{
  "id": "1411.2491",
  "version": "v1",
  "published": "2014-11-10T16:26:40.000Z",
  "updated": "2014-11-10T16:26:40.000Z",
  "title": "A note on measure-geometric Laplacians",
  "authors": [
    "Marc Kesseböhmer",
    "Tony Samuel",
    "Hendrik Weyer"
  ],
  "comment": "9 pages, 10 figures",
  "categories": [
    "math.FA",
    "math.SP"
  ],
  "abstract": "We consider the measure-geometric Laplacians $\\Delta^{\\mu}$ with respect to atomless compactly supported Borel probability measures $\\mu$ as introduced by Freiberg and Z\\\"ahle in 2002 and show that the harmonic calculus of $\\Delta^{\\mu}$ can be deduced from the classical (weak) Laplacian. We explicitly calculate the eigenvalues and eigenfunctions of $\\Delta^{\\mu}$. Further, it is shown that there exists a measure-geometric Laplacian whose eigenfunctions are the Chebyshev polynomials and illustrate our results through specific examples of fractal measures, namely Salem and inhomogeneous self-similar Cantor measures.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-11-10T16:26:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35P20",
      "42B35",
      "47G30",
      "45D05"
    ],
    "keywords": [
      "measure-geometric laplacian",
      "compactly supported borel probability measures",
      "inhomogeneous self-similar cantor measures",
      "fractal measures",
      "specific examples"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1411.2491K"
    }
  }
}