{
  "id": "1211.4446",
  "version": "v1",
  "published": "2012-11-19T14:48:04.000Z",
  "updated": "2012-11-19T14:48:04.000Z",
  "title": "On convergence and growth of sums $\\sum c_k f(kx)$",
  "authors": [
    "Istvan Berkes"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "For a periodic function $f$ with bounded variation and integral zero on its period interval, we show that $\\sum_{k=1}^\\infty c_k^2 (\\log\\log k)^\\gamma <\\infty$, $\\gamma>4$ implies the almost everywhere convergence of $\\sum_{k=1}^\\infty c_k f(n_kx)$ for any increasing sequence $(n_k)$ of integers. We also construct an example showing that the previous condition is not sufficient for $\\gamma<2$. Finally we give an a.e. bound for the growth of sums $\\sum_{k=1}^N f(n_kx)$ differing from the corresponding optimal result for trigonometric sums by a loglog factor.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-11-19T14:48:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "convergence",
      "integral zero",
      "period interval",
      "periodic function",
      "corresponding optimal result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1211.4446B"
    }
  }
}