{
  "abstract": "We study the spectrality of convolution and product measures. Let~$μ$~be a compactly supported Borel probability measure on~$\\Bbb{R}^d$~such that ~$\\mathcal{Z}(\\widehatμ)\\cup\\{0\\}=T$~is an additive subgroup, and let~$ν$~be a Borel probability measure on~$\\Bbb{R}^d$. We prove that~$μ*ν$~is spectral if and only if~$T$~is a spectrum of~$μ$, ~$ν$~is spectral, and~$suppν$~is contained in a translate of the annihilator of~$T$. Moreover, every spectrum of~$μ*ν$~decomposes into full cosets of~$T$, and every choice of one point from each coset is a spectrum of~$ν$. We also obtain a convolution-factor criterion which strengthens a result of Wu. As applications, we characterize the spectrality of~$L_{[0,1]}*ν$~and prove that the spectrality of~$L_{[0,1]^d}*ν$~implies that of~$ν$~without a compact support assumption. For product measures, we give necessary and sufficient conditions for an orthogonal set to be a spectrum by relating its block structure to spectra of the factors. No compact support assumptions are required.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.33987",
  "author_total": 1,
  "authors": [
    {
      "name": "Ting Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2303f28vq85g6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.33987",
  "fulltext_url": null,
  "id": "2303f28vq85g6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.33987",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2604|Mathematical Analysis and Transform Methods"
    ]
  ],
  "openalex": "W7223833277",
  "pdf_url": "https://arxiv.org/pdf/2610.33987",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2303f28vq85g6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.33987",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.33987"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Spectrality of convolution and product measures: a block decomposition approach",
  "title_text": "Spectrality of convolution and product measures: a block decomposition approach",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/2303f28vq85g6",
  "venue": "arXiv",
  "versions": []
}