{
  "id": "0811.4440",
  "version": "v1",
  "published": "2008-11-26T21:59:57.000Z",
  "updated": "2008-11-26T21:59:57.000Z",
  "title": "Continuous Wavelets on Compact Manifolds",
  "authors": [
    "Daryl Geller",
    "Azita Mayeli"
  ],
  "doi": "10.1007/s00209-008-0405-7",
  "categories": [
    "math.FA",
    "math.CA",
    "math.SP"
  ],
  "abstract": "Let $\\bf M$ be a smooth compact oriented Riemannian manifold, and let $\\Delta_{\\bf M}$ be the Laplace-Beltrami operator on ${\\bf M}$. Say $0 \\neq f \\in \\mathcal{S}(\\RR^+)$, and that $f(0) = 0$. For $t > 0$, let $K_t(x,y)$ denote the kernel of $f(t^2 \\Delta_{\\bf M})$. We show that $K_t$ is well-localized near the diagonal, in the sense that it satisfies estimates akin to those satisfied by the kernel of the convolution operator $f(t^2\\Delta)$ on $\\RR^n$. We define continuous ${\\cal S}$-wavelets on ${\\bf M}$, in such a manner that $K_t(x,y)$ satisfies this definition, because of its localization near the diagonal. Continuous ${\\cal S}$-wavelets on ${\\bf M}$ are analogous to continuous wavelets on $\\RR^n$ in $\\mathcal{S}(\\RR^n)$. In particular, we are able to characterize the H$\\ddot{o}$lder continuous functions on ${\\bf M}$ by the size of their continuous ${\\mathcal{S}}-$wavelet transforms, for H$\\ddot{o}$lder exponents strictly between 0 and 1. If $\\bf M$ is the torus $\\TT^2$ or the sphere $S^2$, and $f(s)=se^{-s}$ (the ``Mexican hat'' situation), we obtain two explicit approximate formulas for $K_t$, one to be used when $t$ is large, and one to be used when $t$ is small.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-11-26T21:59:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42C40",
      "42B20",
      "58J40",
      "58J35",
      "35P05"
    ],
    "keywords": [
      "continuous wavelets",
      "compact manifolds",
      "smooth compact oriented riemannian manifold",
      "satisfies estimates akin",
      "explicit approximate formulas"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0811.4440G"
    }
  }
}