{
  "id": "2106.07532",
  "version": "v1",
  "published": "2021-06-14T15:57:50.000Z",
  "updated": "2021-06-14T15:57:50.000Z",
  "title": "Hilbert points in Hardy spaces",
  "authors": [
    "Ole Fredrik Brevig",
    "Joaquim Ortega-Cerdà",
    "Kristian Seip"
  ],
  "categories": [
    "math.FA",
    "math.CA",
    "math.CV"
  ],
  "abstract": "A Hilbert point in $H^p(\\mathbb{T}^d)$, for $d\\geq1$ and $1\\leq p \\leq \\infty$, is a nontrivial function $\\varphi$ in $H^p(\\mathbb{T}^d)$ such that $\\| \\varphi \\|_{H^p(\\mathbb{T}^d)} \\leq \\|\\varphi + f\\|_{H^p(\\mathbb{T}^d)}$ whenever $f$ is in $H^p(\\mathbb{T}^d)$ and orthogonal to $\\varphi$ in the usual $L^2$ sense. When $p\\neq 2$, $\\varphi$ is a Hilbert point in $H^p(\\mathbb{T})$ if and only if $\\varphi$ is a nonzero multiple of an inner function. An inner function on $\\mathbb{T}^d$ is a Hilbert point in any of the spaces $H^p(\\mathbb{T}^d)$, but there are other Hilbert points as well when $d\\geq 2$. We investigate the case of $1$-homogeneous polynomials in depth and obtain as a byproduct a new proof of the sharp Khintchin inequality for Steinhaus variables in the range $2<p<\\infty$. We also study briefly the dynamics of a certain nonlinear projection operator that characterizes Hilbert points as its fixed points. We exhibit an example of a function $\\varphi$ that is a Hilbert point in $H^p(\\mathbb{T}^3)$ for $p=2, 4$, but not for any other $p$; this is verified rigorously for $p>4$ but only numerically for $1\\leq p<4$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-14T15:57:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hardy spaces",
      "inner function",
      "characterizes hilbert points",
      "sharp khintchin inequality",
      "nonlinear projection operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}