{
  "id": "2401.05243",
  "version": "v1",
  "published": "2024-01-10T16:15:11.000Z",
  "updated": "2024-01-10T16:15:11.000Z",
  "title": "Frame-like Fourier expansions for finite Borel measures on $\\mathbb{R}$",
  "authors": [
    "Chad Berner"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "It is known that if a finite Borel measure $\\mu$ on $[0,1)$ possesses a frame of exponential functions for $L^{2}(\\mu)$, then $\\mu$ is of pure type. In this paper, we prove the existence of a class of finite Borel measures $\\mu$ on $[0,1)$ that are not of pure type that possess frame-like Fourier expansions for $L^{2}(\\mu)$. We also show properties and classifications of certain measures possessing this type of Fourier expansion. Additionally, we establish a frame-like Fourier expansion for $L^{2}(\\mu)$ where $\\mu$ is a singular Borel probability measure on $\\mathbb{R}$. Finally, we show measures $\\mu$ on $[0,1)$ that possess these frame-like Fourier expansions for $L^{2}(\\mu)$ have all $f\\in L^{2}(\\mu)$ as $L^{2}(\\mu)$ limits of harmonic functions with frame-like coefficients. We also discuss when the inner products of these expansions coincide with model spaces and subspaces of harmonic functions on the disk.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-10T16:15:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42A16",
      "42A20",
      "42C15",
      "46C07"
    ],
    "keywords": [
      "finite borel measure",
      "pure type",
      "harmonic functions",
      "singular borel probability measure",
      "possess frame-like fourier expansions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}