{
  "abstract": "We study sampling discretization of the uniform norm for trigonometric polynomials with frequencies in a hyperbolic cross of level $N$ on $\\T^d$. For every $d,N\\ge2$ and $\\varepsilon\\in(0,1)$, we construct an explicit norming set with at most $\\bigl(C_1(1+\\varepsilon^{-1})\\bigr)^{d-1}N^{1+\\varepsilon}$ points and norming constant at most $\\bigl(C_2(1+\\varepsilon^{-1})\\bigr)^{d-1}$, where $C_1,C_2$ are absolute constants. We also prove that independent Haar-distributed points achieve the same exponent $1+\\varepsilon$: a sample of size at least $C(d,\\varepsilon)N^{1+\\varepsilon}\\log(2/η)$ norms the entire space simultaneously with probability at least $1-η$, with a norming constant bounded by $\\exp(C_d\\varepsilon^{-d})$ and independent of $N$. The deterministic construction combines uniformly stable de la Vallée Poussin sampling operators with a Smolyak-type combination identity. The random result follows from an abstract norming theorem for sums of spaces associated with commuting regular partitions, together with a multiscale approximation of hyperbolic-cross polynomials. Finally, we show that every norming set of constant $B>1$ for the trigonometric polynomials with frequencies in $\\{-1,0,1\\}^d$ has at least $\\bigl(πd/(2e\\log B)\\bigr)^{d/2}$ points. This rules out quadratic bounds in the dimension of the polynomial space with a prefactor growing only polynomially in $d$ when the norming constant is fixed.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.07679",
  "author_total": 2,
  "authors": [
    {
      "name": "Feng Dai"
    },
    {
      "name": "Andriy Prymak"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3c8yjpzgj6k2f.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.07679",
  "fulltext_url": null,
  "id": "3c8yjpzgj6k2f",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.07679",
  "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.CA"
    ],
    [
      "arxiv",
      "math.FA"
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    [
      "arxiv",
      "math.NA"
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      "arxiv",
      "math.PR"
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    [
      "openalex",
      "2612|Mathematical Approximation and Integration"
    ]
  ],
  "openalex": "W7220912307",
  "pdf_url": "https://arxiv.org/pdf/2610.07679",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3c8yjpzgj6k2f.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.07679",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.07679"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Sampling discretization of the uniform norm for hyperbolic-cross trigonometric polynomials",
  "title_text": "Sampling discretization of the uniform norm for hyperbolic-cross trigonometric polynomials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/3c8yjpzgj6k2f",
  "venue": "arXiv",
  "versions": []
}