\phi \mapsto P(\phi )$ is considered in the lemma below.
This leads to the following definition of the differential operator on $U$ called the formal transpose of $P,$ which will be denoted by $P_{*}$ to avoid confusion with the transpose map, that is defined by
For the last line we used integration by parts combined with the fact that $\phi$ and therefore all the functions $f(x)c_{\alpha }(x)\partial ^{\alpha }\phi (x)$ have compact support.^{[note 8]} Continuing the calculation above, for all $\phi \in {\mathcal {D}}(U):$
The Lemma combined with the fact that the formal transpose of the formal transpose is the original differential operator, that is, $P_{**}=P,$^{[21]} enables us to arrive at the correct definition: the formal transpose induces the (continuous) canonical linear operator $P_{*}:C_{c}^{\infty }(U)\to C_{c}^{\infty }(U)$ defined by $\phi \mapsto P_{*}(\phi ).$ We claim that the transpose of this map, ${}^{t}P_{*}:{\mathcal {D}}'(U)\to {\mathcal {D}}'(U),$ can be taken as $D_{P}.$ To see this, for every $\phi \in {\mathcal {D}}(U),$ compute its action on a distribution of the form $D_{f}$ with $f\in C^{\infty }(U)$:
${\begin{aligned}\left\langle {}^{t}P_{*}\left(D_{f}\right),\phi \right\rangle &=\left\langle D_{P_{**}(f)},\phi \right\rangle &&{\text{Using Lemma above with }}P_{*}{\text{ in place of }}P\\&=\left\langle D_{P(f)},\phi \right\rangle &&P_{**}=P\end{aligned}}$
We call the continuous linear operator $D_{P}:={}^{t}P_{*}:{\mathcal {D}}'(U)\to {\mathcal {D}}'(U)$ the differential operator on distributions extending $P$.^{[21]} Its action on an arbitrary distribution $S$ is defined via:
$D_{P}(S)(\phi )=S\left(P_{*}(\phi )\right)\quad {\text{ for all }}\phi \in {\mathcal {D}}(U).$
If $(T_{i})_{i=1}^{\infty }$ converges to $T\in {\mathcal {D}}'(U)$ then for every multi-index $\alpha ,(\partial ^{\alpha }T_{i})_{i=1}^{\infty }$ converges to $\partial ^{\alpha }T\in {\mathcal {D}}'(U).$
Multiplication of distributions by smooth functionsedit
A differential operator of order 0 is just multiplication by a smooth function. And conversely, if $f$ is a smooth function then $P:=f(x)$ is a differential operator of order 0, whose formal transpose is itself (that is, $P_{*}=P$). The induced differential operator $D_{P}:{\mathcal {D}}'(U)\to {\mathcal {D}}'(U)$ maps a distribution $T$ to a distribution denoted by $fT:=D_{P}(T).$ We have thus defined the multiplication of a distribution by a smooth function.
We now give an alternative presentation of the multiplication of a distribution $T$ on $U$ by a smooth function $m:U\to \mathbb {R} .$ The product $mT$ is defined by
$\langle mT,\phi \rangle =\langle T,m\phi \rangle \qquad {\text{ for all }}\phi \in {\mathcal {D}}(U).$
This definition coincides with the transpose definition since if $M:{\mathcal {D}}(U)\to {\mathcal {D}}(U)$ is the operator of multiplication by the function $m$ (that is, $(M\phi )(x)=m(x)\phi (x)$), then
Under multiplication by smooth functions, ${\mathcal {D}}'(U)$ is a module over the ring$C^{\infty }(U).$ 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 $\mathbb {R} ,$ then $m\delta =m(0)\delta ,$ and if $\delta ^{'}$ is the derivative of the delta distribution, then
The bilinear multiplication map $C^{\infty }(\mathbb {R} ^{n})\times {\mathcal {D}}'(\mathbb {R} ^{n})\to {\mathcal {D}}'\left(\mathbb {R} ^{n}\right)$ given by $(f,T)\mapsto fT$ is not continuous; it is however, hypocontinuous.^{[22]}
Example. The product of any distribution $T$ with the function that is identically 1 on $U$ is equal to $T.$
Example. Suppose $(f_{i})_{i=1}^{\infty }$ is a sequence of test functions on $U$ that converges to the constant function $1\in C^{\infty }(U).$ For any distribution $T$ on $U,$ the sequence $(f_{i}T)_{i=1}^{\infty }$ converges to $T\in {\mathcal {D}}'(U).$^{[23]}
If $(T_{i})_{i=1}^{\infty }$ converges to $T\in {\mathcal {D}}'(U)$ and $(f_{i})_{i=1}^{\infty }$ converges to $f\in C^{\infty }(U)$ then $(f_{i}T_{i})_{i=1}^{\infty }$ converges to $fT\in {\mathcal {D}}'(U).$
Problem of multiplying distributionsedit
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.^{[24]} 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 $\operatorname {p.v.} {\frac {1}{x}}$ is the distribution obtained by the Cauchy principal value
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) causal perturbation theory. 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.
Inspired by Lyons' rough path theory,^{[25]}Martin Hairer proposed a consistent way of multiplying distributions with certain structures (regularity structures^{[26]}), 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 functionedit
Let $T$ be a distribution on $U.$ Let $V$ be an open set in $\mathbb {R} ^{n}$ and $F:V\to U.$ If $F$ is a submersion then it is possible to define
$T\circ F\in {\mathcal {D}}'(V).$
This is the composition of the distribution $T$ with $F$, and is also called the pullback of $T$ along $F$, sometimes written
$F^{\sharp }:T\mapsto F^{\sharp }T=T\circ F.$
The pullback is often denoted $F^{*},$ although this notation should not be confused with the use of '*' to denote the adjoint of a linear mapping.
The condition that $F$ be a submersion is equivalent to the requirement that the Jacobian derivative $dF(x)$ 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.^{[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. The uniqueness of this extension is guaranteed since $F^{\#}$ is a continuous linear operator on ${\mathcal {D}}(U).$ 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 $\mathbb {R} ^{n}$ onto an open subset $U$ of $\mathbb {R} ^{n}$ change of variables under the integral gives:
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 $\mathbb {R} ^{n}$ then we denote by $f\ast g$the convolution of $f$ and $g,$ defined at $x\in \mathbb {R} ^{n}$ to be the integral
provided that the integral exists. If $1\leq p,q,r\leq \infty$ are such that ${\textstyle {\frac {1}{r}}={\frac {1}{p}}+{\frac {1}{q}}-1}$ then for any functions $f\in L^{p}(\mathbb {R} ^{n})$ and $g\in L^{q}(\mathbb {R} ^{n})$ we have $f\ast g\in L^{r}(\mathbb {R} ^{n})$ and $\|f\ast g\|_{L^{r}}\leq \|f\|_{L^{p}}\|g\|_{L^{q}}.$^{[29]} If $f$ and $g$ are continuous functions on $\mathbb {R} ^{n},$ at least one of which has compact support, then $\operatorname {supp} (f\ast g)\subseteq \operatorname {supp} (f)+\operatorname {supp} (g)$ and if $A\subseteq \mathbb {R} ^{n}$ then the value of $f\ast g$ on $A$ do not depend on the values of $f$ outside of the Minkowski sum$A-\operatorname {supp} (g)=\{a-s:a\in A,s\in \operatorname {supp} (g)\}.$^{[29]}
Importantly, if $g\in L^{1}(\mathbb {R} ^{n})$ has compact support then for any $0\leq k\leq \infty ,$ the convolution map $f\mapsto f\ast g$ is continuous when considered as the map $C^{k}(\mathbb {R} ^{n})\to C^{k}(\mathbb {R} ^{n})$ or as the map $C_{c}^{k}(\mathbb {R} ^{n})\to C_{c}^{k}(\mathbb {R} ^{n}).$^{[29]}
Translation and symmetryedit
Given $a\in \mathbb {R} ^{n},$ the translation operator $\tau _{a}$ sends $f:\mathbb {R} ^{n}\to \mathbb {C}$ to $\tau _{a}f:\mathbb {R} ^{n}\to \mathbb {C} ,$ defined by $\tau _{a}f(y)=f(y-a).$ This can be extended by the transpose to distributions in the following way: given a distribution $T,$the translation of $T$ by $a$ is the distribution $\tau _{a}T:{\mathcal {D}}(\mathbb {R} ^{n})\to \mathbb {C}$ defined by $\tau _{a}T(\phi ):=\left\langle T,\tau _{-a}\phi \right\rangle .$^{[30]}^{[31]}
Given $f:\mathbb {R} ^{n}\to \mathbb {C} ,$ define the function ${\tilde {f}}:\mathbb {R} ^{n}\to \mathbb {C}$ by ${\tilde {f}}(x):=f(-x).$ Given a distribution $T,$ let ${\tilde {T}}:{\mathcal {D}}(\mathbb {R} ^{n})\to \mathbb {C}$ be the distribution defined by ${\tilde {T}}(\phi ):=T\left({\tilde {\phi }}\right).$ The operator $T\mapsto {\tilde {T}}$ is called the symmetry with respect to the origin.^{[30]}
Convolution of a test function with a distributionedit
Convolution with $f\in {\mathcal {D}}(\mathbb {R} ^{n})$ defines a linear map:
which is continuous with respect to the canonical LF space topology on ${\mathcal {D}}(\mathbb {R} ^{n}).$
Convolution of $f$ with a distribution $T\in {\mathcal {D}}'(\mathbb {R} ^{n})$ can be defined by taking the transpose of $C_{f}$ relative to the duality pairing of ${\mathcal {D}}(\mathbb {R} ^{n})$ with the space ${\mathcal {D}}'(\mathbb {R} ^{n})$ of distributions.^{[32]} If $f,g,\phi \in {\mathcal {D}}(\mathbb {R} ^{n}),$ then by Fubini's theorem
Extending by continuity, the convolution of $f$ with a distribution $T$ is defined by
$\langle f\ast T,\phi \rangle =\left\langle T,{\tilde {f}}\ast \phi \right\rangle ,\quad {\text{ for all }}\phi \in {\mathcal {D}}(\mathbb {R} ^{n}).$
An alternative way to define the convolution of a test function $f$ and a distribution $T$ is to use the translation operator $\tau _{a}.$ The convolution of the compactly supported function $f$ and the distribution $T$ is then the function defined for each $x\in \mathbb {R} ^{n}$ 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 $\mathbb {C} ^{n}$ to $\mathbb {R} ^{n},$ the restriction of an entire function of exponential type in $\mathbb {C} ^{n}$ to $\mathbb {R} ^{n}$), then the same is true of $T\ast f.$^{[30]} If the distribution $T$ has compact support as well, then $f\ast T$ is a compactly supported function, and the Titchmarsh convolution theoremHörmander (1983, Theorem 4.3.3) implies that:
where $\operatorname {ch}$ denotes the convex hull and $\operatorname {supp}$ denotes the support.
Convolution of a smooth function with a distributionedit
Let $f\in C^{\infty }(\mathbb {R} ^{n})$ and $T\in {\mathcal {D}}'(\mathbb {R} ^{n})$ and assume that at least one of $f$ and $T$ has compact support. The convolution of $f$ and $T,$ denoted by $f\ast T$ or by $T\ast f,$ is the smooth function:^{[30]}
Let $M$ be the map $f\mapsto T\ast f$. If $T$ is a distribution, then $M$ is continuous as a map ${\mathcal {D}}(\mathbb {R} ^{n})\to C^{\infty }(\mathbb {R} ^{n})$. If $T$ also has compact support, then $M$ is also continuous as the map $C^{\infty }(\mathbb {R} ^{n})\to C^{\infty }(\mathbb {R} ^{n})$ and continuous as the map ${\mathcal {D}}(\mathbb {R} ^{n})\to {\mathcal {D}}(\mathbb {R} ^{n}).$^{[30]}
If $L:{\mathcal {D}}(\mathbb {R} ^{n})\to C^{\infty }(\mathbb {R} ^{n})$ is a continuous linear map such that $L\partial ^{\alpha }\phi =\partial ^{\alpha }L\phi$ for all $\alpha$ and all $\phi \in {\mathcal {D}}(\mathbb {R} ^{n})$ then there exists a distribution $T\in {\mathcal {D}}'(\mathbb {R} ^{n})$