{
  "id": "2106.00511",
  "version": "v1",
  "published": "2021-06-01T14:16:08.000Z",
  "updated": "2021-06-01T14:16:08.000Z",
  "title": "Completion versus removal of redundancy by perturbation",
  "authors": [
    "Ole Christensen",
    "Marzieh Hasannasab"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "A sequence $\\{g_k\\}_{k=1}^\\infty$ in a Hilbert space $\\cal H$ has the expansion property if each $f\\in \\overline{\\text{span}} \\{g_k\\}_{k=1}^\\infty$ has a representation $f= \\sum_{k=1}^\\infty c_k g_k$ for some scalar coefficients $c_k.$ In this paper we analyze the question whether there exist small norm-perturbations of $\\{g_k\\}_{k=1}^\\infty$ which allow to represent all $f\\in \\cal H;$ the answer turns out to be yes for frame sequences and Riesz sequences, but no for general basic sequences. The insight gained from the analysis is used to address a somewhat dual question, namely, whether it is possible to remove redundancy from a sequence with the expansion property via small norm-perturbations; we prove that the answer is yes for frames $\\{g_k\\}_{k=1}^\\infty$ such that $g_k\\to 0$ as $k\\to \\infty,$ as well as for frames with finite excess. This particular question is motivated by recent progress in dynamical sampling.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-01T14:16:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42C40"
    ],
    "keywords": [
      "redundancy",
      "small norm-perturbations",
      "expansion property",
      "completion",
      "general basic sequences"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}