{
  "id": "1306.3641",
  "version": "v1",
  "published": "2013-06-16T08:36:27.000Z",
  "updated": "2013-06-16T08:36:27.000Z",
  "title": "Remez-Type Inequality for Smooth Functions",
  "authors": [
    "Yosef Yomdin"
  ],
  "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 ball $B^n \\subset {\\mathbb R}^n$ can be bounded through the maximum of its absolute value on any subset $Z\\subset Q^n_1$ of positive $n$-measure $m_n(Z)$. In \\cite{Yom} a stronger version of Remez inequality was obtained: the Lebesgue $n$-measure $m_n$ was replaced by a certain geometric quantity $\\omega_{n,d}(Z)$ satisfying $\\omega_{n,d}(Z)\\geq m_n(Z)$ for any measurable $Z$. The quantity $\\omega_{n,d}(Z)$ can be effectively estimated in terms of the metric entropy of $Z$ and it may be nonzero for discrete and even finite sets $Z$. In the present paper we extend Remez inequality to functions of finite smoothness. This is done by combining the result of \\cite{Yom} with the Taylor polynomial approximation of smooth functions. As a consequence we obtain explicit lower bounds in some examples in the Whitney problem of a $C^k$-smooth extrapolation from a given set $Z$, in terms of the geometry of $Z$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-06-16T08:36:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26D05",
      "30E05",
      "42A05"
    ],
    "keywords": [
      "smooth functions",
      "remez-type inequality",
      "absolute value",
      "explicit lower bounds",
      "taylor polynomial approximation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1306.3641Y"
    }
  }
}