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Injectivity of sampled Gabor phase retrieval in spaces with general integrability conditions

by M. Wellershoff

(Report number 2021-43)

Abstract
It was recently shown that functions in \(L^4([-B,B])\) can be uniquely recovered up to a global phase factor from the absolute values of their Gabor transform sampled on a rectangular lattice. We prove that this remains true if one replaces \(L^4([-B,B])\) by \(L^p([-B,B])\) with \(p \in [2,\infty]\). To do so, we adapt the original proof by Grohs and Liehr and use sampling results in Bernstein spaces with general integrability parameters. Furthermore, we present some modifications of a result of Müntz–Szász type first proven by Zalik. Finally, we consider the implications of our results for more general function spaces obtained by applying the fractional Fourier transform to \(L^p([-B,B])\) and for more general nonuniform sampling sets.

Keywords: Phase retrieval, Gabor transform, Sampling theory, Time-frequency analysis, Müntz–Szász type

@Techreport{W21_985,
  author = {M. Wellershoff},
  title = {Injectivity of sampled Gabor phase retrieval in spaces with general integrability conditions},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2021-43},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2021/2021-43.pdf },
  year = {2021}
}

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