{
  "id": "2011.06971",
  "version": "v1",
  "published": "2020-11-13T15:36:58.000Z",
  "updated": "2020-11-13T15:36:58.000Z",
  "title": "Reconstruction of polytopes from the modulus of the Fourier transform with small wave length",
  "authors": [
    "Konrad Engel",
    "Bastian Laasch"
  ],
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "Let $\\mathcal{P}$ be an $n$-dimensional convex polytope and $\\mathcal{S}$ be a hypersurface in $\\mathbb{R}^n$. This paper investigates potentials to reconstruct $\\mathcal{P}$ or at least to compute significant properties of $\\mathcal{P}$ if the modulus of the Fourier transform of $\\mathcal{P}$ on $\\mathcal{S}$ with wave length $\\lambda$, i.e., $|\\int_{\\mathcal{P}} e^{-i\\frac{1}{\\lambda}\\mathbf{s}\\cdot\\mathbf{x}} \\,\\mathbf{dx}|$ for $\\mathbf{s}\\in\\mathcal{S}$, is given, $\\lambda$ is sufficiently small and $\\mathcal{P}$ and $\\mathcal{S}$ have some well-defined properties. The main tool is an asymptotic formula for the Fourier transform of $\\mathcal{P}$ with wave length $\\lambda$ when $\\lambda \\rightarrow 0$. The theory of X-ray scattering of nanoparticles motivates this study since the modulus of the Fourier transform of the reflected beam wave vectors are approximately measurable in experiments.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-13T15:36:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B10",
      "52B11",
      "81U40"
    ],
    "keywords": [
      "fourier transform",
      "small wave length",
      "reconstruction",
      "dimensional convex polytope",
      "reflected beam wave vectors"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}