{
  "abstract": "We present new short exponential sum approximations of length $N$ for $f_1(x)=\\frac{1}{a+x}$ with $a>0$ on $[0, \\infty)$ and for $f_2(x)= {\\mathrm e}^{-x^2/2σ}$ with $σ>0$ on ${\\mathbb R}$ with geometric error decay $ρ^{-2N}$ for user-defined $N \\ge 2$ and $ρ>1$. The approximations are built over consecutive intervals $[b_j, \\, b_{j+1}) \\subset [0, \\infty)$, $j \\in {\\mathbb N}_{0}$, with interval lengths that depend on $ρ$ and grow exponentially for $f_1$ and are equidistant for $f_2$. All parameters determining the exponential sum approximations on $[b_j, \\, b_{j+1})$ are easily computed from the initial parameters on $[b_0, \\, b_{1})$, ensuring numerical stability. Our method is based on Gauss-Laguerre and Gauss-Hermite quadrature, respectively, applied to suitable parametric integral representations of $f_1$ and $f_2$. This technique ensures consistent relative errors across all intervals. Using the obtained exponential sum approximations, we achieve highly accurate approximations of $\\log(x)$ on $[1,\\infty)$ and of the error function $\\mathrm{erf}(x)$ with predictable geometric error decay. Numerical examples for $N=8$ and $N=10$ clearly illustrate the theoretical error estimates.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.03855",
  "author_total": 3,
  "authors": [
    {
      "name": "Gerlind Plonka"
    },
    {
      "name": "Yannick Riebe"
    },
    {
      "name": "Annie Cuyt"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cjjspk2xn5d8p.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.03855",
  "fulltext_url": null,
  "id": "cjjspk2xn5d8p",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.03855",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2606.03855",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cjjspk2xn5d8p.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2606.03855",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.03855"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Approximation by short exponential sums with geometric error decay based on Gauss quadrature",
  "title_text": "Approximation by short exponential sums with geometric error decay based on Gauss quadrature",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/cjjspk2xn5d8p",
  "venue": "arXiv",
  "versions": []
}