Commit 796fff23 by Fabian Nuraddin Alexander Gabel 💬

### Merge branch 'dev' into 'master'

end of winter term 2021/22 lecture period

See merge request !20
parents f80b3cba 2eea8c9d
Pipeline #129991 passed with stage
in 1 minute and 57 seconds
 ... ... @@ -2,7 +2,7 @@ \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*)] ... ... @@ -96,8 +96,14 @@ \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} ... ...
 ... ... @@ -852,7 +852,7 @@ \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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