# Characterization of delta Distribution

I encountered this problem while solving a problem related to characterization of delta function upto constant multiple.

$\phi \in D(R)$, Space of compactly supported infinitely differentiable functions. If $\phi(0)=0$, Then there exist a $\psi \in D(R)$ such that $\phi = x\psi$

Original question is as follow:

If $xT=0$, where T is a distribution. Then, $T = c\delta$

If above proposition could be proved, then this characterization would follow.

• I know this question is very old, but I've run into the same problem recently. I see how Hadamard's Lemma gives you an infinitely differentiable $\psi$, but why is $\psi$ also compactly supported? – artificial_moonlet Oct 3 '14 at 1:45
• if $\phi =x\psi$ then one of the two functions has compact support iff the other one has... – Dan Petersen Oct 3 '14 at 5:33
• Ah, I misread the question prompt. I didn't realize we were also assuming $\phi(0) = 0$. – artificial_moonlet Oct 3 '14 at 12:28
For the original question, you could use the fact that the support of $T$ can only be at the origin. Then a theorem (I forgot the name, it is in the distribution book of Kolk and Duistermaat) states that we can write
$T=\sum_{i=0}^{n}c_i\delta^{(i)}$
for some $n$. It easily follows that $c_i=0$ for $i>0$.