{
  "id": "1201.2096",
  "version": "v2",
  "published": "2012-01-10T16:11:08.000Z",
  "updated": "2012-08-09T16:26:01.000Z",
  "title": "Fréchet frames, general definition and expansions",
  "authors": [
    "Stevan Pilipović",
    "Diana T. Stoeva"
  ],
  "comment": "A new section is added and a minor revision is done",
  "categories": [
    "math.FA"
  ],
  "abstract": "We define an {\\it $(X_1,\\Theta, X_2)$-frame} with Banach spaces $X_2\\subseteq X_1$, $|\\cdot|_1 \\leq |\\cdot|_2$, and a $BK$-space $(\\Theta, \\snorm[\\cdot])$. Then by the use of decreasing sequences of Banach spaces ${X_s}_{s=0}^\\infty$ and of sequence spaces ${\\Theta_s}_{s=0}^\\infty$, we define a general Fr\\' echet frame on the Fr\\' echet space $X_F=\\bigcap_{s=0}^\\infty X_s$. We give frame expansions of elements of $X_F$ and its dual $X_F^*$, as well of some of the generating spaces of $X_F$ with convergence in appropriate norms. Moreover, we give necessary and sufficient conditions for a general pre-Fr\\' echet frame to be a general Fr\\' echet frame, as well as for the complementedness of the range of the analysis operator $U:X_F\\to\\Theta_F$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-08-09T16:26:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42C15",
      "46A13"
    ],
    "keywords": [
      "fréchet frames",
      "general definition",
      "echet frame",
      "banach spaces",
      "analysis operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1201.2096P"
    }
  }
}