New findings on the Zeros of Fourier Integrals

Abstract

There has been considerable interest over the years in the so-called Fourier phase retrieval problem. Applications abound and it still remains a difficult problem. Based on the analytic properties of bandlimited functions, it is well known that the 1D phase retrieval problem generally has no unique solution. The lack of uniqueness arises from the existence of complex zeros located off the real axis, i.e. in the complex plane. These analytic properties also suggest that in 2D or higher dimensional problems there is a unique solution1. The question is how to find this “unique” solution, especially when only noisy sampled power spectral data are available. Indeed, the meaning of uniqueness needs to be redefined under these circumstances.