{
  "id": "math/0602440",
  "version": "v2",
  "published": "2006-02-20T18:28:50.000Z",
  "updated": "2007-01-09T17:59:53.000Z",
  "title": "Completeness, special functions and uncertainty principles over q-linear grids",
  "authors": [
    "Luis Daniel Abreu"
  ],
  "comment": "15 pages, final version (first and second introductory paragraphs switched, an easier proof of the last theorem)",
  "journal": "J. Phys. A: Math. Gen. 39 (2006) 14567-14580",
  "doi": "10.1088/0305-4470/39/47/004",
  "categories": [
    "math.CA",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We derive completeness criteria for sequences of functions of the form $% f(x\\lambda_{n})$, where $\\lambda_{n}$ is the $nth$ zero of a suitably chosen entire function. Using these criteria, we construct systems of nonorthogonal Fourier-Bessel functions and their $q$-analogues, as well as other complete sets of $q$-special functions. The completeness of certain sets of $q$-Bessel functions is then used to prove that, if a function $f$ and its $q$-Hankel transform both vanish at the points $\\{q^{-n}\\}_{n=1}^{% \\infty}$, $0<q<1$, then $f$ must vanish on the whole $q$-linear grid $% \\{q^{n}\\} _{n=-\\infty}^{\\infty}$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-01-09T17:59:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A30",
      "33D45",
      "42C30",
      "94A11"
    ],
    "keywords": [
      "special functions",
      "uncertainty principles",
      "q-linear grids",
      "suitably chosen entire function",
      "nonorthogonal fourier-bessel functions"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "journal": "Journal of Physics A Mathematical General",
      "year": 2006,
      "month": "Nov",
      "volume": 39,
      "number": 47,
      "pages": 14567
    },
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006JPhA...3914567A"
    }
  }
}