Give an example of distribution $u \in \mathcal{D}'((0,+\infty)) $ that is not extendible to $\mathbb{R}$

Question

i.e.

find a $u \in \mathcal{D}'((0,+\infty))$, such that for any $v \in \mathcal{D}'(\mathbb{R})$,

$v|_{(0, +\infty)} \neq u$. In order to find such an example,

my question:

I tried to prove $e^{1/x^2}$ is an example, i.e. there is a distribution u such that $ u|_{(0, +\infty)} = e^{1/x^2}$, but I cannot find a contradiction.

Answer 1

An example is $u=\sum\limits_{n=1}^\infty \delta^{(n)}_{1/n}$, that ist $u(\varphi)= \sum\limits_{n=1}^\infty (-1)^n \varphi^{(n)}(1/n)$ for $\varphi\in\mathscr D((0,\infty))$. A neat argument that it cannot be extended to $\mathscr D'(\mathbb R)$ (despite the intuitive but not very precise idea that the series ''does not make sense'' for $\varphi\in\mathscr D(\mathbb R)$) is that distributions with compact support in $\mathbb R$ have finite order.

For open $A\subseteq B$ the restriction map $\mathscr D'(B)\to \mathscr D'(A)$ is surjective if and only if $A$ is some union of connected components of $B$.