{
  "abstract": "We prove that if $f_1,\\ldots,f_N$ is an uniformly bounded orthonormal system, then for all $p\\in[1,2)$ there is an estimate for its Kolmogorov widths: $d_n(\\{f_1,\\dots,f_N\\},L_p) \\gtrsim \\min\\{1,n^{-1/p}N^{1/2}\\}$, $n\\le N/2$. In the regime $n\\asymp N$ this provides the lower bound $N^{-α_p}$ with a sharp exponent $α_p:=1/p-1/2$. Besides that, it follows that a good approximation of such ONS requires the dimension $n\\gtrsim N^{p/2}$. Our second result is an approximation theorem for Fourier matrices of finite abelian groups (this includes the usual DFT matrices). For every $η>0$ there is $a=a(η)>0$ such that any Fourier matrix $F$ of sufficiently large order has an approximation of rank $N^{1-a}$ with the row-wise $\\ell_1$-error at most $N^{1/2+η}$; the exponent $1/2$ is sharp. Consequences include the approximation of the same kind for circulant matrices; the approximation of the trigonometric functions $\\exp(2πi\\langle λ,x\\rangle)$, $λ\\in Λ=K\\cap\\mathbb{Z}^d$, $K$ is symmetric convex, with dimension $|Λ|^{1-a}$ and an optimal error $|Λ|^{-α_p+η}$; bounds for widths of weighted Wiener classes.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.37790",
  "author_total": 1,
  "authors": [
    {
      "name": "Yuri Malykhin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/erjdbbq2yrxxf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.37790",
  "fulltext_url": null,
  "id": "erjdbbq2yrxxf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.37790",
  "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": "W7214915982",
  "pdf_url": "https://arxiv.org/pdf/2609.37790",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/erjdbbq2yrxxf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.37790",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.37790"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Low-dimensional approximation of uniformly bounded orthonormal systems",
  "title_text": "Low-dimensional approximation of uniformly bounded orthonormal systems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/erjdbbq2yrxxf",
  "venue": "arXiv",
  "versions": []
}