{
  "id": "0809.0604",
  "version": "v1",
  "published": "2008-09-03T11:59:47.000Z",
  "updated": "2008-09-03T11:59:47.000Z",
  "title": "On the Fourier transform of the symmetric decreasing rearrangements",
  "authors": [
    "Philippe Jaming"
  ],
  "journal": "Annales de l'Institut Fourier 61 (2011) 53-77",
  "categories": [
    "math.CA"
  ],
  "abstract": "Inspired by work of Montgomery on Fourier series and Donoho-Strak in signal processing, we investigate two families of rearrangement inequalities for the Fourier transform. More precisely, we show that the $L^2$ behavior of a Fourier transform of a function over a small set is controlled by the $L^2$ behavior of the Fourier transform of its symmetric decreasing rearrangement. In the $L^1$ case, the same is true if we further assume that the function has a support of finite measure. As a byproduct, we also give a simple proof and an extension of a result of Lieb about the smoothness of a rearrangement. Finally, a straightforward application to solutions of the free Shr\\\"odinger equation is given.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-09-03T11:59:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42A38",
      "42B10",
      "42C20",
      "33C10"
    ],
    "keywords": [
      "symmetric decreasing rearrangement",
      "fourier transform",
      "fourier series",
      "small set",
      "simple proof"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0809.0604J"
    }
  }
}