{
  "abstract": "Given a matrix $A\\in M_{k\\times N}(\\Bbb{F}),$ we examine finite frames $\\{f_j\\}_{j=1}^N$ with vanishing $A$-moments and investigate their connections with the frame dilation property. We establish several equivalent characterizations for the existence of vanishing $A-$moment dilations, including conditions involving dual frames, operator ranges, and linear independence properties associated with the matrix $A$. In particular, we show that every frame with redundancy $k$ admits a dual frame with vanishing $A$-moments for some totally nonsingular matrix $A$. A key takeaway of this is that by invoking such a dual frame as the encoding frame, any frame with redundancy $k$ (erasure robustness is not needed) allows perfect signal reconstructions for any number of $\\ell$-erasures ($\\ell \\leq k$) which only involves inverting a matrix of small size ($\\ell$) instead of inverting the subframe operator which is a matrix of large size ($N-\\ell$). A special case of this type of frames are $k$-th vanishing moment frames which have natural connections with graph frames. We characterize the existence of dual frames with $k-th$ vanishing moments through polynomial cancellation identities and operator theoretical conditions. We further discuss some applications to problems in erasure recovery and graph frames. These results provide a unified framework linking vanishing moment conditions, dual frame theory, dilation theory, and graph theoretic constructions of frames.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.11043",
  "author_total": 4,
  "authors": [
    {
      "name": "Deguang Han"
    },
    {
      "name": "Shankhadeep Mondal"
    },
    {
      "name": "Kai Liu"
    },
    {
      "name": "Omar Nour"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/40hcqj5c06yr9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.11043",
  "fulltext_url": null,
  "id": "40hcqj5c06yr9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.11043",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2604|Mathematical Analysis and Transform Methods"
    ]
  ],
  "openalex": "W7221075369",
  "pdf_url": "https://arxiv.org/pdf/2610.11043",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/40hcqj5c06yr9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.11043",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.11043"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Vanishing Moment Dilations and Applications to Balanced and Graph Frames",
  "title_text": "Vanishing Moment Dilations and Applications to Balanced and Graph Frames",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/40hcqj5c06yr9",
  "venue": "arXiv",
  "versions": []
}