{
  "id": "0904.4471",
  "version": "v2",
  "published": "2009-04-28T19:33:59.000Z",
  "updated": "2009-12-29T18:49:06.000Z",
  "title": "Redundancy for localized and Gabor frames",
  "authors": [
    "Radu Balan",
    "Pete Casazza",
    "Zeph Landau"
  ],
  "comment": "35 pages. Formulation of the Gabor results strengthened. An acknowledgement section added. To appear in the Israel Journal of Mathematics",
  "categories": [
    "math.FA"
  ],
  "abstract": "Redundancy is the qualitative property which makes Hilbert space frames so useful in practice. However, developing a meaningful quantitative notion of redundancy for infinite frames has proven elusive. Though quantitative candidates for redundancy exist, the main open problem is whether a frame with redundancy greater than one contains a subframe with redundancy arbitrarily close to one. We will answer this question in the affirmative for $\\ell^1$-localized frames. We then specialize our results to Gabor multi-frames with generators in $M^1(\\R^d)$, and Gabor molecules with envelopes in $W(C,l^1)$. As a main tool in this work, we show there is a universal function $g(x)$ so that for every $\\epsilon>0$, every Parseval frame $\\{f_i\\}_{i=1}^M$ for an $N$-dimensional Hilbert space $H_N$ has a subset of fewer than $(1+\\epsilon)N$ elements which is a frame for $H_N$ with lower frame bound $g(\\epsilon/(2\\frac{M}{N}-1))$. This work provides the first meaningful quantative notion of redundancy for a large class of infinite frames. In addition, the results give compelling new evidence in support of a general definition of reudndancy given in [7].",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-12-29T18:49:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46C99"
    ],
    "keywords": [
      "redundancy",
      "gabor frames",
      "infinite frames",
      "hilbert space frames",
      "dimensional hilbert space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0904.4471B"
    }
  }
}