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 {$\left\{{\tilde{\varphi }}_{j,k}^{\gamma }\right\}\subseteq H^{-s}\left(R^d\right)$ 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 $\{|⟨f,{\tilde{\varphi }}_{j,k}^{\gamma }⟩|\}$. 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 $\hat{f}$ is L¹-integrable. Moreover, numerical simulations are conducted to test the efficiency of the reconstruction algorithm.