Abstract
Phase retrieval and phaseless reconstruction for Hilbert space frames is a very active area of research. Recently, it was shown that these concepts are equivalent. In this paper, we make a detailed study of a weakening of these concepts to weak phase retrieval and weak phaseless reconstruction. We will give several necessary and/or sufficient conditions for frames to have these weak properties. We will prove three surprising results: (1) Weak phaseless reconstruction is equivalent to phaseless reconstruction. That is, it never was weak; (2) weak phase retrieval is not equivalent to weak phaseless reconstruction; (3) weak phase retrieval requires at least 2m − 2 vectors in an m-dimensional Hilbert space. We also give several examples illustrating the relationship between these concepts.
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Acknowledgement
The second through fifth authors were supported by NSF DMS 1609760, NSF ATD 1321779, and ARO W911NF-16-1-0008.
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Botelho-Andrade, S., Casazza, P.G., Ghoreishi, D., Jose, S., Tremain, J.C. (2017). Weak Phase Retrieval. In: Boche, H., Caire, G., Calderbank, R., März, M., Kutyniok, G., Mathar, R. (eds) Compressed Sensing and its Applications. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-69802-1_7
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DOI: https://doi.org/10.1007/978-3-319-69802-1_7
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