{
  "abstract": "How small can the positivity loss be for an $N$-bandlimited spectral operator that exactly reproduces all modes up to degree $L$? For spherical polynomial approximation on $\\mathbb S^d$, we prove that the smallest possible excess of the uniform operator norm above 1, equivalently the least positivity loss, is of sharp order $\\left({L}/{(N+1)}\\right)^2$ when $1\\le L< N$. The lower bound follows from a Fejér peak test and a concentration estimate for bandlimited kernels, while a matching upper bound is obtained by correcting a positive Jackson operator with a smooth filter. We illustrate the result in three settings. On the circle, taking $N=sL-1$, this determines the sharp order of the generalized-projection constant above 1 and identifies the gap between the $s^{-1}$ excess of delayed de la Vallée--Poussin means and the optimal $s^{-2}$ order. For filtered hyperinterpolation, whose operator norm has long been known to be uniformly bounded, we give a quantitative lower bound on its separation from the positivity threshold 1. Finally, we identify an operator-level obstruction to maximum principles.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02695",
  "author_total": 1,
  "authors": [
    {
      "name": "Hao-Ning Wu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/08engcv6b2smq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02695",
  "fulltext_url": null,
  "id": "08engcv6b2smq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02695",
  "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.CA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "2612|Mathematical Approximation and Integration"
    ]
  ],
  "openalex": "W7207637326",
  "pdf_url": "https://arxiv.org/pdf/2609.02695",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/08engcv6b2smq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2609.02695",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02695"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Positivity loss in bandlimited spectral reproduction on spheres",
  "title_text": "Positivity loss in bandlimited spectral reproduction on spheres",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/08engcv6b2smq",
  "venue": "arXiv",
  "versions": []
}