{
  "id": "1403.3632",
  "version": "v1",
  "published": "2014-03-14T16:27:07.000Z",
  "updated": "2014-03-14T16:27:07.000Z",
  "title": "Convexity, Moduli of Smoothness and a Jackson-Type Inequality",
  "authors": [
    "Zeev Ditzian",
    "Andriy Prymak"
  ],
  "journal": "Acta Math. Hungar. 130 (2011), no. 3, 254-285",
  "doi": "10.1007/s10474-010-0008-8",
  "categories": [
    "math.CA"
  ],
  "abstract": "For a Banach space $B$ of functions which satisfies for some $m>0$ $$ \\max(\\|F+G\\|_B,\\|F-G\\|_B) \\ge (\\|F\\|^s_B + m\\|G\\|^s_B)^{1/s}, \\forall F,G\\in B \\ (*) $$ a significant improvement for lower estimates of the moduli of smoothness $\\omega^r(f,t)_B$ is achieved. As a result of these estimates, sharp Jackson inequalities which are superior to the classical Jackson type inequality are derived. Our investigation covers Banach spaces of functions on $R^d$ or $T^d$ for which translations are isometries or on $S^{d-1}$ for which rotations are isometries. Results for $C_0$ semigroups of contractions are derived. As applications of the technique used in this paper, many new theorems are deduced. An $L_p$ space with $1<p<\\infty $ satisfies $(*)$ where $s=\\max(p,2),$ and many Orlicz spaces are shown to satisfy $(*)$ with appropriate $s.$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-03-14T16:27:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "41A63",
      "41A17",
      "41A25",
      "46B20",
      "47D60"
    ],
    "keywords": [
      "jackson-type inequality",
      "smoothness",
      "investigation covers banach spaces",
      "classical jackson type inequality",
      "sharp jackson inequalities"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1403.3632D"
    }
  }
}