{
  "abstract": "We develop the $L^p$ space and approximation theory for shallow neural networks with $\\mathrm{ReLU}^k$ activations. The central object is the Radon-domain $L^p$ space $\\mathcal{R}L^p_k(Ω)$ containing all functions on a bounded domain $Ω$ that admit a ridge integral representation whose coefficient density belongs to $L^p$ in the Radon domain. In the Hilbert case $p=2$, we prove by elementary Fourier analysis that this space recovers the critical Sobolev space $H^{k+(d+1)/2}(Ω)$. For general $1<p<\\infty$, the identity becomes a sandwich for Bessel-potential Sobolev spaces. The sharp gap of each side is exactly the Seeger--Sogge--Stein loss for the Radon transform as a Fourier integral operator. This also clarifies how the activation regularity and Radon back-projection jointly produce the regularity. As an application, we discretize the integral representation using a deterministic interpolation skeleton plus uniform sampling. This yields high-probability $L^p$ approximation rates and the optimal Hilbert rate $O\\!\\big(n^{-\\frac12-\\frac{2k+1}{2d}}\\big)$ at $p=2$ for linearized neural networks.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.24795",
  "author_total": 2,
  "authors": [
    {
      "name": "Juncai He"
    },
    {
      "name": "Zitong Tian"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/efavp6018ckk2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.24795",
  "fulltext_url": null,
  "id": "efavp6018ckk2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.24795",
  "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",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "2604|Mathematical Analysis and Transform Methods"
    ]
  ],
  "openalex": "W7165770242",
  "pdf_url": "https://arxiv.org/pdf/2606.24795",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/efavp6018ckk2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2606.24795",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.24795"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Sharp Sobolev Sandwich and Approximation Rates of Radon-Domain $L^p$ Ridge Integral Spaces for ReLU$^k$ Networks",
  "title_text": "Sharp Sobolev Sandwich and Approximation Rates of Radon-Domain L^p Ridge Integral Spaces for ReLU^k Networks",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/efavp6018ckk2",
  "venue": "arXiv",
  "versions": []
}