{
  "id": "math-ph/0406057",
  "version": "v2",
  "published": "2004-06-24T11:21:50.000Z",
  "updated": "2006-12-15T12:42:38.000Z",
  "title": "Wavelet Transform on the Circle and the Real Line: A Unified Group-Theoretical Treatment",
  "authors": [
    "Manuel Calixto",
    "Julio Guerrero"
  ],
  "comment": "32 pages, LaTeX, 1 figure. Final version published in ACHA",
  "journal": "Appl. Comput. Harmon. Anal. 21, 204-229 (2006)",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "We present a unified group-theoretical derivation of the Continuous Wavelet Transform (CWT) on the circle $\\mathbb S^1$ and the real line $\\mathbb{R}$, following the general formalism of Coherent States (CS) associated to unitary square integrable (modulo a subgroup, possibly) representations of the group $SL(2,\\mathbb{R})$. A general procedure for obtaining unitary representations of a group $G$ of affine transformations on a space of signals $L^2(X,dx)$ is described, relating carrier spaces $X$ to (first or higher-order) ``polarization subalgebras'' ${\\cal P}_X$. We also provide explicit admissibility and continuous frame conditions for wavelets on $\\mathbb S^1$ and discuss the Euclidean limit in terms of group contraction.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-12-15T12:42:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "real line",
      "unified group-theoretical treatment",
      "affine transformations",
      "general formalism",
      "coherent states"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math.ph...6057C"
    }
  }
}