By generalized functions I mean functionals from $C^\infty_c(U)\to \mathbb{R}$ as defined in wikipedia. I want to compute:
$$ \operatorname{comb}(\omega)*\operatorname{pv}(\frac{1}{\omega}) $$
But the operation is a convolution between two generalized functions. Generally speaking, only linear operations can be extended from the regular functions into the generalized functions, but despite that it is very natural to say:
$$\operatorname{comb}(\omega)*\operatorname{pv}(\frac{1}{\omega}) = \sum_{n\in \mathbb{Z}} \operatorname{pv}(\frac{1}{\omega-n})$$
As that is the way a convolution with $\operatorname{comb}$ affects regular functions. Another way to "compute" this expression without convolving distributions is by:
$$ \mathcal{F}(\operatorname{comb}(\omega)*\operatorname{pv}(\frac{1}{\omega})) = \mathcal{F}(\operatorname{comb}(\omega))\cdot \mathcal{F}(\operatorname{pv}(\frac{1}{\omega}))$$
Where here the Fourier transform of the Cauchy principal value is (proportional to) the sign function, which is a regular function.
I'm aware that more expansive definitions of generalized functions exist, that for example allow multiplication. That fact is mentioned on this wikipedia page. Sadly, I don't have access to their source article so I don't know what those definitions are or if they help me solve my problem.
Is the calculation I want to do allowed? How could I calculate it formally?