{
  "id": "2407.13075",
  "version": "v1",
  "published": "2024-07-18T00:52:17.000Z",
  "updated": "2024-07-18T00:52:17.000Z",
  "title": "Spectral Eigen-subspace and Tree Structure for a Cantor Measure",
  "authors": [
    "Guotai Deng",
    "Yan-Song Fu",
    "Qingcan Kang"
  ],
  "comment": "31 pages, 3 figures",
  "categories": [
    "math.FA"
  ],
  "abstract": "In this work we investigate the question of constructions of the possible Fourier bases $E(\\Lambda)=\\{e^{2\\pi i \\lambda x}:\\lambda\\in\\Lambda\\}$ for the Hilbert space $L^2(\\mu_4)$, where $\\mu_4$ is the standard middle-fourth Cantor measure and $\\Lambda$ is a countable discrete set. We show that the set $$\\mathop \\bigcap_{p\\in 2\\Z+1}\\left\\{\\Lambda\\subset \\R: \\text{$E(\\Lambda)$ and $E(p\\Lambda)$ are Fourier bases for $L^2(\\mu_4)$}\\right\\}$$ has the cardinality of the continuum. We also give other characterizations on the orthonormal set of exponential functions being a basis for the space $L^2(\\mu_4)$ from the viewpoint of measure and dimension. Moreover, we provide a method of constructing explicit discrete set $\\Lambda$ such that $E(\\Lambda)$ and its all odd scaling sets $E(\\Lambda),p\\in2\\Z+1,$ are still Fourier bases for $L^2(\\mu_4)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-18T00:52:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42A10",
      "28A80",
      "42A65"
    ],
    "keywords": [
      "tree structure",
      "spectral eigen-subspace",
      "fourier bases",
      "standard middle-fourth cantor measure",
      "constructing explicit discrete set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}