Phase Retrieval of Real-valued Functions in Sobolev Space

The Sobolev space Hs(.d) with s > d/2 contains many important functions such as the bandlimited or rational ones. In this paper we propose a sequence of measurement functions {{.~.j,k}.H.s(Rd) to the phase retrieval problem for the real-valued functions in Hs(.d). We prove that any real-valued function f . Hs(.d) can be determined, up to a global sign, by the phaseless measurements {|.f,.~.j,k.|}. It is known that phase retrieval is unstable in infinite dimensional spaces with respect to perturbations of the measurement functions. We examine a special type of perturbations that ensures the stability for the phase-retrieval problem for all the real-valued functions in Hs(.d) . C1(.d), and prove that our iterated reconstruction procedure guarantees uniform convergence for any function f . Hs(.d).C1(.d) whose Fourier transform f^ is L1-integrable. Moreover, numerical simulations are conducted to test the efficiency of the reconstruction algorithm.