{
  "id": "2108.02459",
  "version": "v1",
  "published": "2021-08-05T08:49:11.000Z",
  "updated": "2021-08-05T08:49:11.000Z",
  "title": "Higher derivatives of functions vanishing on a given set",
  "authors": [
    "Y. Yomdin"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "Let $f: B^n \\rightarrow {\\mathbb R}$ be a $d+1$ times continuously differentiable function on the unit ball $B^n$, with $\\max_{z\\in B^n} \\Vert f(z) \\Vert=1$. A well-known fact is that if $f$ vanishes on a set $Z\\subset B^n$ with a non-empty interior, then for each $k=1,\\ldots,d+1$ the norm of the $k$-th derivative $||f^{(k)}||$ is at least $M=M(n,k)>0$. \\medskip We show that this fact remains valid for all ``sufficiently dense'' sets $Z$ (including finite ones). The density of $Z$ is measured via the behavior of the covering numbers of $Z$. In particular, the bound $||f^{(k)}||\\ge \\tilde M=\\tilde M(n,k)>0$ holds for each $Z$ with the box (or Minkowski, or entropy) dimension $\\dim_e(Z)$ greater than $n-\\frac{1}{k}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-05T08:49:11.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "higher derivatives",
      "functions vanishing",
      "fact remains valid",
      "non-empty interior",
      "times continuously differentiable function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}