{
  "id": "2004.05579",
  "version": "v1",
  "published": "2020-04-12T10:16:34.000Z",
  "updated": "2020-04-12T10:16:34.000Z",
  "title": "Reconstruction of piecewise-smooth multivariate functions from Fourier data",
  "authors": [
    "David Levin"
  ],
  "comment": "22 pages, 21 figures",
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "In some applications, one is interested in reconstructing a function $f$ from its Fourier series coefficients. The problem is that the Fourier series is slowly convergent if the function is non-periodic, or is non-smooth. In this paper, we suggest a method for deriving high order approximation to $f$ using a Pad\\'e-like method. Namely, by fitting some Fourier coefficients of the approximant to the given Fourier coefficients of $f$. Given the Fourier series coefficients of a function on a rectangular domain in $\\mathbb{R}^d$, assuming the function is piecewise smooth, we approximate the function by piecewise high order spline functions. First, the singularity structure of the function is identified. For example in the 2-D case, we find high accuracy approximation to the curves separating between smooth segments of $f$. Secondly, simultaneously we find the approximations of all the different segments of $f$. We start by developing and demonstrating a high accuracy algorithm for the 1-D case, and we use this algorithm to step up to the multidimensional case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-12T10:16:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65D15",
      "42B05"
    ],
    "keywords": [
      "piecewise-smooth multivariate functions",
      "fourier data",
      "fourier series coefficients",
      "reconstruction",
      "piecewise high order spline functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}