I will unashamedly say that this was at least spurred by homework. However I have gone far beyond the syllabus of the course and still can't find an authoritative answer. And it seems an interesting question to me that I doubt the professor will answer (if I wasn't ashamed to ask).
I am asked to calculate the Fourier transform of the convolution of two signals, for generality:
I have tried two approaches.
First, take the product of the Fourier transforms of the sinusoids. This leads to an expression that contains terms of the form $\delta(\omega-a)\delta(\omega-b)$. According to  and unless I missed it, the product of two distributions, unlike other operations, is not defined.
Secondly calculate the convolution directly. This leads me to an integral of the form:
This also I think is non-convergent.
So am I right to think that this convolution and its Fourier transform are not defined?
 Zemanian: Distribution Theory and Transform Analysis