Phase Retrieval of Real-valued Functions in Sobolev Space

The Sobolev space HS（Rd） 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 { φj^r,k}∈C 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 （Rd） can be determined, up to a global sign, by the phaseless measurements {｜（ f, φj^r,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（Rd） ∩ C1（Rd）, and prove that our iterated reconstruction procedure guarantees uniform convergence for any function f ∈ Hs （Rd）∩ C1 （Rd） whose Fourier transform f is L1-integrable. Moreover, numerical simulations are conducted to test the efficiency of the reconstruction algorithm.