{
  "id": "1809.10786",
  "version": "v1",
  "published": "2018-09-27T22:26:37.000Z",
  "updated": "2018-09-27T22:26:37.000Z",
  "title": "Fast SGL Fourier transforms for scattered data",
  "authors": [
    "Christian Wülker"
  ],
  "categories": [
    "math.NA"
  ],
  "abstract": "Spherical Gauss-Laguerre (SGL) basis functions, i. e., normalized functions of the type $L_{n-l-1}^{(l + 1/2)}(r^2) r^l Y_{lm}(\\vartheta,\\varphi)$, $|m| \\leq l < n \\in \\mathbb{N}$, $L_{n-l-1}^{(l + 1/2)}$ being a generalized Laguerre polynomial, $Y_{lm}$ a spherical harmonic, constitute an orthonormal polynomial basis of the space $L^2$ on $\\mathbb{R}^3$ with radial Gaussian (multivariate Hermite) weight $\\exp(-r^2)$. We have recently described fast Fourier transforms for the SGL basis functions based on an exact quadrature formula with certain grid points in $\\mathbb{R}^3$. In this paper, we present fast SGL Fourier transforms for scattered data. The idea is to employ well-known basal fast algorithms to determine a three-dimensional trigonometric polynomial that coincides with the bandlimited function of interest where the latter is to be evaluated. This trigonometric polynomial can then be evaluated efficiently using the well-known non-equispaced FFT (NFFT). We proof an error estimate for our algorithms and validate their practical suitability in extensive numerical experiments.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-09-27T22:26:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65T50",
      "F.2.1",
      "G.1.2"
    ],
    "keywords": [
      "fast sgl fourier transforms",
      "scattered data",
      "employ well-known basal fast algorithms",
      "trigonometric polynomial",
      "basis functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}