{
  "id": "0911.1937",
  "version": "v1",
  "published": "2009-11-10T16:21:22.000Z",
  "updated": "2009-11-10T16:21:22.000Z",
  "title": "Remez-Type Inequality for Discrete Sets",
  "authors": [
    "Y. Yomdin"
  ],
  "comment": "22 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "The classical Remez inequality bounds the maximum of the absolute value of a polynomial $P(x)$ of degree $d$ on $[-1,1]$ through the maximum of its absolute value on any subset $Z$ of positive measure in $[-1,1]$. Similarly, in several variables the maximum of the absolute value of a polynomial $P(x)$ of degree $d$ on the unit cube $Q^n_1 \\subset {\\mathbb R}^n$ can be bounded through the maximum of its absolute value on any subset $Z\\subset Q^n_1$ of positive $n$-measure. The main result of this paper is that the $n$-measure in the Remez inequality can be replaced by a certain geometric invariant $\\omega_d(Z)$ which can be effectively estimated in terms of the metric entropy of $Z$ and which may be nonzero for discrete and even finite sets $Z$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-11-10T16:21:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "remez-type inequality",
      "discrete sets",
      "absolute value",
      "classical remez inequality bounds",
      "finite sets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0911.1937Y"
    }
  }
}