{
  "id": "1809.09726",
  "version": "v1",
  "published": "2018-09-25T21:11:52.000Z",
  "updated": "2018-09-25T21:11:52.000Z",
  "title": "Sharp Remez inequality",
  "authors": [
    "S. Tikhonov",
    "P. Yuditskii"
  ],
  "comment": "12 pages, 1 figure",
  "categories": [
    "math.CA"
  ],
  "abstract": "Let an algebraic polynomial $P_n(\\zeta)$ of degree $n$ be such that $|P_n(\\zeta)|\\le 1$ for $\\zeta\\in E\\subset\\mathbb{T}$ and $|E|\\ge 2\\pi -s$. We prove the sharp Remez inequality $$ \\sup_{\\zeta\\in\\mathbb{T}}|P_n(\\zeta)|\\le \\mathfrak{T}_{n}\\left(\\sec \\frac{s} 4\\right),$$ where $\\mathfrak{T}_{n}$ is the Chebyshev polynomial of degree $n$. The equality holds if and only if $$ P_n(e^{iz})=e^{i(nz/2+c_1)}\\mathfrak{T}_n\\left(\\sec\\frac s 4\\cos \\frac {z-c_0} 2\\right), \\quad c_0,c_1\\in\\mathbb{R}. $$ This gives the solution of the long-standing problem on the sharp constant in the Remez inequality for trigonometric polynomials.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-09-25T21:11:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "41A17",
      "41A44",
      "30C35",
      "41A50"
    ],
    "keywords": [
      "sharp remez inequality",
      "algebraic polynomial",
      "trigonometric polynomials",
      "equality holds",
      "sharp constant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}