{
  "id": "2107.10348",
  "version": "v1",
  "published": "2021-07-21T20:31:41.000Z",
  "updated": "2021-07-21T20:31:41.000Z",
  "title": "How many Fourier coefficients are needed?",
  "authors": [
    "Mihail N. Kolountzakis",
    "Effie Papageorgiou"
  ],
  "comment": "16 pages, 1 figure",
  "categories": [
    "math.CA",
    "cs.NA",
    "math.NA"
  ],
  "abstract": "We are looking at families of functions or measures on the torus (in dimension one and two) which are specified by a finite number of parameters $N$. The task, for a given family, is to look at a small number of Fourier coefficients of the object, at a set of locations that is predetermined and may depend only on $N$, and determine the object. We look at (a) the indicator functions of at most $N$ intervals of the torus and (b) at sums of at most $N$ complex point masses on the two-dimensional torus. In the first case we reprove a theorem of Courtney which says that the Fourier coefficients at the locations $0, 1, \\ldots, N$ are sufficient to determine the function (the intervals). In the second case we produce a set of locations of size $O(N \\log N)$ which suffices to determine the measure.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-21T20:31:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "41A05",
      "41A27",
      "42A15",
      "42A16"
    ],
    "keywords": [
      "fourier coefficients",
      "complex point masses",
      "first case",
      "second case",
      "two-dimensional torus"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}