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:
$\mathrm{F}\left\lbrace\sin^3\left(at+b\right)*\cos^3\left(ct+d\right)\right\rbrace$.
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 [1] 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:
$\int_{-\infty}^{\infty}\sin^3\left(a\tau+b\right)\cos^3\left(c(\tau-t)+d\right)\mathrm{d}\tau$
This also I think is non-convergent.
So am I right to think that this convolution and its Fourier transform are not defined?
[1] Zemanian: Distribution Theory and Transform Analysis