{
  "abstract": "Multi-Threshold (MT) sampling fixes voltage levels and records when a waveform crosses them. The resulting signal-generated event stream is governed not by a prescribed clock but by the nonuniform geometry of the realized crossing set. Within classical nonuniform sampling, we develop a retained-event MT theory for static, uniformly spaced thresholds. For a realized retained MT time set, Beurling's weak-limit theorem gives the exact necessary-and-sufficient condition for stable sampling of $PW_Ω$. For a fixed signal class, the same principle gives the a priori criterion: uniform stable MT sampling is equivalent to Bernstein uniqueness for every set in the class weak-limit hull. This criterion reduces to computable forms under structured priors. Periodic templates give finite fiber/rank tests, periodic-gap streams give a Shannon-type complete-interpolation boundary and a stable-sampling density rule, and finite-state gap priors reduce noncritical certification to a maximum-cycle-mean condition. Kadec-type local certificates, retained-density bounds, and finite-matrix tests provide practical sufficient conditions for finite active records, while conditional perturbation bounds quantify how jitter, front-end noise, and threshold error consume a clean sampling margin under event correspondence. The theory also identifies the unavoidable boundary of this model: unrestricted finite-energy retained MT records have zero full-line lower Beurling density and therefore cannot provide a universal prior-free sampling theorem for $PW_Ω$. Within retained-event point sampling, the MT recovery question is resolved precisely: exact recovery is governed by event-set geometry, computable guarantees require structural priors or finite-dimensional/tail assumptions, and the unrestricted static finite-energy problem is impossible.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22180",
  "author_total": 2,
  "authors": [
    {
      "name": "Ao Qiu"
    },
    {
      "name": "Qingguo Xie"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/93gbepkpg7jzx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22180",
  "fulltext_url": null,
  "id": "93gbepkpg7jzx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22180",
  "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",
      "cs.IT"
    ],
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      "arxiv",
      "eess.SP"
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      "arxiv",
      "math.IT"
    ],
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      "arxiv",
      "physics.ins-det"
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      "arxiv",
      "physics.med-ph"
    ],
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      "openalex",
      "2604|Mathematical Analysis and Transform Methods"
    ]
  ],
  "openalex": "W7214026349",
  "pdf_url": "https://arxiv.org/pdf/2609.22180",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/93gbepkpg7jzx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:19Z",
      "native_id": "2609.22180",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22180"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    },
    {
      "field": "physics",
      "origin": "arxiv:physics.ins-det",
      "subfield": "applied-instrumentation"
    },
    {
      "field": "physics",
      "origin": "arxiv:physics.med-ph",
      "subfield": "biological-medical"
    }
  ],
  "title": "A Nonuniform-Sampling Theory for Static Multi-Threshold Sampling",
  "title_text": "A Nonuniform-Sampling Theory for Static Multi-Threshold Sampling",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/93gbepkpg7jzx",
  "venue": "arXiv",
  "versions": []
}