{
  "abstract": "Let $R$ be a normalized root system in $\\mathbb{R}^d$ with reflection group $G$, let $k$ be a $G$-invariant multiplicity function, and let $\\mathcal{F}_k$ be the associated Dunkl transform. We write $k_α=k(α)>0$ and $\\mathrm{d}μ_k(x)=w(x)\\,\\mathrm{d}x$, where $w(x)=\\prod_{α\\in R_+}|\\langle x,α\\rangle|^{2k_α}$. We study observability, Hölder-type interpolation, and spectral inequalities for the Dunkl heat equation on $\\mathbb{R}^d$. We establish a Bernstein inequality for ordinary derivatives and a Logvinenko--Sereda theorem with a spectral constant of the form $e^{C(1+N)}$ for functions whose Dunkl transforms are supported in $\\overline{B(0,N)}$. We characterize observable sets as the measurable sets that are thick with respect to $μ_k$, and prove the equivalence of the observability, Hölder-type interpolation, and spectral inequalities.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28993",
  "author_total": 3,
  "authors": [
    {
      "name": "Xingyu Zhao"
    },
    {
      "name": "Longben Wei"
    },
    {
      "name": "Zhiwen Duan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f3dtrp00qsfrq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28993",
  "fulltext_url": null,
  "id": "f3dtrp00qsfrq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28993",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2604|Mathematical Analysis and Transform Methods"
    ]
  ],
  "openalex": "W7214330100",
  "pdf_url": "https://arxiv.org/pdf/2609.28993",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f3dtrp00qsfrq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.28993",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28993"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Observability Inequalities and the Logvinenko--Sereda Theorem for the Dunkl Transform",
  "title_text": "Observability Inequalities and the Logvinenko--Sereda Theorem for the Dunkl Transform",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/f3dtrp00qsfrq",
  "venue": "arXiv",
  "versions": []
}