{
  "id": "1306.0632",
  "version": "v1",
  "published": "2013-06-04T01:42:49.000Z",
  "updated": "2013-06-04T01:42:49.000Z",
  "title": "Beurling Spectra of Functions on Locally Compact Abelian Groups",
  "authors": [
    "B. Basit",
    "A. J. Pryde"
  ],
  "comment": "17 pages. Corrections and compliment to Monash Analysis Paper 88 (1993)",
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $G$ be a locally compact abelian topological group. For locally bounded measurable functions $\\varphi: G\\to\\Bbb {C}$ we discuss notions of spectra for $\\varphi$ relative to subalgebras of $L^{1}(G)$. In particular we study polynomials on $G$ and determine their spectra. We also characterize the primary ideals of certain Beurling algebras $L_{w}^{1}(\\Bbb Z)$ on the group of integers $\\Bbb Z$. This allows us to classify those elements of $L_{w}^{1}(G)$ that have finite spectrum. If $\\varphi$ is a uniformly continuous function whose differences are bounded, there is a Beurling algebra naturally associated with $\\varphi$. We give a condition on the spectrum of $\\varphi$ relative to this algebra which ensures that $\\varphi$ is bounded. Finally we give spectral conditions on a bounded function on $\\Bbb R$ that ensure that its indefinite integral is bounded.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-06-04T01:42:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "43A15",
      "43A45",
      "46J20"
    ],
    "keywords": [
      "locally compact abelian groups",
      "beurling spectra",
      "beurling algebra",
      "locally compact abelian topological group",
      "finite spectrum"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1306.0632B"
    }
  }
}