Problem set 7

Problem 1¶

Let $U$ be an open interval on $\mathbb{R}$. Let $T$ be a distribution on $U$. If $T' = 0$, then $T$ is a constant, i.e. for every $\varphi\in \mathscr{C}_c^{\infty}(U)$, $(T,\varphi) = c\varphi$ for some constant $c$.

Problem 2¶

Let $T\in \mathcal D'$, $\phi\in \mathscr{C}_c^{\infty}$. Define the convolution of $T*\phi$ to be a function given by $(T*\phi)(x)\mapsto (T(y),\phi(x - y))$.

(1) Show that under the assumptions, $T*\phi$ is smooth.

(2) Show that $T*\phi(x) = (T(y),\tau_x\phi^{-}(y))$.

(3) Let $\phi\in \mathscr{C}_c^{l}(\mathbb{R})$, $\psi \in \mathscr{C}_c^{0}(\mathbb{R})$. Consider the Riemann sum $R_h(x):=\sum_{k\in \mathbb{Z}} \phi(x - kh)\psi(kh)h$ of $\phi*\psi(x) = \int_{-\infty}^{+\infty}\phi(x - y)\psi(y)dy$. Show that $R_h(x)\to \phi*\psi(x)$ in $\mathscr{C}_c^{l}$ when $h\to 0$.

Problem 3¶

Let $\psi_\epsilon\in \mathscr{C}_c^{\infty}(\mathbb{R})$ with $\int \psi_\epsilon \equiv 1$ for every ε and $supp(\psi_\epsilon)\to \{0\}$ (in the sense that $\bigcap supp \psi_\epsilon = \{0\}$). For $T\in \mathcal D'$, by problem 2, $T*\psi_\epsilon\in \mathscr{C}^{\infty}(\mathbb{R})$. Show that $(T*\psi_\epsilon,\varphi) \to (T,\varphi)$ for every $\varphi\in \mathcal D$ when $\epsilon \to 0$.

Problem 4¶

Show that $T_n\to T$ in $\mathcal D'$ implies $T'_n\to T'$ in $\mathcal D'$.