Holograms made using only the phase of an object's Fourier transform (FT) have been demonstrated by others to contain enough information to produce a noisy but recognizable reconstruction. However, no mathematical analysis has been available. In this paper, a theory is developed that explains many features observed in numerical experiments. It was found that splitting the original function into a sum of even and odd parts results in a useful equation for the FT phase. This made it possible for example to see mathematically why edge enhancement occurs and why the phase from one object and the amplitude from another will produce an image of the former. It was also found that zero crossings in the FT of the even part of the input function play a dominant role in determining the nature of a reconstruction.