Commit 796fff23 authored by Fabian Nuraddin Alexander Gabel's avatar Fabian Nuraddin Alexander Gabel 💬
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end of winter term 2021/22 lecture period

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\begin{ex}[(Distribution Theory)]
\label{ex:distribution}
\index{Distribution}
In Section~\ref{sec:distributions}, we introduced the space of test functions $\mathcal{D} \coloneqq \CC^\infty(\Omega)$, $\Omega \subset \R^d$ and the corresponding dual space $\mathcal{D}'$.
In Section~\ref{sec:distributions}, we introduced the space of test functions $\mathcal{D} \coloneqq \CC^\infty(\Omega)$, $\Omega \subset \R^d$, and the corresponding dual space $\mathcal{D}'$.
An element $F \in \mathcal{D}'$ is called \emph{distribution} and is a linear continuous mapping from $\mathcal{D}$ to $\R$.
In this exercise, we will derive further properties of distributions.
\begin{enumerate}[label=(\alph*)]
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\emph{Note: We have even more: Every function $x \mapsto |x|^p$ is convex for $p \geq 1$ and strictly convex for $p > 1$, see e.g. \cite[Example 6.23]{bredies}}
\end{enumerate}
\item Let $J \colon U \to \mathbb{R}$ be convex and $\lambda \geq 0$.
Show that $\lambda J \colon U \to \mathbb{R}, u \mapsto \lambda \cdot J(u)$ is also convex.
\item Let $J, K \colon U \to \mathbb{R}$ be convex.
Show that $J + K \colon U \to \mathbb{R}, u \mapsto J(u) + K(u)$ is also convex.
\item Let $\Phi \colon U \to V$ be \emph{affine linear}\index{Affine linear}, i.e. $\Phi(u) = c + Au$ for a linear operator $A \colon U \to V$ and a constant $c \in V$.
If $J \colon V \to \R$ is \emph{convex}, then also the composition $J \circ \Phi$ is convex.
\item Let $J \colon U \to\R$ be convex and $\varphi \colon [0,\infty) \to \R$ be convex and \emph{monotone increasing}, i.e.\@ $\varphi(x) \leq \varphi(y)$ if $x \leq y$.
Then $\varphi \circ J$ is also convex.
\end{enumerate}
\end{ex}
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\vgap{20mm}
\item Inversion of the previous point:
\begin{equation*}
g(x) = x_1^{\alpha_1} \cdots x_d^{\alpha_d} f(x) \quad\implies\quad \hat g(z) = \ii^{\alpha_1 + \cdots + \alpha_d} \frac{\displaystyle \partial^{\alpha_1 + \cdots + \alpha_d}}{\displaystyle \partial x_1^{\alpha_1} \cdots \partial x_d^{\alpha_d}} \hat f(z)
g(x) = x_1^{\alpha_1} \cdots x_d^{\alpha_d} f(x) \quad\implies\quad \hat g(z) = \ii^{\alpha_1 + \cdots + \alpha_d} \frac{\displaystyle \partial^{\alpha_1 + \cdots + \alpha_d}}{\displaystyle \partial z_1^{\alpha_1} \cdots \partial z_d^{\alpha_d}} \hat f(z)
\end{equation*}
\item\label{it:faltungssatz} ~\\[-14mm]
\begin{flalign*}
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