{
  "abstract": "Let $\\mathbb{D}$ be the ridge dictionary generated by $\\operatorname{ReLU}^k$ on a bounded Lipschitz domain $Ω\\subset\\mathbb{R}^d$. We establish sharp embeddings between Besov spaces and the associated variation space $\\mathcal L_1(\\mathbb{D})$ in the quasi-Banach range $0<p\\leq 1$. Specifically, \\[ B^s_{p,q}(Ω)\\hookrightarrow \\mathcal L_1(\\mathbb{D}) \\] when $s\\geq k+d/p$ for $0<q\\leq 1$, and when $s>k+d/p$ for $1<q\\le\\infty$. A rescaled-bump construction shows that this smoothness threshold is sharp. Conversely, for $0<p<1$, \\[ \\mathcal L_1(\\mathbb{D})\\hookrightarrow B^{k+1}_{p,2}(Ω), \\] and both the smoothness $k+1$ and the fine index $2$ are optimal. The forward embedding converts Besov regularity of functions and solutions of partial differential equations into quantitative finite-width approximation bounds and greedy-algorithm convergence guarantees for shallow $\\operatorname{ReLU}^k$ networks. The proofs use Littlewood-Paley localization, Fourier-Radon representations, measure-valued derivatives, and vector-valued singular integrals.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.00680",
  "author_total": 2,
  "authors": [
    {
      "name": "Yuwen Li"
    },
    {
      "name": "Yupeng Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/efh4cq1cmbtsv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.00680",
  "fulltext_url": null,
  "id": "efh4cq1cmbtsv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.00680",
  "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": "W7206164276",
  "pdf_url": "https://arxiv.org/pdf/2609.00680",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/efh4cq1cmbtsv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2609.00680",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.00680"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sharp embeddings between quasi-Banach Besov spaces and shallow ReLU variation spaces",
  "title_text": "Sharp embeddings between quasi-Banach Besov spaces and shallow ReLU variation spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/efh4cq1cmbtsv",
  "venue": "arXiv",
  "versions": []
}