{
  "abstract": "In this paper, we introduce a new set of $\\mathbb{L}_2$-orthogonal polynomials on the hemisphere $\\mathbb{S}^d_+:=\\{\\mathbf{x}\\in\\mathbb{R}^{d+1}:\\|\\mathbf{x}\\|=1, \\mathbf{x}\\cdot\\mathbf{e}_{d+1}\\geq0\\}$ and establish $\\mathbb{L}_1$ Marcinkiewicz-Zygmund inequalities on the hemisphere for even and odd spherical polynomials. Hemispherical $t$-designs provide equal weight cubature rules on the hemisphere which are exact for polynomials up to degree $t$. We propose variational characterizations of hemispherical $t$-designs by the $\\mathbb{L}_2$-orthogonal polynomials. Moreover, based on the Marcinkiewicz-Zygmund inequalities and the $\\mathbb{L}_2$-orthogonal polynomials, we prove that for each $n\\geq C_d t^d$, there exists a hemispherical $t$-design on the hemisphere with $n$ points, where $C_d$ is a constant depending only on $d$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.39171",
  "author_total": 2,
  "authors": [
    {
      "name": "Chao Li"
    },
    {
      "name": "Xiaojun Chen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bjkxbg90d0716.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.39171",
  "fulltext_url": null,
  "id": "bjkxbg90d0716",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.39171",
  "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": "W7215103667",
  "pdf_url": "https://arxiv.org/pdf/2609.39171",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bjkxbg90d0716.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.39171",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.39171"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Marcinkiewicz-Zygmund inequalities for hemispherical $t$-designs",
  "title_text": "Marcinkiewicz-Zygmund inequalities for hemispherical t-designs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/bjkxbg90d0716",
  "venue": "arXiv",
  "versions": []
}