{
  "abstract": "In 2002 Nielsen and Zhou conjectured that the shifted wavelet packet system associated with a Daubechies filter of length at least four fails to be a Schauder basis of $L^p(\\mathbb{R})$ for $1\\le p\\le\\infty$, $p\\ne2$, and that the packets are not uniformly bounded in $p$-mean across scales for any $p>2$. The basis obstruction throughout the range $1<p<\\infty$, $p\\ne2$, was previously established for length four, via an explicit computation; the full $p$-mean assertion remained open. We prove both for every Daubechies filter of length at least four: the high-pass transition matrices satisfy $ρ_pρ_{p'}>2$ for $1<p<\\infty$, $p\\ne2$, and for $p>2$ the full family of transition matrices has $p$-norm joint spectral radius exceeding the Haar value $4^{1/p}$. The key step is structural: an affine piece of the matrix pressure forces constant spectral radius, which by a theorem of Protasov and Voynov forces simultaneous orthogonality of the transition matrices, incompatible with the double zero of the high-pass symbol. The results extend to every real spectral factor with the Daubechies magnitude response, including the least-asymmetric filters.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32904",
  "author_total": 1,
  "authors": [
    {
      "name": "Morten Nielsen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c1ferq5gcecvx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32904",
  "fulltext_url": null,
  "id": "c1ferq5gcecvx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32904",
  "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.FA"
    ],
    [
      "openalex",
      "2604|Mathematical Analysis and Transform Methods"
    ]
  ],
  "openalex": "W7214711771",
  "pdf_url": "https://arxiv.org/pdf/2609.32904",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c1ferq5gcecvx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.32904",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32904"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Mean size and Schauder-basis properties of Daubechies wavelet packets",
  "title_text": "Mean size and Schauder-basis properties of Daubechies wavelet packets",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/c1ferq5gcecvx",
  "venue": "arXiv",
  "versions": []
}