{
  "id": "2005.13802",
  "version": "v1",
  "published": "2020-05-28T06:34:42.000Z",
  "updated": "2020-05-28T06:34:42.000Z",
  "title": "Spectral properties of some unions of linear spaces",
  "authors": [
    "Chun-Kit Lai",
    "Bochen Liu",
    "Hal Prince"
  ],
  "categories": [
    "math.FA",
    "math.CA"
  ],
  "abstract": "We consider \\textit{additive spaces}, consisting of two intervals of unit length or two general probability measures on ${\\mathbb R}^1$, positioned on the axes in ${\\mathbb R}^2$, with a natural additive measure $\\rho$. We study the relationship between the exponential frames, Riesz bases, and orthonormal bases of $L^2(\\rho)$ and those of its component spaces. We find that the existence of exponential bases depends strongly on how we position our measures on ${\\mathbb R}^1$. We show that non-overlapping additive spaces possess Riesz bases, and we give a necessary condition for overlapping spaces. We also show that some overlapping additive spaces of Lebesgue type have exponential orthonormal bases, while some do not. A particular example is the \"L\" shape at the origin, which has a unique orthonormal basis up to translations of the form \\[ \\left\\{e^{2 \\pi i (\\lambda_1 x_1 + \\lambda_2 x_2)} : (\\lambda_1, \\lambda_2) \\in \\Lambda \\right\\}, \\] where \\[ \\Lambda = \\{ (n/2, -n/2) \\mid n \\in {\\mathbb Z} \\}. \\]",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-28T06:34:42.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "spectral properties",
      "linear spaces",
      "orthonormal basis",
      "additive spaces possess riesz bases",
      "non-overlapping additive spaces possess riesz"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}