{
  "abstract": "Vector-valued multiscale representations are essential when signals or fields take values in $\\mathbb{R}^m$ and component interactions carry meaningful information. Most multiwavelet and super-wavelet constructions are formulated in scalar Hilbert-space settings and typically produce channelwise scalar coefficients followed by recombination. We develop an intrinsic framework for vector-valued wavelets on $L^2(\\mathbb{R}^d,\\mathbb{R}^m)$ by endowing this space with a natural $\\mathbb{M}_m(\\mathbb{R})$-valued inner product, thereby turning it into a Hilbert $\\mathbb{M}_m(\\mathbb{R})$-module. This module viewpoint yields matrix-valued coefficients that encode cross-component interactions and provides canonical reconstruction through a Parseval-type identity. Within this setting, we introduce a constructive lifting procedure that builds separable multivariate vector-valued wavelet bases in $L^2(\\mathbb{R}^d,\\mathbb{R}^m)$ from scalar wavelet bases while preserving compact support, vanishing moments, and regularity.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30589",
  "author_total": 2,
  "authors": [
    {
      "name": "Hicham Tarif"
    },
    {
      "name": "Nadir Maaroufi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1v60fbmzas45c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30589",
  "fulltext_url": null,
  "id": "1v60fbmzas45c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30589",
  "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.FA"
    ],
    [
      "openalex",
      "2604|Mathematical Analysis and Transform Methods"
    ]
  ],
  "openalex": "W7204908728",
  "pdf_url": "https://arxiv.org/pdf/2608.30589",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1v60fbmzas45c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.30589",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30589"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Vector-Valued Wavelet Bases as Hilbert $\\mathbb{M}_m(\\mathbb{R})$-Module Bases: A Construction from Scalar Wavelets",
  "title_text": "Vector-Valued Wavelet Bases as Hilbert 𝕄_m(ℝ)-Module Bases: A Construction from Scalar Wavelets",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/1v60fbmzas45c",
  "venue": "arXiv",
  "versions": []
}