Reconstructing Real-Valued Functions from Unsigned Coefficients with Respect to Wavelet and Other Frames

In this paper we consider the following problem of phase retrieval: given a collection of real-valued band-limited functions $$\{\psi _{\lambda }\}_{\lambda \in \Lambda }\subset L^2(\mathbb {R}^d)$${ψλ}λ∈Λ⊂L2(Rd) that constitutes a semi-discrete frame, we ask whether any real-valued function $$f \in L^2(\mathbb {R}^d)$$f∈L2(Rd) can be uniquely recovered from its unsigned convolutions $${\{|f *\psi _\lambda |\}_{\lambda \in \Lambda }}$${|f∗ψλ|}λ∈Λ. We find that under some mild assumptions on the semi-discrete frame and if f has exponential decay at $$\infty $$∞, it suffices to know $$|f *\psi _\lambda |$$|f∗ψλ| on suitably fine lattices to uniquely determine f (up to a global sign factor). We further establish a local stability property of our reconstruction problem. Finally, for two concrete examples of a (discrete) frame of $$L^2(\mathbb {R}^d)$$L2(Rd), $$d=1,2$$d=1,2, we show that through sufficient oversampling one obtains a frame such that any real-valued function with exponential decay can be uniquely recovered from its unsigned frame coefficients.