{
  "abstract": "Spherical harmonics have been studied extensively and are used in many fields of application. We consider the probably earliest application of spherical harmonics, namely for a representation of the solution of the Laplace equation on the $d$-dimensional unit sphere where the boundary data are represented in a spherical harmonics basis. We consider general $d \\geq 2$. For the solution $u$ of the Laplace equation the fundamental maximum principle holds. If one approximates the boundary data in the finite dimensional space spanned by all spherical harmonics of degree at most $N$, the resulting discrete solution, denoted by $u_N$, does in general not satisfy the maximum principle. In this paper we derive sharp estimates for $r_N^{\\rm max}$ such that the maximum principle holds for $u_N(x)$ for all $x \\in \\Bbb{R}^d$ with $\\|x\\| \\leq r_N^{\\rm max}$. We show that for $d=3$ one has $r_N^{\\rm max} \\gtrsim 1- 3 N^{-1} \\ln\\big( 2^\\frac13 N \\big) $. Similar estimates are derived for $d \\neq 3$. We also study the sharpness of these estimates.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.13540",
  "author_total": 1,
  "authors": [
    {
      "name": "Arnold Reusken"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e8ec4s9myqpdb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.13540",
  "fulltext_url": null,
  "id": "e8ec4s9myqpdb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.13540",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "2612|Mathematical Approximation and Integration"
    ]
  ],
  "openalex": "W7221549126",
  "pdf_url": "https://arxiv.org/pdf/2610.13540",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e8ec4s9myqpdb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.13540",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.13540"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "On a maximum principle in discretization with spherical harmonics",
  "title_text": "On a maximum principle in discretization with spherical harmonics",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/e8ec4s9myqpdb",
  "venue": "arXiv",
  "versions": []
}