Celebratio Mathematica — Daubechies

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R. Al-Aifari, I. Daubechies, P. Grohs, and G. Thakur: “Reconstructing real-valued functions from unsigned coefficients with respect to wavelet and other frames,” J. Fourier Anal. Appl. 23 : 6 (December 2017), pp. 1480–1494. MR 3735589 ArXiv 1601.07579 article

In this paper we consider the following problem of phase retrieval: given a collection of real-valued band-limited functions \[ \{\psi_{\lambda}\}_{\lambda\in\Lambda} \subset L^2(\mathbb{R}^d) \] that constitutes a semi-discrete frame, we ask whether any real-valued function \( f\in L^2(\mathbb{R}^d) \) can be uniquely recovered from its unsigned convolutions \[ \{|f*\psi_{\lambda}|\}_{\lambda\in\Lambda} .\] We find that under some mild assumptions on the semi-discrete frame and if \( f \) has exponential decay at \( \infty \), it suffices to know \( |f*\psi_{\lambda}| \) 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 \( L^2(\mathbb{R}^d) \), \( d=1,2 \) 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.

Rima Alaifari	Related
Philipp Grohs	Related
Gaurav S. Thakur	Related