{
  "id": "2204.14176",
  "version": "v1",
  "published": "2022-04-29T16:05:49.000Z",
  "updated": "2022-04-29T16:05:49.000Z",
  "title": "The uncertainty principle for the short-time Fourier transform on finite cyclic groups: cases of equality",
  "authors": [
    "Fabio Nicola"
  ],
  "comment": "12 pages",
  "categories": [
    "math.FA",
    "math-ph",
    "math.MP"
  ],
  "abstract": "A well-known version of the uncertainty principle on the cyclic group $\\mathbb{Z}_N$ states that for any couple of functions $f,g\\in\\ell^2(\\mathbb{Z}_N)\\setminus\\{0\\}$, the short-time Fourier transform $V_g f$ has support of cardinality at least $N$. This result can be regarded as a time-frequency version of the celebrated Donoho-Stark uncertainty principle on $\\mathbb{Z}_N$. Unlike the Donoho-Stark principle, however, a complete identification of the extremals is still missing. In this note we provide an answer to this problem by proving that the support of $V_g f$ has cardinality $N$ if and only if it is a coset of a subgroup of order $N$ of $\\mathbb{Z}_N\\times \\mathbb{Z}_N$. Also, we completely identify the corresponding extremal functions $f,g$. Besides translations and modulations, the symmetries of the problem are encoded by certain metaplectic operators associated with elements of ${\\rm SL}(2,\\mathbb{Z}_{N/a})$, where $a$ is a divisor of $N$. Partial generalizations are given to finite Abelian groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-29T16:05:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "short-time fourier transform",
      "finite cyclic groups",
      "finite abelian groups",
      "celebrated donoho-stark uncertainty principle",
      "partial generalizations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}