{
  "id": "1704.04673",
  "version": "v1",
  "published": "2017-04-15T18:19:34.000Z",
  "updated": "2017-04-15T18:19:34.000Z",
  "title": "Sufficient conditions for convergence of multiple Fourier series with $J_k$-lacunary sequence of rectangular partial sums in terms of Weyl multipliers",
  "authors": [
    "I. L. Bloshanskii",
    "S. K. Bloshanskaya",
    "D. A. Grafov"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "We obtain sufficient conditions for convergence (almost everywhere) of multiple trigonometric Fourier series of functions $f$ in $L_2$ in terms of Weyl multipliers. We consider the case where rectangular partial sums of Fourier series $S_n(x;f)$ have indices $n=(n_1,\\dots,n_N) \\in \\mathbb Z^N$, $N\\ge 3$, in which $k$ $(1\\leq k\\leq N-2)$ components on the places $\\{j_1,\\dots,j_k\\}=J_k \\subset \\{1,\\dots,N\\} = M$ are elements of (single) lacunary sequences (i.e., we consider the, so called, multiple Fourier series with $J_k$-lacunary sequence of partial sums). We prove that for any sample $J_k\\subset M$ the Weyl multiplier for convergence of these series has the form $W(\\nu)=\\prod \\limits_{j=1}^{N-k} \\log(|\\nu_{{\\alpha}_j}|+2)$, where $\\alpha_j\\in M\\setminus J_k $, $\\nu=(\\nu_1,\\dots,\\nu_N)\\in{\\mathbb Z}^N$. So, the \"one-dimensional\" Weyl multiplier -- $\\log(|\\cdot|+2)$ -- presents in $W(\\nu)$ only on the places of \"free\" (nonlacunary) components of the vector $\\nu$. Earlier, in the case where $N-1$ components of the index $n$ are elements of lacunary sequences, convergence almost everywhere for multiple Fourier series was obtained in 1977 by M.Kojima in the classes $L_p$, $p>1$, and by D.K.Sanadze, Sh.V.Kheladze in Orlizc class. Note, that presence of two or more \"free\" components in the index $n$ (as follows from the results by Ch.Fefferman (1971)) does not guarantee the convergence almost everywhere of $S_n(x;f)$ for $N\\geq 3$ even in the class of continuous functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-15T18:19:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42C10",
      "42B05"
    ],
    "keywords": [
      "multiple fourier series",
      "rectangular partial sums",
      "weyl multiplier",
      "lacunary sequence",
      "sufficient conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}