Editing Distribution (mathematics) (section)

====Convolution of distributions====

It is also possible to define the convolution of two distributions <math>S</math> and <math>T</math> on <math>\R^n,</math> provided one of them has compact support. Informally, to define <math>S \ast T</math> where <math>T</math> has compact support, the idea is to extend the definition of the convolution <math>\,\ast\,</math> to a linear operation on distributions so that the associativity formula
<math display=block>S \ast (T \ast \phi) = (S \ast T) \ast \phi</math>
continues to hold for all test functions <math>\phi.</math><ref>{{harvnb|Hörmander|1983|loc=§IV.2}} proves the uniqueness of such an extension.</ref>

It is also possible to provide a more explicit characterization of the convolution of distributions.{{sfn|Trèves|2006|loc=Chapter 27}} Suppose that <math>S</math> and <math>T</math> are distributions and that <math>S</math> has compact support. Then the linear maps
<math display=block>\begin{alignat}{9}
\bullet \ast \tilde{S} : \,& \mathcal{D}(\R^n) && \to    \,&& \mathcal{D}(\R^n) && \quad \text{ and } \quad && \bullet \ast \tilde{T} : \,&& \mathcal{D}(\R^n) && \to    \,&& \mathcal{D}(\R^n) \\
    & f                       && \mapsto\,&& f \ast \tilde{S}    &&       &&     && f                       && \mapsto\,&& f \ast \tilde{T} \\
\end{alignat}</math>
are continuous. The transposes of these maps:
<math display=block>{}^{t}\left(\bullet \ast \tilde{S}\right) : \mathcal{D}'(\R^n) \to \mathcal{D}'(\R^n) \qquad {}^{t}\left(\bullet \ast \tilde{T}\right) : \mathcal{E}'(\R^n) \to \mathcal{D}'(\R^n)</math>
are consequently continuous and it can also be shown that{{sfn|Trèves|2006|pp=284-297}}
<math display=block>{}^{t}\left(\bullet \ast \tilde{S}\right)(T) = {}^{t}\left(\bullet \ast \tilde{T}\right)(S).</math>

This common value is called {{em|the '''convolution''' of <math>S</math> and <math>T</math>}} and it is a distribution that is denoted by <math>S \ast T</math> or <math>T \ast S.</math> It satisfies <math>\operatorname{supp} (S \ast T) \subseteq \operatorname{supp}(S) + \operatorname{supp}(T).</math>{{sfn|Trèves|2006|pp=284-297}} If <math>S</math> and <math>T</math> are two distributions, at least one of which has compact support, then for any <math>a \in \R^n,</math> <math>\tau_a(S \ast T) = \left(\tau_a S\right) \ast T = S \ast \left(\tau_a T\right).</math>{{sfn|Trèves|2006|pp=284-297}} If <math>T</math> is a distribution in <math>\R^n</math> and if <math>\delta</math> is a [[Dirac measure]] then <math>T \ast \delta = T = \delta \ast T</math>;{{sfn|Trèves|2006|pp=284-297}} thus <math>\delta</math> is the [[identity element]] of the convolution operation. Moreover, if <math>f</math> is a function then <math>f \ast \delta^{\prime} = f^{\prime} = \delta^{\prime} \ast f</math> where now the associativity of convolution implies that <math>f^{\prime} \ast g = g^{\prime} \ast f</math> for all functions <math>f</math> and <math>g.</math>

Suppose that it is <math>T</math> that has compact support. For <math>\phi \in \mathcal{D}(\R^n)</math> consider the function
<math display=block>\psi(x) = \langle T, \tau_{-x} \phi \rangle.</math>

It can be readily shown that this defines a smooth function of <math>x,</math> which moreover has compact support. The convolution of <math>S</math> and <math>T</math> is defined by
<math display=block>\langle S \ast T, \phi \rangle = \langle S, \psi \rangle.</math>

This generalizes the classical notion of [[convolution]] of functions and is compatible with differentiation in the following sense: for every multi-index <math>\alpha.</math>
<math display=block>\partial^\alpha(S \ast T) = (\partial^\alpha S) \ast T = S \ast (\partial^\alpha T).</math>

The convolution of a finite number of distributions, all of which (except possibly one) have compact support, is [[associative]].{{sfn|Trèves|2006|pp=284-297}}

This definition of convolution remains valid under less restrictive assumptions about <math>S</math> and <math>T.</math><ref>See for instance {{harvnb|Gel'fand|Shilov|1966–1968|loc=v. 1, pp. 103–104}} and {{harvnb|Benedetto|1997|loc=Definition 2.5.8}}.</ref>

The convolution of distributions with compact support induces a continuous bilinear map <math>\mathcal{E}' \times \mathcal{E}' \to \mathcal{E}'</math> defined by <math>(S,T) \mapsto S * T,</math> where <math>\mathcal{E}'</math> denotes the space of distributions with compact support.{{sfn|Trèves|2006|p=423}} However, the convolution map as a function <math>\mathcal{E}' \times \mathcal{D}' \to \mathcal{D}'</math> is {{em|not}} continuous{{sfn|Trèves|2006|p=423}} although it is separately continuous.{{sfn|Trèves|2006|p=294}} The convolution maps <math>\mathcal{D}(\R^n) \times \mathcal{D}' \to \mathcal{D}'</math> and <math>\mathcal{D}(\R^n) \times \mathcal{D}' \to \mathcal{D}(\R^n)</math> given by <math>(f, T) \mapsto f * T</math> both {{em|fail}} to be continuous.{{sfn|Trèves|2006|p=423}} Each of these non-continuous maps is, however, [[separately continuous]] and [[hypocontinuous]].{{sfn|Trèves|2006|p=423}}