{
  "id": "0906.1162",
  "version": "v1",
  "published": "2009-06-05T16:28:13.000Z",
  "updated": "2009-06-05T16:28:13.000Z",
  "title": "Systems formed by translates of one element in $L_p(\\mathbb R)$",
  "authors": [
    "E. Odell",
    "B. Sari",
    "Th. Schlumprecht",
    "B. Zheng"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $1\\le p <\\infty$, $f\\in L_p(\\real)$ and $\\Lambda\\subseteq \\real$. We consider the closed subspace of $L_p(\\real)$, $X_p (f,\\Lambda)$, generated by the set of translations $f_{(\\lambda)}$ of $f$ by $\\lambda \\in\\Lambda$. If $p=1$ and $\\{f_{(\\lambda)} :\\lambda\\in\\Lambda\\}$ is a bounded minimal system in $L_1(\\real)$, we prove that $X_1 (f,\\Lambda)$ embeds almost isometrically into $\\ell_1$. If $\\{f_{(\\lambda)} :\\lambda\\in\\Lambda\\}$ is an unconditional basic sequence in $L_p(\\real)$, then $\\{f_{(\\lambda)} : \\lambda\\in\\Lambda\\}$ is equivalent to the unit vector basis of $\\ell_p$ for $1\\le p\\le 2$ and $X_p (f,\\Lambda)$ embeds into $\\ell_p$ if $2<p\\le 4$. If $p>4$, there exists $f\\in L_p(\\real)$ and $\\Lambda \\subseteq \\zed$ so that $\\{f_{(\\lambda)} :\\lambda\\in\\Lambda\\}$ is unconditional basic and $L_p(\\real)$ embeds isomorphically into $X_p (f,\\Lambda)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-06-05T16:28:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "translates",
      "unconditional basic sequence",
      "unit vector basis",
      "bounded minimal system"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0906.1162O"
    }
  }
}