{
  "abstract": "We study Volterra memory terms with locally integrable completely monotone kernels on a finite time interval $(0,\\Tend)$ and ask how much $L^{2}$ coercivity they retain at high temporal frequencies. The main tool is an exact formula for the Rayleigh quotients of the cosine modes $ψ_n(t)=(2/\\Tend)^{1/2}\\cos(2πnt/\\Tend)$. It shows that these quotients lie between $(1-κ_n/(2πn))\\,m(2πn/\\Tend)$ and $m(2πn/\\Tend)$, where $m$ is the real memory symbol and $κ_n\\in[1-e^{-2πn},1]$ is the best constant valid for all such kernels. Consequently, for non-constant kernels the algebraic decay rate of the quotients does not depend on $\\Tend$ and coincides with the decay index $ρ\\in[0,2]$ of $m$; any value in $[0,2]$ occurs. The Gaussian kernel $e^{-t^{2}}$, which is of positive type but not completely monotone, shows that this may fail otherwise: its quotients decay like $n^{-4}$, while its symbol decays faster than any power. We also show that the largest Rayleigh quotient on $(0,\\Tend)$ is a continuous and strictly increasing function of $\\Tend$, so that a kernel of total mass larger than one has exactly one critical horizon. Finally, because the memory operator on $(0,\\Tend)$ is compact, it does not contribute to the uniform $L^{2}$ coercivity constant, and for $k_n(t)=ne^{-nt}$ the associated operators converge to the identity strongly but not in norm. The diffusion equation with memory serves as a model throughout.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.12482",
  "author_total": 1,
  "authors": [
    {
      "name": "Hiroki Ishizaka"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/372d2gva61yz1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.12482",
  "fulltext_url": null,
  "id": "372d2gva61yz1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.12482",
  "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.AP"
    ],
    [
      "openalex",
      "2207|Stability and Controllability of Differential Equations"
    ]
  ],
  "openalex": "W7168336812",
  "pdf_url": "https://arxiv.org/pdf/2607.12482",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/372d2gva61yz1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2607.12482",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.12482"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "High-frequency coercivity loss for completely monotone memory on bounded time intervals",
  "title_text": "High-frequency coercivity loss for completely monotone memory on bounded time intervals",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/372d2gva61yz1",
  "venue": "arXiv",
  "versions": []
}