{
  "id": "1502.05806",
  "version": "v3",
  "published": "2015-02-20T09:08:01.000Z",
  "updated": "2015-04-10T03:36:17.000Z",
  "title": "Fully discrete needlet approximation on the sphere",
  "authors": [
    "Yu Guang Wang",
    "Quoc T. Le Gia",
    "Ian H. Sloan",
    "Robert S. Womersley"
  ],
  "comment": "30 pages, 6 figures, updated the introduction, included references of Dai and Mhaskar, added a remark for Theorem 4.3",
  "categories": [
    "math.NA"
  ],
  "abstract": "Spherical needlets are highly localised radial polynomials on the sphere $\\mathbb{S}^{d}\\subset\\mathbb{R}^{d+1}$, $d\\geq2$, with centers at the nodes of a suitable quadrature rule. The original semidiscrete spherical needlet approximation has coefficients defined by inner product integrals. We use an appropriate quadrature rule to construct a fully discrete version. We prove that the fully discrete spherical needlet approximation is equivalent to filtered hyperinterpolation, that is to a filtered Fourier-Laplace series partial sum with inner products replaced by appropriate quadrature sums. We establish $\\mathbb{L}_{p}$-error bounds and rates of convergence, $2\\leq p\\leq \\infty$, for the fully discrete needlet approximation of functions in Sobolev spaces $\\mathbb{W}_{p}^{s}(\\mathbb{S}^{d})$ for $s>d/p$. The power of the needlet approximation for local approximation is shown by a numerical experiment that uses low-level needlets globally together with high-level needlets in a local region.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-02-20T09:08:01.000Z",
      "abstract": "Spherical needlets are highly localised radial polynomials on the sphere $\\mathbb{S}^{d}\\subset\\mathbb{R}^{d+1}$, $d\\geq2$, with centers at the nodes of a suitable quadrature rule. They provide a multiscale decomposition for real $\\mathbb{L}_{2}$ functions on $\\mathbb{S}^{d}$. The original spherical needlet decomposition has its coefficients defined by inner product integrals. In this paper, we use an additional quadrature rule to construct a fully discrete version of the original semidiscrete spherical needlet approximation. We prove that the fully discrete spherical needlet approximation is equivalent to filtered hyperinterpolation, that is to a filtered Fourier-Laplace series partial sum with inner products replaced by appropriate quadrature sums. From this, we establish $\\mathbb{L}_{p}$-error bounds, $2\\leq p\\leq \\infty$, for the fully discrete needlet approximation of functions in Sobolev spaces $\\mathbb{W}_p^s(\\mathbb{S}^{d})$ for $s>d/p$. In particular, the $\\mathbb{L}_p$ error for the fully discrete approximation loses convergence order compared to the semidiscrete needlet approximation only by the exponent $d/p+\\epsilon$ for $\\epsilon>0$, as expected from the embedding of $\\mathbb{W}_p^s(\\mathbb{S}^{d})$ in $C(\\mathbb{S}^{d})$. The theory is illustrated numerically by the approximation of a function of known smoothness, using symmetric spherical designs (for both the needlet quadrature and the inner product quadrature). The power of the needlet approximation for local approximation is shown by a numerical experiment that uses low-level needlets globally together with high-level needlets in a local region.",
      "comment": "30 pages, 6 figures",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-04-10T03:36:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42C40",
      "33C55",
      "65D32",
      "43A90"
    ],
    "keywords": [
      "fully discrete needlet approximation",
      "fourier-laplace series partial sum",
      "approximation loses convergence order",
      "discrete approximation loses convergence",
      "semidiscrete spherical needlet approximation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150205806W"
    }
  }
}