{
  "id": "1604.01302",
  "version": "v1",
  "published": "2016-04-05T15:36:24.000Z",
  "updated": "2016-04-05T15:36:24.000Z",
  "title": "Wiener's problem for positive definite functions",
  "authors": [
    "Dmitry Gorbachev",
    "Sergey Tikhonov"
  ],
  "comment": "17 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "We study the sharp constant $W_{n}(D)$ in Wiener's inequality for positive definite functions \\[ \\int_{\\mathbb{T}^{n}}|f|^{2}\\,dx\\le W_{n}(D)|D|^{-1}\\int_{D}|f|^{2}\\,dx,\\quad D\\subset \\mathbb{T}^{n}. \\] N. Wiener proved that $W_{1}([-\\delta,\\delta])<\\infty$, $\\delta\\in (0,1/2)$. E. Hlawka showed that $W_{n}(D)\\le 2^{n}$, where $D$ is an origin-symmetric convex body. We sharpen Hlawka's estimates for $D$ being the ball $B^{n}$ and the cube $I^{n}$. In particular, we prove that $W_{n}(B^{n})\\le 2^{(0.401\\ldots +o(1))n}$. We also obtain a lower bound of $W_{n}(D)$. Moreover, for a cube $ D=\\frac1q I^{n}$ with $q=3,4,\\ldots,$ we obtain that $W_{n}(D)=2^{n}$. Our proofs are based on the interrelation between Wiener's problem and the problems of Tur\\'an and Delsarte.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-04-05T15:36:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B05",
      "42B10"
    ],
    "keywords": [
      "positive definite functions",
      "wieners problem",
      "sharpen hlawkas estimates",
      "origin-symmetric convex body",
      "sharp constant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160401302G"
    }
  }
}