{
  "id": "1602.05722",
  "version": "v1",
  "published": "2016-02-18T09:00:40.000Z",
  "updated": "2016-02-18T09:00:40.000Z",
  "title": "On partition of unities generated by entire functions and Gabor frames in $\\ltrd$ and $\\ell^2(\\mzd)$",
  "authors": [
    "Ole Christensen",
    "Hong Oh Kim",
    "Rae Young Kim"
  ],
  "comment": "appears in Journal of Fourier Analysis and Applications, 2016",
  "categories": [
    "math.FA"
  ],
  "abstract": "We characterize the entire functions $P$ of $d$ variables, $d\\ge 2,$ for which the $\\mzd$-translates of $P\\chi_{[0,N]^d}$ satisfy the partition of unity for some $N\\in \\mn.$ In contrast to the one-dimensional case, these entire functions are not necessarily periodic. In the case where $P$ is a trigonometric polynomial, we characterize the maximal smoothness of $P\\chi_{[0,N]^d},$ as well as the function that achieves it. A number of especially attractive constructions are achieved, e.g., of trigonometric polynomials leading to any desired (finite) regularity for a fixed support size. As an application we obtain easy constructions of matrix-generated Gabor frames in $\\ltrd,$ with small support and high smoothness. By sampling this yields dual pairs of finite Gabor frames in $\\ell^2(\\mzd).$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-02-18T09:00:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42C15"
    ],
    "keywords": [
      "entire functions",
      "trigonometric polynomial",
      "finite gabor frames",
      "yields dual pairs",
      "one-dimensional case"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160205722C"
    }
  }
}