\chapter{A Convolution Formula for Vector-Valued Tempered Distributions}
\label{chapter:convolution}

\blockcquote{Bisognano1975}{%
  The extension to vector-valued tempered distributions is trivial.
}
Recall that the class $\schwartz{\RR^n}$ of complex-valued Schwartz functions on $\RR^n$
is closed under convolution, a operation that assigns to functions $f$ and $g$ a third one, $f * g$,
given by
\begin{equation*}
  (f*g)(x) = \int f(x-y) g(y) \, dy
  \qquad x \in \RR^n.
\end{equation*}

\begin{definition}{Convolution of a Distribution with a Test Function}{}
  Let $u \in \tempdistrib{\RR^n}$ be tempered distribution and
  let $f \in \schwartz{\RR^n}$ be a Schwartz test function.
  Then the \emph{convolution} of $u$ with $f$ is
  the tempered distribution $u * f \in \tempdistrib{\RR^n}$ defined by
  \begin{equation*}
    (u * f)(g) \defequal u(\tilde{f} * g) \qquad g \in \schwartz{\RR^n},
  \end{equation*}
  where $\tilde{f}(x) = f(-x)$ for all $x \in \RR^n$.
\end{definition}
It is well-known that the convolution can be expressed by the integral
\begin{equation*}
  (u * f)(g) = \int u(\tau_x \tilde{f}@@) g(x) \, dx
\end{equation*}
emphasizing its character of a smoothing operation.
The purpose of this appendix is to state and prove
a vector-valued version of this formula.

Let $X$ be a complex Banach space.
Denote by $C^{\infty}(\RR^n,X)$  the vector space of all functions $f : \RR^n \to X$
such that the derivatives $\partial^{\alpha} f$ exist and are continuous for all multi-indices $\alpha \in \NN^n$.
We define the space $\schwartz{\RR^n,X}$ of \emph{$X$-valued Schwartz functions} to be the vector space
\begin{equation*}
  \schwartz{\RR^n,X} \defequal \braces{f \in C^{\infty}(\RR^n,X) \vcentcolon \norm{f}_{\alpha,\beta} < \infty \forall \alpha,\beta \in \NN^n}
\end{equation*}
equipped with the locally convex topology induced by the family of seminorms
\begin{equation*}
  \norm{f}_{\alpha,\beta} = \sup_{x \in \RR^n} \abs{x^{\alpha}} \norm{\partial^{\beta} f(x)}_X.
\end{equation*}
We define the space $\tempdistrib{\RR^n,X}$ of \emph{$X$-valued tempered distributions} to be the vector space
\begin{equation*}
  \tempdistrib{\RR^n,X} \defequal \BoundedLinearOperators[\big]{\schwartz{\RR^n},X}.
\end{equation*}
equipped with the bounded convergence topology.

\begin{proposition}{Vector-Valued Convolution Formula}{}
  Let $v \in \tempdistrib{\RR^n,X}$ be tempered distribution with values in a Banach space $X$, and
  let $f \in \schwartz{\RR^n}$ be a Schwartz test function. Then one has
  \begin{equation*}
    (v * f)(g) = \int v(\tau_x \tilde{f}@@) g(x) \, dx \qquad g \in \schwartz{\RR^n}.
  \end{equation*}
\end{proposition}

Der Beweis ist in Arbeit ;)

%\nomenclature[B]{$\BoundedLinearOperators{X,Y}$}{bounded linear operators from $X$ to $Y$\nomnorefpage}

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