{
  "id": "math/9512211",
  "version": "v1",
  "published": "1995-12-01T00:00:00.000Z",
  "updated": "1995-12-01T00:00:00.000Z",
  "title": "A Hilbert space of Dirichlet series and systems of dilated functions in $L^2(0,1)$",
  "authors": [
    "Håkan Hedenmalm",
    "Peter Lindqvist",
    "Kristian Seip"
  ],
  "journal": "Duke Math. J. 86 (1997), 1--37",
  "categories": [
    "math.FA"
  ],
  "abstract": "For a function $\\varphi$ in $L^2(0,1)$, extended to the whole real line as an odd periodic function of period 2, we ask when the collection of dilates $\\varphi(nx)$, $n=1,2,3,\\ldots$, constitutes a Riesz basis or a complete sequence in $L^2(0,1)$. The problem translates into a question concerning multipliers and cyclic vectors in the Hilbert space $\\cal H$ of Dirichlet series $f(s)=\\sum_n a_nn^{-s}$, where the coefficients $a_n$ are square summable. It proves useful to model $\\cal H$ as the $H^2$ space of the infinite-dimensional polydisk, or, which is the same, the $H^2$ space of the character space, where a character is a multiplicative homomorphism from the positive integers to the unit circle. For given $f$ in $\\cal H$ and characters $\\chi$, $f_\\chi(s)=\\sum_na_n\\chi(n)n^{-s}$ is a vertical limit function of $f$. We study certain probabilistic properties of these vertical limit functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "1995-12-01T00:00:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "dirichlet series",
      "hilbert space",
      "dilated functions",
      "vertical limit function",
      "odd periodic function"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1995math.....12211H"
    }
  }
}