Fourier analysis: multiplication of a chirp signal by a function in the frequency space.

I have a chirp signal H (t), from the time t = 0 to t = 280. Meanwhile, the frequency of the signal f goes from 10 Hz to 730 Hz. I also have a frequency function F (f). Now I want to modulate the signal H (t) using F (f). For that, my idea was to do FT in H (t) and then multiply it by F (f). After that, an inverse FT.
But I'm not sure how to do this in Mathematica.

Can anyone help with this?