{
  "abstract": "Finite-aperture near-field sensing leads to a super-resolution geometry fundamentally different from the translation-invariant Fourier setting. In the Fresnel regime, wavefront curvature introduces a range-dependent quadratic aperture phase, so distinguishability is governed by incomplete quadratic exponential sums of the form \\( \\sum_{n=0}^{N_r-1} a_n e^{i(ω_1 n+ω_2 n^2)} \\), rather than by angular separation alone. We develop a deterministic recovery theory for sparse measures with ranges on a finite grid and continuous angles, and introduce a support-uniform quadratic-phase aperture criterion replacing classical minimum separation. Under this criterion, total-variation minimization exactly recovers every sparse measure in the admissible support class, uniformly over all nonzero complex amplitudes. The proof develops nonasymptotic support-uniform bounds for finite quadratic sums and combines them with a gauged Hermite dual certificate controlling interpolation, local curvature, and off-support leakage. We further construct a finite-harmonic Bessel-Vandermonde lift with explicit truncation error. In the far-field limit, the quadratic phase disappears and the theory reduces to Fourier-type angular super-resolution.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.31299",
  "author_total": 3,
  "authors": [
    {
      "name": "Sajad Daei"
    },
    {
      "name": "Gabor Fodor"
    },
    {
      "name": "Mikael Skoglund"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7db2xaq75vz89.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.31299",
  "fulltext_url": null,
  "id": "7db2xaq75vz89",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.31299",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2206|Sparse and Compressive Sensing Techniques"
    ]
  ],
  "openalex": "W7214616111",
  "pdf_url": "https://arxiv.org/pdf/2609.31299",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7db2xaq75vz89.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.31299",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.31299"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A Mathematical Theory of Near-Field Super-Resolution",
  "title_text": "A Mathematical Theory of Near-Field Super-Resolution",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/7db2xaq75vz89",
  "venue": "arXiv",
  "versions": []
}