Phase Retrieval of Real-valued Functions in Sobolev Space

The Sobolev space H^s(R^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 {ϕ_j,k^γ} ⊆ H^-s(R^d) to the phase retrieval problem for the real-valued functions in H^s(R^d). We prove that any real-valued function f ∈ H^s(R^d) can be determined, up to a global sign, by the phaseless measurements {|<f, ϕ_j,k^γ>|}. 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 H^s(R^d) ∩ C¹(R^d), and prove that our iterated reconstruction procedure guarantees uniform convergence for any function f ∈ H^s(R^d) ∩ C¹(R^d) whose Fourier transform f is L¹-integrable. Moreover, numerical simulations are conducted to test the efficiency of the reconstruction algorithm.