{
  "abstract": "It has remained open to identify the necessary and sufficient conditions for the $L^2$ convergence of hyperinterpolation since it was introduced by Sloan in 1995. We show that the $L^1$-$L^2$ Marcinkiewicz-Zygmund (MZ) condition, together with the asymptotic functional approximation property for polynomials, is the answer. We further prove that the optimal $L^1$-$L^2$ MZ constant coincides with the operator norm of the hyperinterpolation operator, and it admits a natural Banach space duality interpretation. With an explicit construction, we also show that Pólya's classical conditions for quadrature convergence are not sufficient for the $L^2$ convergence of hyperinterpolation. This reveals a fundamental distinction between the convergence of linear functionals (quadrature formulas) and that of linear operators (hyperinterpolation operators). We establish a strict logical hierarchy for the stability and accuracy conditions governing the convergence of quadrature and hyperinterpolation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.00525",
  "author_total": 3,
  "authors": [
    {
      "name": "Congpei An"
    },
    {
      "name": "Xiannan Hu"
    },
    {
      "name": "Xiaoming Yuan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2s92e743bmrr6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.00525",
  "fulltext_url": null,
  "id": "2s92e743bmrr6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.00525",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "2604|Mathematical functions and polynomials"
    ]
  ],
  "openalex": "W7172430861",
  "pdf_url": "https://arxiv.org/pdf/2608.00525",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2s92e743bmrr6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2608.00525",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.00525"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "From Pólya's Conditions to a Complete Characterization of the $L^2$ Convergence of Hyperinterpolation",
  "title_text": "From Pólya's Conditions to a Complete Characterization of the L^2 Convergence of Hyperinterpolation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/2s92e743bmrr6",
  "venue": "arXiv",
  "versions": []
}