{
  "schema_version": 1,
  "problem_number": "OWR-720-011",
  "title": "The Canonical Normalized Tight Frame of a Wavelet Frame Need Not Have Wavelet Structure: An Explicit Riesz Wavelet for a Question of Han and Frank",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "D. Han asked in 2003 (Trans. Amer. Math. Soc. 355, Question 4) whether S^{−1/2}ψ is always a wavelet frame when ψ is a wavelet frame with frame operator S; M. Frank restated the question in 2004 (Oberwolfach Rep. 1): does the canonical normalized tight frame {S^{−1/2}D^jT_kψ} of a wavelet frame always have wavelet structure? Han proved that it does for semi-orthogonal wavelet frames, and Bownik, Jasper and Speegle (2011) asserted, without an example or a proof, that this fails for some frame wavelets. We give an explicit example with complete proofs. For an orthonormal dyadic wavelet ψ of L²(ℝ) and 0 < ε < 1, the function η = ψ + εDψ generates a Riesz wavelet basis with optimal frame bounds (1 − ε)² and (1 + ε)²; it is the classical example of a Riesz wavelet whose canonical dual frame is not a wavelet system (Daubechies; Chui; Chui and Shi). Its canonical normalized tight frame is the orthonormal basis {Uψ_{j,k}}, where U is the unitary polar factor of I + εB and Bψ_{j,k} = ψ_{j+1,2k}. We compute U exactly: B is a bilateral shift on the span of the ψ_{j,0}, where U is multiplication by (1 + εz)/|1 + εz|, and a unilateral shift on the other invariant subspaces, where the matrix of U is (u_{n−m} − u_{n+m+2})_{n,m≥0} with explicit binomial series u_i, all nonzero and of known sign. Consequently ‖S^{−1/2}T_1η − T_1S^{−1/2}η‖ ≥ ε(1/2 − 3ε²/8), so S^{−1/2} is not in Han's local commutant, and the set {S^{−1/2}D^jT_kη} is not of the form {D^{rj}T_{ck}φ_l} for any integer r ≥ 1, nonzero dyadic rational c and family (φ_l). For the Shannon wavelet and 1/√2 ≤ ε < 1, S^{−1/2}η does not even generate a Bessel system. The second part of Han's question, his question whether S^{−1} in the local commutant implies S^{−1/2} in it, redundant wavelet frames and quasi-affine systems are not treated. This is an unrefereed note.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.FA",
    "math.CA"
  ],
  "keywords": [
    "wavelet frames",
    "Riesz wavelets",
    "canonical normalized tight frame",
    "canonical Parseval frame",
    "symmetric (Löwdin) orthogonalization",
    "polar decomposition",
    "local commutant",
    "Toeplitz and Hankel matrices",
    "Shannon wavelet",
    "counterexample",
    "Oberwolfach Reports problems",
    "UnsolvedMath",
    "OWR-720-011",
    "math.FA",
    "math.CA",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-17",
  "publication_date": "2026-10-17",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-17",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-720-011/",
  "pdf_url": "https://eulersolve.org/papers/owr-720-011/paper.pdf?v=4c94d59915ef",
  "doi": "10.5281/zenodo.23438995",
  "zenodo_record_url": "https://zenodo.org/records/23438995",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Answers Han's Question 4 (2003) and Frank's question (2004) negatively by an explicit Riesz wavelet; Bownik, Jasper and Speegle (2011) had asserted the negative answer without an example or proof.",
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    "source.zip": {
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    "verification_report.md": {
      "sha256": "0a54cf6adff91139609a40d87bbc8db9fb86a620f651b743eae9ddedba8f7e5f"
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  "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."
}