{
  "id": "2112.10136",
  "version": "v2",
  "published": "2021-12-19T12:48:05.000Z",
  "updated": "2022-06-17T10:30:59.000Z",
  "title": "Injectivity of sampled Gabor phase retrieval in spaces with general integrability conditions",
  "authors": [
    "Matthias Wellershoff"
  ],
  "comment": "31 pages; improved presentation",
  "categories": [
    "math.FA"
  ],
  "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 M\\\"untz-Sz\\'asz 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.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-06-17T10:30:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "94A12",
      "94A20"
    ],
    "keywords": [
      "sampled gabor phase retrieval",
      "general integrability conditions",
      "injectivity",
      "general nonuniform sampling sets",
      "fractional fourier transform"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}