How to correctly convolve two delta functions?

[badkitty]

How do I correctly compute the convolution of two delta functions? For example, if I want to compute $\delta(\omega)\otimes\delta(\omega)$, I should integrate $$\int_{-\infty}^\infty \delta(\omega-\Omega)\delta(\Omega) d\Omega$$
This integrand "fires" at two places: $\Omega = 0$ and $\Omega = \omega$, and evaluates to $\delta(\omega)$ in either case. That leads me to say that $\delta(\omega)\otimes\delta(\omega) = 2\delta(\omega)$, which somehow feels wrong... I want it to be just $\delta(\omega)$.

Anyone know the correct treatment here?

 

[RUber]

In your example, $\int_{-\infty}^\infty \delta(\omega-\Omega) \delta(\Omega) \, d\Omega$ is only non-zero if both $\omega - \Omega = 0$ and $\omega = 0$.
So for this, you end up with $\delta(\omega)$.