{
  "abstract": "For a real-valued continuous periodic function $f$, let \\[ A_r f(x)=\\frac{1}{2r}\\int_{x-r}^{x+r}f(t)\\,dt,\\qquad A_0f(x)=f(x), \\] and let $r_f(x)$ be the smallest radius at which $\\sup_{r\\geq 0}A_r f(x)$ is attained. We call $r_f$ the frequency function of $f$. Steinerberger proved that a two-valued condition involving both $r_f$ and $r_{-f}$ characterizes a single sinusoidal function, and asked if the same result holds under the assumption that $r_f$ has finite range (or even range with Lebesgue measure zero). In this paper, we prove that a stronger finite-valued condition for $r_f$ characterizes a sinusoidal function for the case when $f$ is periodic, smooth, and belongs to a quasianalytic Denjoy-Carleman class. However, we also show that the conjecture of Steinerberger is not true in general, since we can construct non-sinusoidal periodic smooth functions in non-quasianalytic Denjoy-Carleman classes such that their frequency functions have finite range.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.24953",
  "author_total": 2,
  "authors": [
    {
      "name": "Carlos Garzón"
    },
    {
      "name": "José Madrid"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/27cftcrve8akp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.24953",
  "fulltext_url": null,
  "id": "27cftcrve8akp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.24953",
  "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.CA"
    ],
    [
      "openalex",
      "2604|Mathematical Analysis and Transform Methods"
    ]
  ],
  "openalex": "W7223005136",
  "pdf_url": "https://arxiv.org/pdf/2610.24953",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/27cftcrve8akp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.24953",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.24953"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On the frequency function of periodic quasianalytic and non-quasianalytic functions",
  "title_text": "On the frequency function of periodic quasianalytic and non-quasianalytic functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/27cftcrve8akp",
  "venue": "arXiv",
  "versions": []
}