Evolving Radiation Pattern (Sampling Kernel) due to Diffraction

It is well known that the far-field radiation pattern of an acoustic source is proportional to the spatial Fourier transform of the source's radiating amplitude distribution. In optics, this is called Fraunhofer diffraction theory (Goodman, 2005). By linearity of the Fourier transform, it follows that if the speaker array emits valid sampling kernels near the array (i.e., the radiation patterns overlap and add to a constant overall gain for a plane wave), then the far-field patterns will similarly overlap and add to a constant gain. All points in between must be valid as well, being progressive diffractions of the source distribution, but a rigorous proof with quantified approximation error would be nice to see (there are always terms to neglect, and it is good to be mindful of them).