{
  "id": "0811.1342",
  "version": "v1",
  "published": "2008-11-09T14:57:53.000Z",
  "updated": "2008-11-09T14:57:53.000Z",
  "title": "On localization properties of Fourier transforms of hyperfunctions",
  "authors": [
    "A. G. Smirnov"
  ],
  "comment": "21 pages, final version, accepted for publication in J. Math. Anal. Appl",
  "journal": "J. Math. Anal. Appl. 351 (2009) 57-69",
  "doi": "10.1016/j.jmaa.2008.10.003",
  "categories": [
    "math.FA",
    "math.CV"
  ],
  "abstract": "In [Adv. Math. 196 (2005) 310-345] the author introduced a new generalized function space $\\mathcal U(R^k)$ which can be naturally interpreted as the Fourier transform of the space of Sato's hyperfunctions on $R^k$. It was shown that all Gelfand--Shilov spaces $S^{\\prime 0}_\\alpha(R^k)$ ($\\alpha>1$) of analytic functionals are canonically embedded in $\\mathcal U(R^k)$. While the usual definition of support of a generalized function is inapplicable to elements of $S^{\\prime 0}_\\alpha(R^k)$ and $\\mathcal U(R^k)$, their localization properties can be consistently described using the concept of {\\it carrier cone} introduced by Soloviev [Lett. Math. Phys. 33 (1995) 49-59; Comm. Math. Phys. 184 (1997) 579-596]. In this paper, the relation between carrier cones of elements of $S^{\\prime 0}_\\alpha(R^k)$ and $\\mathcal U(R^k)$ is studied. It is proved that an analytic functional $u\\in S^{\\prime 0}_\\alpha(R^k)$ is carried by a cone $K\\subset R^k$ if and only if its canonical image in $\\mathcal U(R^k)$ is carried by $K$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-11-09T14:57:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fourier transform",
      "localization properties",
      "carrier cone",
      "analytic functional",
      "gelfand-shilov spaces"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0811.1342S"
    }
  }
}