Editing Distribution (mathematics) (section)

====Convolution of a test function with a distribution====

Convolution with <math>f \in \mathcal{D}(\R^n)</math> defines a linear map:
<math display=block>\begin{alignat}{4}
C_f : \,& \mathcal{D}(\R^n) && \to    \,&& \mathcal{D}(\R^n) \\
        & g                 && \mapsto\,&& f \ast g \\
\end{alignat}</math>
which is [[continuous function|continuous]] with respect to the canonical [[LF space]] topology on <math>\mathcal{D}(\R^n).</math>

Convolution of <math>f</math> with a distribution <math>T \in \mathcal{D}'(\R^n)</math> can be defined by taking the transpose of <math>C_f</math> relative to the duality pairing of <math>\mathcal{D}(\R^n)</math> with the space <math>\mathcal{D}'(\R^n)</math> of distributions.{{sfn|Trèves|2006|loc=Chapter 27}} If <math>f, g, \phi \in \mathcal{D}(\R^n),</math> then by [[Fubini's theorem]]
<math display=block>\langle C_fg, \phi \rangle = \int_{\R^n}\phi(x)\int_{\R^n}f(x-y) g(y) \,dy \,dx = \left\langle g,C_{\tilde{f}}\phi \right\rangle.</math>

Extending by continuity, the convolution of <math>f</math> with a distribution <math>T</math> is defined by
<math display=block>\langle f \ast T, \phi \rangle = \left\langle T, \tilde{f} \ast \phi \right\rangle, \quad \text{ for all } \phi \in \mathcal{D}(\R^n).</math>

An alternative way to define the convolution of a test function <math>f</math> and a distribution <math>T</math> is to use the translation operator <math>\tau_a.</math> The convolution of the compactly supported function <math>f</math> and the distribution <math>T</math> is then the function defined for each <math>x \in \R^n</math> by
<math display=block>(f \ast T)(x) = \left\langle T, \tau_x \tilde{f} \right\rangle.</math>

It can be shown that the convolution of a smooth, compactly supported function and a distribution is a smooth function. If the distribution <math>T</math> has compact support, and if <math>f</math> is a polynomial (resp. an exponential function, an analytic function, the restriction of an entire analytic function on <math>\Complex^n</math> to <math>\R^n,</math> the restriction of an entire function of exponential type in <math>\Complex^n</math> to <math>\R^n</math>), then the same is true of <math>T \ast f.</math>{{sfn|Trèves|2006|pp=284-297}} If the distribution <math>T</math> has compact support as well, then <math>f\ast T</math> is a compactly supported function, and the [[Titchmarsh convolution theorem]] {{harvtxt|Hörmander|1983|loc=Theorem 4.3.3}} implies that:
<math display=block>\operatorname{ch}(\operatorname{supp}(f \ast T)) = \operatorname{ch}(\operatorname{supp}(f)) + \operatorname{ch} (\operatorname{supp}(T))</math>
where <math>\operatorname{ch}</math> denotes the [[convex hull]] and <math>\operatorname{supp}</math> denotes the support.