{
  "id": "1108.5237",
  "version": "v1",
  "published": "2011-08-26T04:39:32.000Z",
  "updated": "2011-08-26T04:39:32.000Z",
  "title": "Comparison of spectra of absolutely regular distributions and applications",
  "authors": [
    "Bolis Basit",
    "Alan J. Pryde"
  ],
  "comment": "24 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "We study the reduced Beurling spectra $sp_{\\Cal {A},V} (F)$ of functions $F \\in L^1_{loc} (\\jj,X)$ relative to certain function spaces $\\Cal{A}\\st L^{\\infty}(\\jj,X)$ and $V\\st L^1 (\\r)$ and compare them with other spectra including the weak Laplace spectrum. Here $\\jj$ is $\\r_+$ or $\\r$ and $X$ is a Banach space. If $F$ belongs to the space $ \\f'_{ar}(\\jj,X)$ of absolutely regular distributions and has uniformly continuous indefinite integral with $0\\not\\in sp_{\\A,\\f(\\r)} (F)$ (for example if F is slowly oscillating and $\\A$ is $\\{0\\}$ or $C_0 (\\jj,X)$), then $F$ is ergodic. If $F\\in \\f'_{ar}(\\r,X)$ and $M_h F (\\cdot)= \\int_0^h F(\\cdot+s)\\,ds$ is bounded for all $h > 0$ (for example if $F$ is ergodic) and if $sp_{C_0(\\r,X),\\f} (F)=\\emptyset$, then ${F}*\\psi \\in C_0(\\r,X)$ for all $\\psi\\in \\f(\\r)$. We show that tauberian theorems for Laplace transforms follow from results about reduced spectra. Our results are more general than previous ones and we demonstrate this through examples",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-08-26T04:39:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A10",
      "44A10",
      "47A35",
      "43A60"
    ],
    "keywords": [
      "absolutely regular distributions",
      "applications",
      "comparison",
      "weak laplace spectrum",
      "function spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1108.5237B"
    }
  }
}