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Phase Retrieval for Wide Band Signals

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Abstract

This study investigates the phase retrieval problem for wide-band signals. We solve the following problem: given \(f\in L^2(\mathbb {R})\) with Fourier transform in \(L^2(\mathbb {R},e^{2c|x|}\,\text{ d }x)\), we find all functions \(g\in L^2(\mathbb {R})\) with Fourier transform in \(L^2(\mathbb {R},e^{2c|x|}\,\text{ d }x)\), such that \(|f(x)|=|g(x)|\) for all \(x\in \mathbb {R}\). To do so, we first translate the problem to functions in the Hardy spaces on the disc via a conformal bijection, and take advantage of the inner-outer factorization. We also consider the same problem with additional constraints involving some transforms of f and g, and determine if these constraints force uniqueness of the solution.

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Acknowledgements

This study has been carried out with financial support from the French State, managed by the French National Research Agency (ANR) in the frame of the “Investments for the Future” Programme IdEx Bordeaux-CPU (ANR-10-IDEX-03-02). This paper was completed during the first author’s visit at the Schrödinger Institute, Vienna, during the workshop “Operator Related Function Theory”. We kindly acknowledge ESI’s hospitality. The research of the second author is partially supported by the project ANR-18-CE40-0035 and the Joint French-Russian Research Project PRC-CNRS/RFBR 2017–2019. The third author is supported by the CHED-PhilFrance scholarship from Campus France and the Commission of Higher Education (CHED), Philippines. We would like to thank the referees for their helpful comments and suggestions. We truly appreciate the time they spent to check for corrections. Some results in this paper have been announced in [23]. We also thank the referees of that announcement for their helpful comments that also led to improvements here.

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Authors and Affiliations

  1. Université de Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251, 33400, Talence, France

    Philippe Jaming, Karim Kellay & Rolando Perez III

  2. Institute of Mathematics, University of the Philippines Diliman, 1101, Quezon City, Philippines

    Rolando Perez III

Authors
  1. Philippe Jaming
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  2. Karim Kellay
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  3. Rolando Perez III
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Correspondence to Philippe Jaming.

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Communicated by Gabriel Peyre.

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Jaming, P., Kellay, K. & Perez, R. Phase Retrieval for Wide Band Signals. J Fourier Anal Appl 26, 54 (2020). https://doi.org/10.1007/s00041-020-09767-1

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  • DOI: https://doi.org/10.1007/s00041-020-09767-1

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