{
  "id": "2109.09643",
  "version": "v1",
  "published": "2021-09-20T15:53:10.000Z",
  "updated": "2021-09-20T15:53:10.000Z",
  "title": "Fourier coefficients of functions in power-weighted $L_2$-spaces and conditionality constants of bases in Banach spaces",
  "authors": [
    "Jose L. Ansorena"
  ],
  "categories": [
    "math.FA",
    "math.CA"
  ],
  "abstract": "We prove that, given $2<p<\\infty$, the Fourier coefficients of functions in $L_2(\\mathbb{T}, \\lvert t \\rvert^{1-2/p}\\, dt)$ belong to $\\ell_p$, and that, given $1<p<2$, the Fourier series of sequences in $\\ell_p$ belong $L_2(\\mathbb{T}, \\lvert t \\rvert^{2/p-1}\\, dt)$. Then, we apply these results to the study of conditional Schauder bases and conditional almost greedy bases in Banach spaces. Specifically, we prove that, for every $1<p<\\infty$ and every $0\\le \\alpha<1$, there is a Schauder basis of $\\ell_p$ whose conditionality constants grow as $(m^\\alpha)_{m=1}^\\infty$, and there is an almost greedy basis of $\\ell_p$ whose conditionality constants grow as $((\\log m)^\\alpha)_{m=2}^\\infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-20T15:53:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42A16",
      "46B15",
      "41A65"
    ],
    "keywords": [
      "fourier coefficients",
      "banach spaces",
      "conditionality constants grow",
      "greedy basis",
      "schauder basis"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}