Prove that $f$ is a linear combination of $\delta^{(k)}$ for $k=0, 1\ldots, n-1$.

Question

Let $f$ is a tempered distribution such that $x^nf =0$ for an integer $n$. Prove that $f$ is a linear combination of $\delta^{(k)}$ for $k=0, 1\ldots, n-1$.

$\delta$ is Dirac delta function.

Please help me. I have no idea.

Answer 1

Let $\phi \in \mathcal S(\mathbb R)$ and $P = \sum_{k=0}^{n-1} \frac{\phi^{(k)}(0)}{k!} x^k$ its Taylor polynomial with degree $k-1$ around $0$.

Then, there is a $\mathcal C^{+\infty}$ function $\psi$ such that [see here]: $$\phi = P + x^n \psi$$

Let $\chi$ be a smooth compactly supported function that is $1$ on a neighborhood of $0$. Then, $\chi P$ and $\chi x^n \psi \in \mathcal S(\mathbb R)$.

Therefore : \begin{align} \langle f,\phi\rangle &= \langle f , \chi \phi\rangle \\ &= \langle f, \chi P\rangle +\langle x^n f, \psi\rangle \\ &= \langle f,\chi P\rangle \end{align}

Therefore $\langle f,\phi\rangle$ only depends (linearly) on $\phi(0), \phi^{(1)}(0),\ldots, \phi^{(n-1)}(0)$, ie $f$ is a linear combination of $\delta_0,\delta^{(1)}_0,\ldots,\delta_0^{(n-1)}$