{
  "id": "1910.02777",
  "version": "v1",
  "published": "2019-10-07T13:25:44.000Z",
  "updated": "2019-10-07T13:25:44.000Z",
  "title": "Wiener algebras and trigonometric series in a coordinated fashion",
  "authors": [
    "E. Liflyand",
    "R. Trigub"
  ],
  "comment": "20 pages",
  "categories": [
    "math.CA",
    "math.CV"
  ],
  "abstract": "Let $W_0(\\mathbb R)$ be the Wiener Banach algebra of functions representable by the Fourier integrals of Lebesgue integrable functions. It is proven in the paper that, in particular, a trigonometric series $\\sum\\limits_{k=-\\infty}^\\infty c_k e^{ikt}$ is the Fourier series of an integrable function if and only if there exists a $\\phi\\in W_0(\\mathbb R)$ such that $\\phi(k)=c_k$, $k\\in\\mathbb Z$. If $f\\in W_0(\\mathbb R)$, then the piecewise linear continuous function $\\ell_f$ defined by $\\ell_f(k)=f(k)$, $k\\in\\mathbb Z$, belongs to $W_0(\\mathbb R)$ as well. Moreover, $\\|\\ell_f\\|_{W_0}\\le \\|f\\|_{W_0}$. Similar relations are established for more advanced Wiener algebras. These results are supplemented by numerous applications. In particular, new necessary and sufficient conditions are proved for a trigonometric series to be a Fourier series and new properties of $W_0$ are established.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-07T13:25:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42A38",
      "42A32",
      "42A50",
      "42A82"
    ],
    "keywords": [
      "trigonometric series",
      "coordinated fashion",
      "fourier series",
      "wiener banach algebra",
      "fourier integrals"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}