{
  "id": "1511.05718",
  "version": "v1",
  "published": "2015-11-18T10:13:24.000Z",
  "updated": "2015-11-18T10:13:24.000Z",
  "title": "Functions of exponential growth in a half-plane, sets of uniqueness and the M\"untz--Sz'asz problem for the Bergman space",
  "authors": [
    "Marco M. Peloso",
    "Maura Salvatori"
  ],
  "categories": [
    "math.CV",
    "math.FA"
  ],
  "abstract": "We introduce and study some new spaces of holomorphic functions on the right half-plane. In a previous work, S. Krantz, C. Stoppato and the first named author formulated the M\"untz--Sz'asz problem for the Bergman space, that is, the problem to characterize the sets of complex powers that form a complete set the unweighted Bergman space of a disc. In this paper, we construct a space of holomorphic functions on the right half-plane, whose sets of uniqueness correspond exactly to the sets of powers that are a complete set in Bergman space. We show that this space is a reproducing kernel Hilbert space and we prove a Paley--Wiener type theorem among several other structural properties. Moreover, we determine a sufficient condition on a set of powers to be a set of uniqueness for this space, thus providing a sufficient condition for the solution of the M\"untz--Sz'asz problem for the Bergman space.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-18T10:13:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30H99",
      "46E22",
      "30C15",
      "30C40"
    ],
    "keywords": [
      "untz-szasz problem",
      "exponential growth",
      "uniqueness",
      "holomorphic functions",
      "right half-plane"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151105718P"
    }
  }
}