{
  "id": "1203.4124",
  "version": "v4",
  "published": "2012-03-19T14:55:40.000Z",
  "updated": "2013-08-26T13:02:03.000Z",
  "title": "Oscillation and the mean ergodic theorem for uniformly convex Banach spaces",
  "authors": [
    "Jeremy Avigad",
    "Jason Rute"
  ],
  "doi": "10.1017/etds.2013.90",
  "categories": [
    "math.DS",
    "math.FA",
    "math.LO"
  ],
  "abstract": "Let B be a p-uniformly convex Banach space, with p >= 2. Let T be a linear operator on B, and let A_n x denote the ergodic average (1 / n) sum_{i< n} T^n x. We prove the following variational inequality in the case where T is power bounded from above and below: for any increasing sequence (t_k)_{k in N} of natural numbers we have sum_k || A_{t_{k+1}} x - A_{t_k} x ||^p <= C || x ||^p, where the constant C depends only on p and the modulus of uniform convexity. For T a nonexpansive operator, we obtain a weaker bound on the number of epsilon-fluctuations in the sequence. We clarify the relationship between bounds on the number of epsilon-fluctuations in a sequence and bounds on the rate of metastability, and provide lower bounds on the rate of metastability that show that our main result is sharp.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2013-08-26T13:02:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A30",
      "03F60"
    ],
    "keywords": [
      "uniformly convex banach spaces",
      "mean ergodic theorem",
      "oscillation",
      "p-uniformly convex banach space",
      "ergodic average"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1203.4124A"
    }
  }
}