{
  "id": "2404.03076",
  "version": "v1",
  "published": "2024-04-03T21:36:21.000Z",
  "updated": "2024-04-03T21:36:21.000Z",
  "title": "Completeness of systems of inner functions",
  "authors": [
    "Nazar Miheisi"
  ],
  "categories": [
    "math.FA",
    "math.CV"
  ],
  "abstract": "Let $\\mathcal{Q}$ denote the class of inner functions $\\vartheta$ such that the measure $\\sum_{w\\in\\vartheta^{-1}(z)} (1-|w|)\\delta_w$ is a Carleson measure for all $z$ in the complement of a set of logarithmic capacity zero. For $\\vartheta,\\varphi\\in\\mathcal{Q}$, we give a simple sufficient condition for the system $\\vartheta^m,\\; \\varphi^n$, $m,n\\in\\mathbb{Z}$, to be complete in the weak-$^*$ topology of $L^\\infty(\\mathbb{T})$. To be precise, we show that this system is complete whenever there is an arc $I$ of the unit circle $\\mathbb{T}$ such that $\\vartheta$ is univalent on $I$ and $\\varphi$ is univalent on $\\mathbb{T}\\setminus I$. As an application of this result, we describe a class of analytic curves $\\Gamma$ such that $(\\Gamma, \\mathcal{X})$ is a Heisenberg uniqueness pair, where $\\mathcal{X}$ is the lattice cross $\\{(m,n)\\in\\mathbb{Z}^2:\\, mn=0\\}$. Our main result extends a theorem of Hedenmalm and Montes-Rodr\\'iguez for atomic inner functions with one singularity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-04-03T21:36:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30B60",
      "30J05"
    ],
    "keywords": [
      "completeness",
      "atomic inner functions",
      "main result extends",
      "logarithmic capacity zero",
      "heisenberg uniqueness pair"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}