{
  "id": "2308.10439",
  "version": "v1",
  "published": "2023-08-21T03:23:47.000Z",
  "updated": "2023-08-21T03:23:47.000Z",
  "title": "On the Approximation of Singular Functions by Series of Non-integer Powers",
  "authors": [
    "Mohan Zhao",
    "Kirill Serkh"
  ],
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "In this paper, we describe an algorithm for approximating functions of the form $f(x)=\\int_{a}^{b} x^{\\mu} \\sigma(\\mu) \\, d \\mu$ over $[0,1] \\subset \\mathbb{R}$, where $0<a<b<\\infty$ and $\\sigma(\\mu)$ is some signed Radon measure over $[a,b]$ or some distribution supported on $[a,b]$. Given the desired accuracy $\\epsilon$ and the values of $a$ and $b$, our method determines a priori a collection of non-integer powers $\\{t_j\\}_{j=1}^N$, so that the functions are approximated by series of the form $f(x)\\approx \\sum_{j=1}^N c_j x^{t_j}$, where the expansion coefficients can be found by solving a square, low-dimensional Vandermonde-like linear system using the collocation points $\\{x_j\\}_{j=1}^N$, also determined a priori by $\\epsilon$ and the values of $a$ and $b$. We prove that our method has a small uniform approximation error which is proportional to $\\epsilon$ multiplied by some small constants. We demonstrate the performance of our algorithm with several numerical experiments, and show that the number of singular powers and collocation points grows as $N=O(\\log{\\frac{1}{\\epsilon}})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-21T03:23:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "41A30",
      "41A58",
      "65D15"
    ],
    "keywords": [
      "non-integer powers",
      "singular functions",
      "small uniform approximation error",
      "low-dimensional vandermonde-like linear system",
      "collocation points grows"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}