{
  "id": "2311.11800",
  "version": "v1",
  "published": "2023-11-20T14:33:16.000Z",
  "updated": "2023-11-20T14:33:16.000Z",
  "title": "The «test space and pairing» idea for frames and some generalized characterizations and topological properties of Euclidean continuous frames",
  "authors": [
    "Nizar El Idrissi",
    "Samir Kabbaj",
    "Brahim Moalige"
  ],
  "comment": "14 pages",
  "journal": "Nizar El Idrissi, Samir Kabbaj, and Brahim Moalige. Some characterizations of frames in $\\ell^2(I; H)$ and topological applications. Proyecciones (Antofagasta), 41(5):1141-1152, 2022",
  "doi": "10.22199/issn.0717-6279-4043",
  "categories": [
    "math.FA",
    "math.GN"
  ],
  "abstract": "We introduce the \"test space and pairing\" idea for frames and apply it to the $\\ell^2$ and $L^2$ spaces. First, we show that for every $J \\neq \\emptyset$, the notions of a classical $I$-frame with values in $H$ and a $J$-extended classical $I$-frame with values in $H$ are the same. The definition of a $J$-extended classical $I$-frame with values in $H$, $u = (u_i)_{i \\in I}$, utilizes the \"test space and pairing\" idea by replacing the usual \"test space\" $H$ with $\\ell^2(J;H)$ and the usual \"pairing\" ${P_u : (v ; (u_i)_{i \\in I}) \\in H \\times \\mathcal{F}_I^H \\mapsto (\\langle v , u_i \\rangle)_{i \\in I} \\in \\ell^2(I;\\mathbb{F})}$ with ${P_u^J : ((v_j)_{j \\in J} ; (u_i)_{i \\in I}) \\in \\ell^2(J;H) \\times \\mathcal{F}_I^H \\mapsto (\\langle v_j , u_i \\rangle)_{(i,j) \\in I \\times J} \\in \\ell^2(I \\times J;\\mathbb{F})}$. Secondly, we prove a similar result when the space $\\ell^2(J;H)$ is replaced with the space $L^2(Y,\\nu;H)$ and the frame $u = (u_x)_{x \\in X}$ is $(X,\\mu)$-continuous. Besides, we define the $J$-extended and $(Y,\\nu)$-extended analysis, synthesis, and frame operators of the frame $u$ and note that they are just natural block-diagonal operators. After that, we generalize quite straightforwardly the well-known characterizations of Euclidean finite frames to the corresponding characterizations of Euclidean continuous frames. One tool that we use in this endeavor is some rewritings of the quotients $N(v ; (u_x)_{x \\in X})$ and $N( (v_y)_{y \\in Y}) ; (u_x)_{x \\in X} )$. Besides, we give a simple sufficient condition for having a frame with values in $\\mathbb{F}^2$ and use it to provide an example of a classical $\\mathbb{N}^*$-frame with values in $\\mathbb{C}^2$. Finally, we generalize some topological properties of the set of frames and Parseval frames from the Euclidean finitely indexed case to the Euclidean continuous one.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-20T14:33:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42C15",
      "54D99"
    ],
    "keywords": [
      "euclidean continuous frames",
      "topological properties",
      "generalized characterizations",
      "test space",
      "simple sufficient condition"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}