{
  "id": "1602.03366",
  "version": "v1",
  "published": "2016-02-10T13:29:41.000Z",
  "updated": "2016-02-10T13:29:41.000Z",
  "title": "Hermite polynomials, linear flows on the torus, and an uncertainty principle for roots",
  "authors": [
    "Diogo Oliveira e Silva",
    "Stefan Steinerberger"
  ],
  "comment": "22 pages, 4 figures",
  "categories": [
    "math.CA"
  ],
  "abstract": "We study a recent result of Bourgain, Clozel and Kahane, a version of which states that a sufficiently nice function $f:\\mathbb{R} \\rightarrow \\mathbb{R}$ that coincides with its Fourier transform and vanishes at the origin has a root in the interval $(c, \\infty)$, where the optimal $c$ satisfies $0.41 \\leq c \\leq 0.64$. A similar result holds in higher dimensions. We improve the one-dimensional result to $0.45 \\leq c \\leq 0.594$, and the lower bound in higher dimensions. We also prove that if an extremizer exists, then it has to have infinitely many double roots. The main ingredient here is a new structure statement about Hermite polynomials which relates their pointwise evaluation to linear flows on the torus, and applies to other families of orthogonal polynomials as well.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-02-10T13:29:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33C45",
      "42B10"
    ],
    "keywords": [
      "hermite polynomials",
      "linear flows",
      "uncertainty principle",
      "higher dimensions",
      "similar result holds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}