{
  "id": "math/0302193",
  "version": "v1",
  "published": "2003-02-17T19:20:03.000Z",
  "updated": "2003-02-17T19:20:03.000Z",
  "title": "On pointwise estimates of positive definite functions with given support",
  "authors": [
    "Mihail N. Kolountzakis",
    "Szilard Gy. Revesz"
  ],
  "comment": "19 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "The following problem originated from a question due to Paul Turan. Suppose $\\Omega$ is a convex body in Euclidean space $\\RR^d$ or in $\\TT^d$, which is symmetric about the origin. Over all positive definite functions supported in $\\Omega$, and with normalized value 1 at the origin, what is the largest possible value of their integral? From this Arestov, Berdysheva and Berens arrived to pose the analogous pointwise extremal problem for intervals in $\\RR$. That is, under the same conditions and normalizations, and for any particular point $z\\in\\Omega$, the supremum of possible function values at $z$ is to be found. However, it turns out that the problem for the real line has already been solved by Boas and Kac, who gave several proofs and also mentioned possible extensions to $\\RR^d$ and non-convex domains as well. We present another approach to the problem, giving the solution in $\\RR^d$ and for several cases in $\\TT^d$. In fact, we elaborate on the fact that the problem is essentially one-dimensional, and investigate non-convex open domains as well. We show that the extremal problems are equivalent to more familiar ones over trigonometric polynomials, and thus find the extremal values for a few cases. An analysis of the relation of the problem for the space $\\RR^d$ to that for the torus $\\TT^d$ is given, showing that the former case is just the limiting case of the latter. Thus the hiearachy of difficulty is established, so that trigonometric polynomial extremal problems gain recognition again.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2003-02-17T19:20:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B10",
      "42B10",
      "26D15",
      "42A82",
      "42A05"
    ],
    "keywords": [
      "positive definite functions",
      "pointwise estimates",
      "polynomial extremal problems gain recognition",
      "trigonometric polynomial extremal problems gain"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math......2193K"
    }
  }
}