{
  "id": "1811.12353",
  "version": "v1",
  "published": "2018-11-29T18:03:37.000Z",
  "updated": "2018-11-29T18:03:37.000Z",
  "title": "Unconditional Frames of Translates in $L_p(\\mathbb{R}^d)$",
  "authors": [
    "Miguel Berasategui",
    "Daniel Carando"
  ],
  "comment": "19 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "We show that, for $1<p \\le 2$, the space $L_p(\\mathbb{R}^d)$ does not admit unconditional Schauder frames $\\left\\lbrace f_i,f_i'\\right\\rbrace_{i\\in\\mathbb{N}}$ where $\\left\\lbrace f_i\\right\\rbrace$ is a sequence of translates of finitely many functions and $\\left\\lbrace f_i'\\right\\rbrace$ is seminormalized. In fact, the only subspaces of $L_p(\\mathbb{R}^d)$ admitting such Banach frames are those isomorphic to $\\ell_p$. On the other hand, if $2<p<+\\infty$ and $\\left\\lbrace \\lambda_{i}\\right\\rbrace_{i\\in\\mathbb{N}}\\subseteq \\mathbb{R}^d$ is an unbounded sequence, there is a subsequence $\\left\\lbrace \\lambda_{m_i}\\right\\rbrace_{i\\in\\mathbb{N}}$, a function $f\\in L_p(\\mathbb{R}^d)$, and a seminormalized sequence of bounded functionals $\\left\\lbrace f_i'\\right\\rbrace_{i\\in\\mathbb{N}}$ such that $\\left\\lbrace T_{\\lambda_{m_i}}f,f_i'\\right\\rbrace_{i\\in\\mathbb{N}}$ is an unconditional Schauder frame for $L_p(\\mathbb{R}^d)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-29T18:03:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B20"
    ],
    "keywords": [
      "unconditional frames",
      "translates",
      "admit unconditional schauder frames",
      "banach frames",
      "unbounded sequence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}