{
  "id": "2110.07988",
  "version": "v2",
  "published": "2021-10-15T10:23:28.000Z",
  "updated": "2021-11-27T09:40:32.000Z",
  "title": "A note on exponential Riesz bases",
  "authors": [
    "Andrei Caragea",
    "Dae Gwan Lee"
  ],
  "journal": "Sampl. Theory Signal Process. Data Anal. Vol. 20, Article number: 13 (2022), Open Access",
  "doi": "10.1007/s43670-022-00031-9",
  "categories": [
    "math.CA"
  ],
  "abstract": "We prove that if $I_\\ell = [a_\\ell,b_\\ell)$, $\\ell=1, \\ldots, L$, are disjoint intervals in $[0,1)$ with the property that the numbers $1, a_1, \\ldots, a_L, b_1, \\ldots, b_L$ are linearly independent over $\\mathbb{Q}$, then there exist pairwise disjoint sets $\\Lambda_\\ell \\subset \\mathbb{Z}$, $\\ell=1, \\ldots, L$, such that for every $J \\subset \\{ 1, \\ldots , L \\}$, the system $\\{e^{2\\pi i \\lambda x} : \\lambda\\in \\cup_{\\ell \\in J} \\, \\Lambda_\\ell \\}$ is a Riesz basis for $L^2 ( \\cup_{\\ell \\in J} \\, I_\\ell)$. Also, we show that for any disjoint intervals $I_\\ell$, $\\ell=1, \\ldots, L$, contained in $[1,N)$ with $N \\in \\mathbb{N}$, the orthonormal basis $\\{e^{2\\pi i n x} : n \\in \\mathbb{Z} \\}$ of $L^2[0,1)$ can be complemented by a Riesz basis $\\{e^{2\\pi i \\lambda x} : \\lambda\\in\\Lambda\\}$ for $L^2(\\cup_{\\ell=1}^L \\, I_{\\ell})$ with some set $\\Lambda \\subset (\\frac{1}{N} \\mathbb{Z}) \\backslash \\mathbb{Z}$, in the sense that their union $\\{e^{2\\pi i \\lambda x} : \\lambda\\in \\mathbb{Z} \\cup \\Lambda\\}$ is a Riesz basis for $L^2 ( [0,1) \\cup I_1 \\cup \\cdots \\cup I_L )$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2021-11-27T09:40:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42C15"
    ],
    "keywords": [
      "riesz basis",
      "exponential riesz bases",
      "disjoint intervals",
      "pairwise disjoint sets",
      "orthonormal basis"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Springer"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}