{
  "id": "2010.05107",
  "version": "v1",
  "published": "2020-10-10T22:07:25.000Z",
  "updated": "2020-10-10T22:07:25.000Z",
  "title": "Kolmogorov widths of Besov classes $B^1_{1,θ}$ and products of octahedra",
  "authors": [
    "Yuri Malykhin"
  ],
  "categories": [
    "math.FA",
    "math.CA"
  ],
  "abstract": "In this paper we find the orders of decay for Kolmogorov widths of some Besov classes related to $W^1_1$ (the behaviour of the widths for $W^1_1$ remains unknown): $$ d_n(B^1_{1,\\theta}[0,1],L_q[0,1])\\asymp n^{-1/2}\\log^{\\max(\\frac12,1-\\frac{1}{\\theta})}n,\\quad 2<q<\\infty. $$ The proof relies on the lower bound for widths of product of octahedra in a special norm (maximum of two weighted $\\ell_q$ norms). This bound generalizes the theorem of B.S.~Kashin on widths of octahedra in $\\ell_q^N$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-10T22:07:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "kolmogorov widths",
      "besov classes",
      "proof relies",
      "lower bound",
      "remains unknown"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}