{
  "id": "1403.4168",
  "version": "v4",
  "published": "2014-03-17T17:02:36.000Z",
  "updated": "2016-07-14T02:50:46.000Z",
  "title": "Fourier and Beyond: Invariance Properties of a Family of Integral Transforms",
  "authors": [
    "Cameron L. Williams",
    "Bernhard G. Bodmann",
    "Donald J. Kouri"
  ],
  "comment": "14 pages",
  "categories": [
    "math.FA",
    "math.CA"
  ],
  "abstract": "The Fourier transform is typically seen as closely related to the additive group of real numbers, its characters and its Haar measure. In this paper, we propose an alternative viewpoint; the Fourier transform can be uniquely characterized by an intertwining relation with dilations and by having a Gaussian as an eigenfunction. This broadens the perspective to an entire family of Fourier-like transforms that are uniquely identified by the same dilation property and having Gaussian-like functions as eigenfunctions. We show that these transforms share many properties with the Fourier transform, particularly unitarity, periodicity and eigenvalues. We also establish short-time analogues of these transforms and show a reconstruction property and an orthogonality relation for the short-time transforms.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-07-09T03:41:29.000Z",
      "abstract": "This paper presents a family of transforms which share many properties with the Fourier transform. We prove that these transforms are isometries on $L^2(\\mathbb{R})$ and have the same scaling property. The transforms can be chosen to leave Gaussian-like functions invariant. We also establish short-time analogs of these transforms.",
      "comment": "15 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v4",
      "updated": "2016-07-14T02:50:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "invariance properties",
      "integral transforms",
      "leave gaussian-like functions invariant",
      "fourier transform",
      "establish short-time analogs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1403.4168W"
    }
  }
}