Prove there do not exists such distribution.

Question

If $u\in \mathcal{D}'(0, \infty)$ such that $$\langle u,\varphi\rangle=\sum^{\infty}_{n=1}\varphi^{(n)}\Big(\frac{1}{n}\Big)$$ Prove that there do not exists any $v\in \mathcal{D}'(\mathbb{R})$ whose restriction to $(0,\infty)$ is $u$.

Answer 1

Let $\eta\in\mathcal{D}(\mathbb{R})$ be such that $\eta(x)=1$ for $x\in(0,1)$. Let $\varphi\in\mathcal{D}(\mathbb{R})$ be defined by $\varphi(x)=\eta(x)\,e^x$. If $v\in\mathcal{D'}(\mathbb{R})$ is an extension of $u$, what would $\langle v,\varphi\rangle$ be?