{
  "id": "2109.14454",
  "version": "v2",
  "published": "2021-09-29T14:40:23.000Z",
  "updated": "2022-02-07T00:17:27.000Z",
  "title": "Discretizing $L_p$ norms and frame theory",
  "authors": [
    "Daniel Freeman",
    "Dorsa Ghoreishi"
  ],
  "comment": "17 pages, version 2",
  "categories": [
    "math.FA",
    "cs.NA",
    "math.NA"
  ],
  "abstract": "Given an $N$-dimensional subspace $X$ of $L_p([0,1])$, we consider the problem of choosing $M$-sampling points which may be used to discretely approximate the $L_p$ norm on the subspace. We are particularly interested in knowing when the number of sampling points $M$ can be chosen on the order of the dimension $N$. For the case $p=2$ it is known that $M$ may always be chosen on the order of $N$ as long as the subspace $X$ satisfies a natural $L_\\infty$ bound, and for the case $p=\\infty$ there are examples where $M$ may not be chosen on the order of $N$. We show for all $1\\leq p<2$ that there exist classes of subspaces of $L_p([0,1])$ which satisfy the $L_\\infty$ bound, but where the number of sampling points $M$ cannot be chosen on the order of $N$. We show as well that the problem of discretizing the $L_p$ norm of subspaces is directly connected with frame theory. In particular, we prove that discretizing a continuous frame to obtain a discrete frame which does stable phase retrieval requires discretizing both the $L_2$ norm and the $L_1$ norm on the range of the analysis operator of the continuous frame.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-02-07T00:17:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E30",
      "46B15",
      "42C15",
      "65J05"
    ],
    "keywords": [
      "frame theory",
      "sampling points",
      "discretizing",
      "continuous frame",
      "stable phase retrieval"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}