{
  "schema_version": 1,
  "problem_number": "AMR-155-0008",
  "title": "The Functions ψ_b of Bownik and Weber Are Frame Wavelets for 1/8 < b ≤ 1/6",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For 0 < b < 1 let ψ_b ∈ L²(R) be given by hat ψ_b = 1_{(−1,−b)∪(b,1)}. Bownik and Weber (Studia Math. 159 (2003)) determined for which b the dyadic affine system of ψ_b is a frame, except for 1/8 < b ≤ 1/6, where they believed it to be a frame and called b = 1/6 the hardest case; Bownik and Rzeszotnik list this question as open in their survey of open problems in wavelet theory. We prove that ψ_b is a frame wavelet for every b ∈ (1/8, 1/6], with frame bounds 1/54 and 6 (not sharp). With the results of Bownik and Weber, ψ_b is a frame wavelet exactly for b ∈ (0, 1/6] ∪ [1/3, 1/2]. The proof uses the equivalence of affine and quasi-affine frames (Ron and Shen; Chui, Shi and Stöckler) and the fiberization of shift-invariant systems (Ron and Shen; Bownik). Every fiber of the dual Gramian is, after a change of signs, the Laplacian of a forest plus a nonnegative diagonal term; for 1/8 < b ≤ 1/6 no vertex with one child has a child with one child, a flow gives an isoperimetric inequality, and the discrete Cheeger inequality gives the lower bound. A computer-assisted computation from one of the verification runs improves it to 1/16. No earlier answer was found in the sources we searched. This is an unrefereed note.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.FA",
    "math.CA"
  ],
  "keywords": [
    "frame wavelet",
    "affine system",
    "quasi-affine system",
    "dual Gramian",
    "fiberization",
    "shift-invariant systems",
    "trees",
    "isoperimetric inequality",
    "Cheeger inequality",
    "frame wavelet sets",
    "Bownik-Weber",
    "open problems in wavelet theory",
    "UnsolvedMath",
    "AMR-155-0008",
    "math.FA",
    "math.CA",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-18",
  "publication_date": "2026-10-18",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-18",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-155-0008/",
  "pdf_url": "https://eulersolve.org/papers/amr-155-0008/paper.pdf?v=634530584127",
  "doi": "10.5281/zenodo.23452011",
  "zenodo_record_url": "https://zenodo.org/records/23452011",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The frame bounds 1/54 and 6 are not sharp (a computer-assisted computation gives 1/16); dual frame wavelets for this range are not treated. No earlier answer was found in the sources searched; no priority is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "6345305841276616b6d78fb187198ca70f768417836e9584f5130ff8b5a31d8d"
    },
    "source.zip": {
      "sha256": "f5751dddf5685e2b3d00d508491bd388610e54bb08075c6ca61e346b34203d77"
    },
    "verification_report.md": {
      "sha256": "c6725faec9da42007b792ce1357d38754deee6bae37f11cac6f280be68a7c757"
    }
  },
  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}