{
  "abstract": "We study the computational cost and stability of positive cubature and hyperinterpolation on a truncated conic surface with a Jacobi-type weight. A classical quadratic disk-to-cone map is formulated as a measure-preserving $\\mathbb Z_2$ quotient. It identifies degree-$m$ conic traces with even disk polynomials of degree at most $2m$ and gives a two-way correspondence between positive degree-$2n$ cone cubature and centrally symmetric positive degree-$(4n+1)$ disk cubature. Consequently, Möller's lower bound and the node excess above it transfer exactly, producing compressed non-product cone rules from near-minimal disk formulas. The quotient also transfers reproducing kernels and hyperinterpolation operators. Every positive degree-$2n$ cone rule yields an exact $L^2$ sampling isometry on $Π_n(V)$, so the weighted sampling matrix has condition number one, independently of the number and geometry of its nodes. The associated Lebesgue constants satisfy the rule-independent bound $Λ_{n,γ}^V\\ge c_γn^{γ+1}$ for every $γ>-1$. For nonnegative half-integer $γ$, the matching upper bound holds, giving the sharp universal order $Λ_{n,γ}^V\\asymp n^{γ+1}$; in particular, $Λ_n^V\\asymp n$ when $γ=0$. Thus node compression preserves exact $L^2$ conditioning and, in this parameter range, optimal-order $L^\\infty$ stability. Low-degree near-minimal rules and higher-degree optimized disk schemes illustrate the reduction in sampling cost.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.36430",
  "author_total": 2,
  "authors": [
    {
      "name": "Congpei An"
    },
    {
      "name": "Yan Ge"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9353a06zf0zd4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.36430",
  "fulltext_url": null,
  "id": "9353a06zf0zd4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.36430",
  "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",
      "math.CA"
    ],
    [
      "openalex",
      "2604|Mathematical functions and polynomials"
    ]
  ],
  "openalex": "W7214897688",
  "pdf_url": "https://arxiv.org/pdf/2609.36430",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9353a06zf0zd4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2609.36430",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.36430"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Positive Cubature Compression and Optimal Hyperinterpolation Stability on a Conic Surface",
  "title_text": "Positive Cubature Compression and Optimal Hyperinterpolation Stability on a Conic Surface",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/9353a06zf0zd4",
  "venue": "arXiv",
  "versions": []
}