{
  "abstract": "We construct a probability measure on the circle with independent convolution powers and uniformly bounded symmetric Fourier partial sums in $L^1$. Every positive convolution power is therefore singular with respect to Haar measure. The measure is Rajchman. The construction answers a question of Andersson concerning measures with bounded Fourier partial sums outside the radical of $L^1$, and contrasts with his theorem that the classical examples of Weiss, Katznelson, and Brown-Hewitt have convolution squares in $L^2$. Our proof combines sparse probability blocks, an endpoint $L\\log L$ estimate, and a diagonal generalized Riesz product. Exact factorization of local moments permits the simultaneous separation of all convolution powers by Brown's singularity criterion. By the classical spectral theory of independent-power measures, its spectrum is the closed unit disc. In particular, it has non-natural spectrum and induces a non-decomposable convolution operator on $L^1$. The image of the measure in $M(\\mathbb T)/L^1(\\mathbb T)$ has spectral radius one, and the closed unital algebra it generates in $M(\\mathbb T)$ is isometrically isomorphic to $\\ell^1(\\mathbb N_0)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34587",
  "author_total": 1,
  "authors": [
    {
      "name": "Przemysław Ohrysko"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9tcvj0qq5tbev.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34587",
  "fulltext_url": null,
  "id": "9tcvj0qq5tbev",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34587",
  "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"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7223810534",
  "pdf_url": "https://arxiv.org/pdf/2610.34587",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9tcvj0qq5tbev.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34587",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34587"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Independent powers and bounded Fourier partial sums in $L^1$",
  "title_text": "Independent powers and bounded Fourier partial sums in L^1",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/9tcvj0qq5tbev",
  "venue": "arXiv",
  "versions": []
}