{
  "abstract": "We study the eigen-structure of the penalty matrix arising from cubic smoothing splines on equally spaced knots. Using purely matrix-theoretic arguments, we show that its positive eigenvalues are simple, that the associated eigenvectors alternate between even and odd, and that the eigenvector for the $k$th largest eigenvalue has exactly $k+1$ sign changes. The approach provides a direct and transparent alternative to existing variational proofs of the oscillation property. These results show that equally spaced knots support a spline basis with both a parity structure and an oscillation pattern.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.29781",
  "author_total": 4,
  "authors": [
    {
      "name": "Shih-Hao Huang"
    },
    {
      "name": "Jephian C. -H. Lin"
    },
    {
      "name": "ShengLi Tzeng"
    },
    {
      "name": "Tzu-Lun Yuan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3aaab3dwdv2wj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.29781",
  "fulltext_url": null,
  "id": "3aaab3dwdv2wj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.29781",
  "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.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "1703|Matrix Theory and Algorithms"
    ]
  ],
  "openalex": "W7204874706",
  "pdf_url": "https://arxiv.org/pdf/2608.29781",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3aaab3dwdv2wj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.29781",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.29781"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Sturm-Liouville-Type Parity and Oscillation of a Cubic Spline Eigenbasis",
  "title_text": "Sturm-Liouville-Type Parity and Oscillation of a Cubic Spline Eigenbasis",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:54Z",
  "url": "https://research.draconitic.com/paper/3aaab3dwdv2wj",
  "venue": "arXiv",
  "versions": []
}