One general statement can be made. If one finds more than one solution to a given deconvolution problem fitting a given data set, then subtracting any two solutions should give a function whose visibility has to vanish everywhere on the data set. Such a brightness distribution, which contains only unmeasured spatial frequencies, is appropriately called an ``invisible distribution''. Our extra- /inter- polation problem consists in finding the right invisible distribution to add to the visible one!
One constraint often mentioned is the positivity of the brightness of each pixel. To see how powerful this can be, take a sky with just one point source at the field centre. The total flux and two visibilities on baselines suffice to pin down the map completely. The only possible value for all the remaining visibilities is equal to these numbers, which are themselves equal. One cannot add any invisible distribution to this because it is bound to go negative somewhere in the vast empty spaces around our source. But this is an extreme case. The power of positivity diminishes as the field gets filled with emisssion.
Another interesting case is when the emission is known to be confined to a window in the map plane. Define a function inside the window and zero outside. Let be its Fourier transform. Multiplying the map by makes no difference. In Fourier space, this condition is quite non-trivial, viz . Notice how the convolution on the right transfers information from measured to unmeasured parts of the plane, and couples them.