{
  "id": "1507.06781",
  "version": "v1",
  "published": "2015-07-24T08:45:28.000Z",
  "updated": "2015-07-24T08:45:28.000Z",
  "title": "Moment problem for symmetric algebras of locally convex spaces",
  "authors": [
    "M. Ghasemi",
    "M. Infusino",
    "S. Kuhlmann",
    "M. Marshall"
  ],
  "comment": "14 pages",
  "categories": [
    "math.FA",
    "math.AG"
  ],
  "abstract": "It is explained how a locally convex (lc) topology $\\tau$ on a real vector space $V$ extends naturally to a locally multiplicatively convex (lmc) topology $\\overline{\\tau}$ on the symmetric algebra $S(V)$. This allows application of the results on lmc topological algebras obtained by Ghasemi, Kuhlmann and Marshall in [J. Funct. Analysis, 266 no.2 (2014) 1041-1049] to obtain representations of $\\overline{\\tau}$-continuous linear functionals $L: S(V)\\rightarrow \\mathbb{R}$ satisfying $L(\\sum S(V)^{2d}) \\subseteq [0,\\infty)$ (more generally, of $\\overline{\\tau}$-continuous linear functionals $L: S(V)\\rightarrow \\mathbb{R}$ satisfying $L(M) \\subseteq [0,\\infty)$ for some $2d$-power module $M$ of $S(V)$) as integrals with respect to uniquely determined Radon measures $\\mu$ supported by special sorts of closed balls in the dual space of $V$. The result is simultaneously more general and less general than the corresponding result of Berezansky, Kondratiev and \\v{S}ifrin in [Mathematical Physics and Applied Mathematics, 12, Kluwer Academic Publishers, 1995], [Ukrain. Mat. \\v{Z}., 23 (1971) 291-306]. It is more general because $V$ can be any locally convex topological space (not just a separable nuclear space), the result holds for arbitrary $2d$-powers (not just squares), and no assumptions of quasi-analyticity are required. It is less general because it is necessary to assume that $L : S(V) \\rightarrow \\mathbb{R}$ is $\\overline{\\tau}$-continuous (not just that $L$ is continuous on the homogeneous parts of degree $k$ of $S(V)$, for each $k\\ge 0$).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-07-24T08:45:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "44A60",
      "14P99"
    ],
    "keywords": [
      "locally convex spaces",
      "symmetric algebra",
      "moment problem",
      "continuous linear functionals",
      "kluwer academic publishers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150706781G"
    }
  }
}