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I want to show, that for $u' \in \mathcal{D}'(\mathbb{R}^n)$

supp $u$ = $\{ 0 \}$ iff there exist numbers $m \in \mathbb{N}, c_{\alpha} \in \mathbb{K}$ such that $u = \sum_{|\alpha| \le m} c_{\alpha} D^{\alpha} \delta_0$

($\mathbb{K}$ equals $\mathbb{R}$ or $\mathbb{C}$). One direction is simple, but I find the other difficult.

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I provide some hints.

  1. $u$ has a finite order.
  2. If $k$ is an integer and $\chi\in\mathcal D(\Bbb R^n)$, with $\chi=1$ in a neighborhood of $0$ then for all $\phi\in\mathcal D(\Omega)$ we can find functions $g_{\alpha}$ such that $$\phi(x)=\sum_{|\beta|\leq k}\frac{x^k}{\beta!}\partial^{\beta}\phi(0)\chi(x)+\sum_{|\alpha|=k+1}x^{\alpha}g_{\alpha}(x).$$
  3. Let $\psi\in\mathcal D(\mathbb R^n)$ such that $\partial^{\alpha}\psi(0)=0$ if $|\alpha|\leq k$. We have to show that $u(\psi)=0$. Show that if $|\alpha|\leq k$, we can find a constant $C$ such that for all $x\in\Bbb R^n$: $$|\partial^{\alpha}\psi(x)|\leq C|x|^{k+1-|\alpha|}.$$
  4. Let $\zeta$ a smooth function which is equal to $0$ in a neighborhood of $0$ and $1$ outside $B(0,1)$. Show that the sequence $\psi_j(x):=\psi(x)\zeta(jx)$ converges to $\psi$ for the topology of $\mathcal D^k(\Bbb R^n)$.
  5. Conclude that $u\psi=0$.
  6. Conclude the problem.
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