{
  "id": "1503.04856",
  "version": "v1",
  "published": "2015-03-16T21:19:20.000Z",
  "updated": "2015-03-16T21:19:20.000Z",
  "title": "Fourier Series for Singular Measures",
  "authors": [
    "John E. Herr",
    "Eric S. Weber"
  ],
  "comment": "12 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "Using the Kaczmarz algorithm, we prove that for any singular Borel probability measure $\\mu$ on $[0,1)$, every $f\\in L^2(\\mu)$ possesses a Fourier series of the form $f(x)=\\sum_{n=0}^{\\infty}c_ne^{2\\pi inx}$. We show that the coefficients $c_{n}$ can be computed in terms of the quantities $\\widehat{f}(n) = \\int_{0}^{1} f(x) e^{-2\\pi i n x} d \\mu(x)$. We also demonstrate a Shannon-type sampling theorem for functions that are in a sense $\\mu$-bandlimited.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-03-16T21:19:20.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fourier series",
      "singular measures",
      "singular borel probability measure",
      "kaczmarz algorithm",
      "shannon-type sampling theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150304856H"
    }
  }
}