Reconstructing real-valued functions from unsigned frame coefficients

In this talk we consider the following problem of phase retrieval in L2: Given a collection of real-valued bandlimited functions that constitutes a semi-discrete frame, we ask whether any real-valued function f can be uniquely recovered from the magnitudes of its convolutions with the frame elements.
We find that under some mild assumptions on the semi-discrete frame and if f decays exponentially at infinity, it suffices to know the unsigned measurements on suitably fine lattices to uniquely determine f (up to a global sign factor).
We further present a local stability property of our reconstruction problem. Finally, for two concrete examples of a (discrete) frame, we show that through sufficient oversampling one obtains a frame such that any function can be uniquely recovered from its unsigned frame coefficients.