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 {ψλ}L2(Rd) that constitutes a semi-discrete frame, we ask whether any real-valued function f∈ L²(R^d) can be uniquely recovered from its unsigned convolutions { | f∗ ψλ| } λ∈Λ. We find that under some mild assumptions on the semi-discrete frame and if f has exponential decay at ∞, it suffices to know | 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 L²(R^d) , 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.

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Author Ingrid Daubechies Mathematics