I have a question regarding discrete inverse Fourier transform, and no answer I found on the internet seem to be satisfying. This might be because I do not fully get some of them, so please excuse my ignorance.
Suppose you have full knowledge of a frequency-domain function $\hat{f}(\omega)$. It is a continuous, well-behaved bump function. You want to compute its inverse Fourier transform $f(t) = \Re \int_{-\infty}^{+\infty} \hat{f}(\omega) e^{j\omega t} d\omega$, but this cannot be done analytically, so you compute a discrete inverse Fourier transform. You know the interval $ \left[ \omega_{\min}, \omega_{\max} \right]$ on which it is reasonable to numerically compute the IDFT, so you now have to sample $\hat{f}(\omega)$ on this interval.
My question is the following: is there an equivalent of the sampling theorem for the inverse Fourier transform ? Is there a generic rule to sample my frequency-domain signal, as there is for time-domain ? I noticed that I obtain something similar to aliasing when I undersample $\hat{f}(\omega)$.
Side note : all of this is used in a finite element code to compute a complex source. As such the IDFT will be performed at least hundreds of thousand of times, so the question to limit its cost.
Thanks in advance !