{
  "id": "1803.08011",
  "version": "v1",
  "published": "2018-03-21T16:56:46.000Z",
  "updated": "2018-03-21T16:56:46.000Z",
  "title": "Wasserstein Distance, Fourier Series and Applications",
  "authors": [
    "Stefan Steinerberger"
  ],
  "categories": [
    "math.CA",
    "math.NT",
    "math.PR"
  ],
  "abstract": "We study the Wasserstein metric $W_p$, a notion of distance between two probability distributions, from the perspective of Fourier Analysis and discuss applications. In particular, we bound the Earth Mover Distance $W_1$ between the distribution of quadratic residues in a finite field $\\mathbb{F}_p$ and uniform distribution by $\\lesssim p^{-1/2}$ (the Polya-Vinogradov inequality implies $\\lesssim p^{-1/2} \\log{p}$). We also show for continuous $f:\\mathbb{T} \\rightarrow \\mathbb{R}_{}$ with mean value 0 $$ (\\mbox{number of roots of}~f) \\cdot \\left( \\sum_{k=1}^{\\infty}{ \\frac{ |\\widehat{f}(k)|^2}{k^2}}\\right)^{\\frac{1}{2}} \\gtrsim \\frac{\\|f\\|^{2}_{L^1(\\mathbb{T})}}{\\|f\\|_{L^{\\infty}(\\mathbb{T})}}.$$ Moreover, we show that for a Laplacian eigenfunction $-\\Delta_g \\phi_{\\lambda} = \\lambda \\phi_{\\lambda}$ on a compact Riemannian manifold $W_p\\left(\\max\\left\\{\\phi_{\\lambda}, 0\\right\\}dx, \\max\\left\\{-\\phi_{\\lambda}, 0\\right\\} dx\\right) \\lesssim_p \\sqrt{\\log{\\lambda}/\\lambda} \\|\\phi_{\\lambda}\\|_{L^1}^{1/p}$ which is at most a factor $\\sqrt{\\log{\\lambda}}$ away from sharp. Several other problems are discussed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-03-21T16:56:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "wasserstein distance",
      "fourier series",
      "applications",
      "compact riemannian manifold",
      "polya-vinogradov inequality implies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}