{
  "id": "2405.17943",
  "version": "v1",
  "published": "2024-05-28T08:17:14.000Z",
  "updated": "2024-05-28T08:17:14.000Z",
  "title": "Shift-invariant subspaces of Sobolev type",
  "authors": [
    "Aleksandar Aksentijević",
    "Suzana Aleksić"
  ],
  "comment": "This paper has seven pages and thirteen references",
  "categories": [
    "math.FA"
  ],
  "abstract": "This paper has the characteristics of a review paper in which results of shift-invariant subspaces of Sobolev type are summarized without proofs. The structure of shift-invariant spaces $V_s$, $s\\in\\mathbb{R}$, generated by at most countable family of generators, which are subspaces of Sobolev spaces $H^s(\\mathbb{R}^n)$, are announced in \\cite{aap} and Bessel sequences, frames and Riesz families of such spaces are characterized. With the Fourier multiplier $\\left(1-\\frac{\\Delta}{4\\pi^2}\\right)^{s/2}f=\\mathcal{F}^{-1}\\big((1+|t|^2)^{s/2}\\widehat{f}(t)\\big)$, we are able to extend notions and theorems in \\cite{MB} to spaces of the Sobolev type.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-28T08:17:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42C15",
      "42B30"
    ],
    "keywords": [
      "sobolev type",
      "shift-invariant subspaces",
      "extend notions",
      "review paper",
      "sobolev spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}