Dirac delta function divided by Dirac delta function

Question

Is the following defined: (Dirac delta function divided by Dirac delta function)

$$f = \frac{\delta}{\delta} = ?$$

Answer 1

I don't know if the following is what you are looking for, but: To give the division a sense, what you can do is look for functions $\phi \in \mathcal E^0(\mathbb R)$ (that is continuous functions $\mathbb R \to \mathbb R$ that fulfill $\phi \delta = \delta$. As for any $\psi \in \mathcal D(\mathbb R)$ we have $$(\phi \delta)(\psi) =\delta(\phi\psi) = \phi(0)\psi(0) = \phi(0)\delta(\psi), $$ that is $\phi \delta = \phi(0)\delta$, so we have $$\phi \delta = \delta \iff \phi(0) = 1. $$