{
  "id": "1512.03183",
  "version": "v1",
  "published": "2015-12-10T09:23:21.000Z",
  "updated": "2015-12-10T09:23:21.000Z",
  "title": "On the Fourier transform of function of two variables which depend only on the maximum of these variables",
  "authors": [
    "R. M. Trigub"
  ],
  "comment": "30 pages; the paper is in Russian, with the title, abstract and key words translated",
  "categories": [
    "math.CA"
  ],
  "abstract": "For functions $f(x_{1},x_{2})=f_{0}\\big(\\max\\{|x_{1}|,|x_{2}|\\}\\big)$ from $L_{1}(\\mathbb{R}^{2})$, sufficient and necessary conditions for the belonging of their Fourier transform $\\widehat{f}$ to $L_{1}(\\mathbb{R}^{2})$ as well as of a function $t\\cdot \\sup\\limits_{y_{1}^{2}+y_{2}^{2}\\geq t^{2}}\\big|\\widehat{f}(y_{1},y_{2})\\big|$ to $L_{1}(\\mathbb{R}^{1}_{+})$. As for the positivity of $\\widehat{f}$ on $\\mathbb{R}^{2}$, it is completely reduced to the same question on $\\mathbb{R}^{1}$ for a function $f_{1}(x)=|x|f_{0}\\big(|x|\\big)+\\int\\limits_{|x|}^{\\infty}f_{0}(t)dt$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-12-10T09:23:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B10"
    ],
    "keywords": [
      "fourier transform",
      "necessary conditions",
      "positivity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "ru",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151203183T"
    }
  }
}