Discrete inverse Fourier transform 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 !
The Fourier Transform is (mostly) symmetric in both directions so the same sampling rules apply. 

The minimum frequency resolution is determined by the length in the time domain, the same way that the minimum time resolution is determined by the "length" in the frequency domain. 

That creates a bit of theoretical problem: in order for the required frequency resolution to be finite, the length in the time domain has to be finite as well. However, a signal that has finite length in one domain has infinite length in the other. So you can't actually sample a signal in both domains at the same time. 

In practice, you need to work around this by replacing "must be zero outside the supported interval" by "must be small enough outside the supported interval". So you accept some residual aliasing in either domain, provided you can make it small enough for your specific application