From c66c3bc73d5d627ec7051e9ada6316c98ae072e0 Mon Sep 17 00:00:00 2001
From: Justin Gassner <justin.gassner@mailbox.org>
Date: Wed, 29 May 2024 14:04:59 +0200
Subject: weiter

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+\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}
+
+\chapterbib
+\cleardoublepage
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