{
  "id": "2401.08042",
  "version": "v1",
  "published": "2024-01-16T01:47:54.000Z",
  "updated": "2024-01-16T01:47:54.000Z",
  "title": "Exponential bases for parallelepipeds with frequency sets lying in the integer lattice",
  "authors": [
    "Dae Gwan Lee",
    "Goetz E. Pfander",
    "David Walnut"
  ],
  "comment": "34 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "The existence of a Fourier basis with frequencies in $\\mathbb{R}^d$ for the space of square integrable functions supported on a given parallelepiped in $\\mathbb{R}^d$ is well understood since the 1950s. In this paper, we consider the restriction to integer frequencies. That is, we give sufficient, and some necessary criteria, for a parallelepiped in $\\mathbb{R}^d$ to permit a Fourier basis (orthonormal or Riesz) for the respective space of square integrable functions whose frequency set is constrained to be a subset of $\\mathbb{Z}^d$. A simple rescaling argument provides criteria on parallelepipeds to permit frequency sets contained in a prescribed set $a_1\\mathbb{Z} \\times a_2\\mathbb{Z} \\times \\ldots \\times a_d\\mathbb{Z}$, a restriction relevant in many applications.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-16T01:47:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B05",
      "42C15"
    ],
    "keywords": [
      "frequency sets lying",
      "exponential bases",
      "integer lattice",
      "parallelepipeds",
      "square integrable functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}