# What is the requirement to reconstruct a spatial domain signal if we sample in the frequency domain?

I've come across an interesting question with regarding to signal reconstruction.

The sampling theorem states that a signal in the time or spatial domain must be sampled at twice a rate twice the bandwidth in order to perfectly reconstruct it. The only requirement of the input signal is that it has finite bandwith.

However, what is the analog to this if we instead sampled in the frequency domain?

My best guess is that the signal in the spatial domain must also be finite (i.e. image must not be infinitely large). My reasoning for this is that windowing of the input signal causes frequencies to "smear" thus giving them a width in which we can sample. If we were to have an infinite sinusoid input signal, the frequency component would be an impulse which would be impossible to sample.

However, I'm struggling to formal define this frequency sampling interval in terms of the input signal size.