# Verification report — OWR-720-011 (canonical normalized tight frame of a wavelet frame; Han's Question 4, Frank's question)

Verification date: 2026-10-17.

**Verdict.** The note answers the question **negatively**, with complete proofs; the scope is as follows.
- **Settled.** For every orthonormal dyadic wavelet ψ of L²(R) and every 0 < ε < 1, the Riesz wavelet η = ψ + εDψ
  has a canonical normalized tight frame E = {S^{-1/2}D^jT_kη} = {Uψ_{j,k}} (U the unitary polar factor of I + εB,
  Bψ_{j,k} = ψ_{j+1,2k}) that is not a wavelet system:
  - indexed reading (Han's own gloss, S^{-1/2} in the local commutant C_η(U_{D,T})):
    ‖S^{-1/2}T_1η − T_1S^{-1/2}η‖ = (1 − u_0² − u_1²)^{1/2} ≥ ε(1/2 − 3ε²/8) > ε/8 (Theorem 1.1(c), Proposition 4.1);
  - set reading: E ≠ {D^{rj}T_{ck}φ_l : j, k ∈ Z, l ∈ L} for every family (φ_l), every integer r ≥ 1 and every nonzero
    dyadic rational c (Theorem 1.1(d));
  - literal reading of Han's question (does the function S^{-1/2}η generate a wavelet frame?): for the Shannon
    wavelet, {D^jT_kS^{-1/2}η} is never a normalized tight frame, and for 1/√2 ≤ ε < 1 not even a Bessel sequence
    (Proposition 1.3).
- **Exact computation** (Theorem 1.2): Uψ_{j,0} = Σ_q u_q ψ_{j+q,0} and, for k ≠ 0,
  Uψ_{j,k} = Σ_{q ≥ −ν(k)} (u_q − u_{q+2ν(k)+2}) ψ_{j+q,2^q k}, where the u_q are the Fourier coefficients of
  (1+εz)/|1+εz|, given by binomial series; all coefficients are nonzero, with explicit signs.
- **Not new.** The example η = ψ + εDψ is the classical example of a Riesz wavelet whose canonical dual frame is not a
  wavelet system (Daubechies 1990; Chui's 1992 tutorial volume; Chui–Shi 1993; as described by Christensen 2014, §2.3,
  and Bownik–Lemvig 2007, §1). The positive answer for semi-orthogonal wavelet frames is Han's (2003, Theorem 1.3).
  Bownik, Jasper and Speegle (Proc. AMS 2011) state, without an example or proof, that the frame wavelets with
  S^{-1/2} in the local commutant form a proper subset of all frame wavelets.
- **Open / not treated.** The second part of Han's Question 4 (under which conditions the answer is yes); Han's
  question whether S^{-1} ∈ C_ψ implies S^{-1/2} ∈ C_ψ (in the example both fail); redundant (non-Riesz) wavelet
  frames; other dilations and dimensions; quasi-affine systems; the literal reading for ψ other than Shannon (e.g.
  Haar) or for ε < 1/√2.

The note is unrefereed.

## Statement checked
- **Primary source of the question.** D. Han, "Approximations for Gabor and wavelet frames", Trans. Amer. Math. Soc.
  355 (2003), no. 8, 3329–3342, doi:10.1090/S0002-9947-03-03047-2 (received 19 February 2002). Read completely in a
  copy of the publisher's version (sha256 `047788e3…87489`).
  - Section 3, p. 3338, Question 4: "Is S^{-1/2}ψ always a wavelet frame?", and, if not, under which conditions.
  - Han's gloss (pp. 3338–3339): the canonical dual is a wavelet frame iff S^{-1} lies in the local commutant
    C_ψ(U_{D,T}); S^{-1/2}ψ is a normalized tight wavelet frame if S^{-1/2} ∈ C_ψ(U_{D,T}); it is not known whether
    S^{-1} ∈ C_ψ implies S^{-1/2} ∈ C_ψ.
  - Theorem 1.3 (statement p. 3333, proof pp. 3339–3340): for semi-orthogonal ψ, S^{-1/2}D^nT_ℓψ = D^nT_ℓS^{-1/2}ψ.
  - Corollary 3.1 (p. 3340): Riesz wavelets with S^{-1/2} ∈ C_ψ. Han's setting: L²(R^d), real expansive matrix,
    lattice Z^d; "wavelet frame" means a single function generating a frame.
- **Restatement (the corpus source).** M. Frank, "Approximation of frames by normalized tight ones", abstract in
  Mini-Workshop: Wavelets and Frames, Oberwolfach Reports 1 (2004), no. 1, 479–544 (Report No. 10/2004),
  doi:10.4171/OWR/2004/10, pp. 529–532; read in the open PDF (sha256 `061421de…582a0d11`).
  - p. 529: S = (θ*θ)^{-1}, the inverse of the usual frame operator, so Frank's S^{1/2}(x_i) is the usual S^{-1/2}x_i.
  - p. 531: it is unknown whether "{S^{1/2}(x_i)}_i has always wavelet structure" for wavelet frames {x_i}_i.
    Frank cites Han (2003) for Theorems 3–5; the sentence before the question concerns wavelet frames without any
    wavelet-type dual, found in discussions of Han with Daubechies, Wexler and Bownik.
  - p. 530: the remark on symmetric orthogonalization of Riesz bases (citing Löwdin).
  - p. 488 (O. Christensen's abstract "Trends in frame theory"): the open problem which conditions make the canonical
    dual a wavelet system.
- **Corpus record.** ulamai/UnsolvedMath, OWR-720-011 (record 30000043; status open in versions 1.6.0 and 1.8.0):
  "Does the canonical normalized tight frame associated with every wavelet frame always have wavelet structure?" —
  a faithful paraphrase of Frank's sentence. The question is Han's Question 4, first part, asked a year earlier.

## Readings
| Reading | Answer | Where |
|---|---|---|
| (I) indexed: S^{-1/2}ψ_{j,k} = D^jT_kφ for all (j,k), i.e. S^{-1/2} ∈ C_ψ(U_{D,T}) (Han's gloss; hypothesis (4.1) of Bownik–Jasper–Speegle, Theorem 4.1) | no, for every orthonormal ψ and every ε ∈ (0,1); quantitative bound | Theorem 1.1(c), Proposition 4.1 |
| (II) as a set, any re-indexing and any number of generators, same dilation and lattice Z | no | Theorem 1.1(d) |
| (II′) as a set, lattices cZ (c a nonzero dyadic rational) and dilations D^r (r ≥ 1) | no | Theorem 1.1(d) |
| E is a shift-invariant system, e.g. quasi-affine | no (E is not T_1-invariant) | Remark 5.4 |
| (III) literal: the function S^{-1/2}η generates a normalized tight wavelet frame | no, for the Shannon wavelet and every ε | Proposition 1.3 |
| (III) literal: the function S^{-1/2}η generates a wavelet frame (or a Bessel system) | no, for the Shannon wavelet and 1/√2 ≤ ε < 1; not decided for ε < 1/√2 or for other ψ | Proposition 1.3 |
| Canonical normalized tight frame of the quasi-affine system of η | not treated | Remark 5.4 |
| Second part of Q4; S^{-1} ∈ C_ψ ⇒ S^{-1/2} ∈ C_ψ?; redundant frames; other dilations | not treated | Section 1.3, Remark 7.1, Remark 7.3 |

## Results in the paper
- **Theorem 1.1.** (a) η_{j,k} = ψ_{j,k} + εψ_{j+1,2k}; Riesz basis with optimal frame bounds (1 − ε)², (1 + ε)²; not
  semi-orthogonal (⟨η_{0,0}, η_{1,0}⟩ = ε). (b) S = AA*, A = I + εB, and S^{-1/2}η_{j,k} = Uψ_{j,k},
  U = A(A*A)^{-1/2} unitary (polar decomposition). (c) the indexed reading fails, with the bound above. (d) the set
  reading fails for lattices cZ and dilations D^r.
- **Lemma 2.1.** B is an isometry; the chain decomposition of Z² into O_0 = {(j,0)} and C_{i,m} = {(i+n, 2^n m)}
  (m odd) reduces B: bilateral shift on H_0, unilateral shift on each chain space. **Lemma 2.2.** U commutes with D
  and with the projections onto the summands.
- **Theorem 1.2 (the polar factor)** with Lemmas 3.1–3.3: the Fourier coefficients γ_k = (−1)^kΓ_{|k|} of
  |1+εe^{iθ}|^{-1} via the binomial series; u_i = γ_i + εγ_{i−1}; on a chain, by the sine transform e_n ↦ sin((n+1)θ),
  ⟨(A*A)^{-1/2}e_μ, e_n⟩ = γ_{n−μ} − γ_{n+μ+2} and ⟨Ue_μ, e_n⟩ = u_{n−μ} − u_{n+μ+2} (Toeplitz minus Hankel);
  termwise sign lemma (Lemma 3.2).
- **Proposition 4.1.** d(ε)² = Σ_{i≥1}Λ_i² + Σ_{i≥2}Δ_i² = 1 − u_0² − u_1²; d(ε) = ε/2 + O(ε³); Table 1
  (d(1/2) = 0.2456370636…).
- **Lemma 5.1.** No nonzero dyadic translate of any Uψ_{j,0} lies in E. **Proposition 5.2.** For e = Uψ_{j0,k0} and
  k ≠ 0: T_k e ∈ E iff k0 ≠ 0 and j0 + ν(k) > ν(k0), and then T_k e = Uψ_{j0,k0+2^{j0}k}. **Remark 5.3.** No uniform
  margin: dist(T_1Uψ_{J,0}, E) ≤ √2 ε^{J+1}(1 − ε²)^{−3/2} → 0.
- **Proposition 1.3** (Shannon): Σ_{j,k}|⟨ψ, D^jT_kS^{-1/2}η⟩|² = 1 + Σ_{i≥1}(2^i − 1)Λ_i².
- **Remarks 7.1–7.3.** The canonical dual Σ_{n=0}^{ν(k)}(−ε)^nψ_{j−n,k/2^n}, with ‖S^{-1}T_1η − T_1S^{-1}η‖ = ε(1−ε²)^{-1/2}
  (classical fact, in this notation); η has no orthonormal dilation in the sense of Bownik–Jasper–Speegle,
  Theorem 4.1; scope.

## Computations (sanity checks; programs and outputs in reproducibility/)
No proof depends on a computation.
- **Program of the note** (`check_note.py`, standard library, about 3.5 s; ALL CHECKS PASSED): β-identities (exact);
  Theorems 1.1(b) and 1.2 as exact formal power series identities up to ε^12 (bilateral part, chains, ten index
  pairs, (AA*)^{-1/2}A = A(A*A)^{-1/2}, unitarity, canonical dual); 18,696 sign comparisons in 100-digit arithmetic for
  fifteen values of ε; Table 1 by two formulas and against the earlier FFT values; index bookkeeping of Lemma 2.1,
  Lemma 5.1 and Proposition 5.2 (84,240 cases); Remark 5.3; the inequalities and sums of Proposition 1.3.
- **Programs with which the results were first obtained** (`original/`): series against FFT (error ≤ 1.2e−16) and
  against finite sections (N = 1500; ≤ 6.1e−15); signs on a grid; d(ε); Haar identities in exact arithmetic.
- **Verification run A** (`verification_run_A/`): quadrature (50 digits) and SVD polar factors of finite sections of
  A (sizes up to 3000; agreement ≤ 3.3e−15); 58,236 sign tests; exact bookkeeping of Proposition 5.2 (25,872 cases,
  no mismatch); Haar functions on a grid of step 2^{−11} (distance 0.245637061 against 0.245637064); Shannon sums.
- **Verification run B** (`verification_run_B/`): exact power series up to ε^24 (three planted errors detected);
  33,066 numerical sign checks; Haar: frame operator = AA* by exact integration; membership of 1,013 translates of
  elements of E decided numerically without the case analysis, all in agreement with Proposition 5.2.
- **Third verification run** (`independent_run_2/`, on the written note): u_i from the factorisation
  g = (1+εz)^{1/2}(1+εz̄)^{−1/2} and Γ_k as hypergeometric values and by quadrature (agreement to 1e−50); the chain
  matrix of Lemma 3.3 through the identities of a polar decomposition, exactly to ε^18, and against finite-section
  polar factors (N = 4000, agreement ≤ 1.7e−14); 34,481 sign tests; Table 1 to ten decimals and d(ε) by three
  routes; Remark 5.3 and Remark 7.1; Lemma 5.1, Proposition 5.2 and Theorem 1.1(d) for the Shannon wavelet from inner
  products of functions (1,005 cases, no disagreement); Proposition 1.3 with Hurwitz-zeta values; the relation of
  Remark 7.2 (sympy).
- **Re-runs.** On 2026-10-17 every program of the package was run again from a copy of the folder; all recorded
  outputs were reproduced exactly, except the line of `own_04_membership_scan.out` that records the running time
  (`reproducibility/RERUN_LOG.txt`). The third verification run re-ran all of them once more from an extracted copy of
  `source.zip`, with byte-identical outputs (`reproducibility/independent_run_2/rerun_report.txt`).

## Independent verification runs
The results were first obtained with proofs and programs. Two independent verification runs, both AI-assisted, then
examined the first written version of the results; each wrote its own programs and re-ran those of the first stage.

| Item | Run A | Run B |
|---|---|---|
| Statement against the sources (Frank 2004; Han 2003 read in a copy of the publisher's version) | CONFIRMED; Han's Question 4 identified as the first appearance | CONFIRMED; same |
| Theorem 1.1(a), (b) | CONFIRMED (re-derived) | CONFIRMED (re-derived; frame operator by integration for Haar) |
| Chain decomposition, models, closed forms, Lemma on signs | CONFIRMED (re-derived; quadrature, SVD) | CONFIRMED (re-derived; exact series to ε^24) |
| Theorem 1.1(c) | CONFIRMED | CONFIRMED |
| Translation lemma and Theorem 1.1(d) | CONFIRMED; shorter proof via Lemma 5.1 with lattices cZ and dilations D^r | CONFIRMED; numerical membership scan; robustness under unimodular factors |
| Literal reading (Shannon) | PROVED (supplement) | PROVED independently (supplement) |
| Known special cases (semi-orthogonal, group-like, canonical dual, Bownik–Weber) | CONSISTENT | CONSISTENT |
| Sources and novelty | PASS WITH FIXES (credit to Han's Question 4; page 488; glosses) | NEEDS FIX (credit), bibliography fixes; no prior answer found |

Neither run found a gap in the mathematics.

**Corrections required by the two runs, and what was done:**
- A1 / F1 (credit): the question is credited to Han's Question 4 (2003, p. 3338) as its first appearance, with Frank
  (2004) as the restatement — abstract, Section 1.1, Corollary 1.4, "Scope and priority".
- A1 / F2 (what is settled): the first part of Q4 is answered in Han's sense (S^{-1/2} ∉ C_η); the second part and
  Han's sub-question are stated as not addressed — abstract, Section 1.3, Remark 7.1, "Scope and priority".
- A2 / F3 (literal reading): the Shannon supplement is included as Proposition 1.3 with a complete proof; the
  undecided cases are named.
- A3: Christensen's open problem on canonical duals is cited on p. 488 of the report.
- A4: Frank does not specify the dilation; Han's matrix is real expansive; the semi-orthogonal identity is cited from
  Han's proof of Theorem 1.3 (p. 3339).
- A5: Frank's sentence before the question is described as concerning frames without any wavelet-type dual and
  attributed by him to discussions of Han with Daubechies, Wexler and Bownik; the references to Daubechies–Han,
  Bownik–Weber and Bownik–Lemvig are given as our pointers; the symmetric-orthogonalization remark cites only Löwdin.
- A6: Bownik–Rzeszotnik (2019) is described only as searched for the question.
- A7: Han (J. Approx. Theory 2004) was read completely (no statement on Question 4); Han–Larson 2009 and 2011 could not
  be read and are listed as not read.
- A8 (optional): adopted — Theorem 1.1(d) for lattices cZ and dilations D^r, any family of generators; E is not
  T_1-invariant (Remark 5.4).
- A9: the frame bounds are called optimal, not attained.
- F4 (bibliography): Oberwolfach Reports vol. 1, no. 1, Report No. 10/2004; Christensen's survey with its published
  title and the arXiv v1 title; Chui's 1992 tutorial volume as cited by Christensen; Daubechies–Bin Han. (The
  Šikić–Speegle–Weiss paper is not cited in the note.)
- F5: Remark 5.3 states and proves that part (d) has no uniform margin.

**Changes made when the note was written, after runs A and B** (not examined by runs A and B; examined first by the
third run, next section):
- The chain coefficients are written as u_{n−μ} − u_{n+μ+2} (Lemma 3.3), with a shorter sign argument; the first
  version gave c_{n,μ} + εc_{n−1,μ} with a case-by-case sign analysis. Checked exactly by `check_note.py` [2].
- Proposition 4.1: d(ε)² = 1 − u_0² − u_1², and the asymptotics. Checked by `check_note.py` [4].
- Remark 5.3 (bound on the distance). Checked by `check_note.py` [6].
- Remark 7.2 (orthonormal dilations) and the reference Bownik–Jasper–Speegle 2011, found in the search for the note.
- Theorem 1.1(d) is proved through Lemma 5.1 (run A's argument); Proposition 5.2 is kept, with the proof rewritten in
  the notation of Theorem 1.2.

## Verification run on the written note (third independent verification run)

A third independent verification run, AI-assisted, examined the written note (13 pages, before the corrections
below), and first the parts that runs A and B had not seen. Its programs and outputs are in
`reproducibility/independent_run_2/` (the folder name refers to the second verification stage); they were written
from the statements of the note before the other programs of the package were opened.

| Area | Verdict | What was checked |
|---|---|---|
| Statement and sources | CORRECT | Frank's abstract pp. 529–532 and Christensen's p. 488 (own download of the open report, same sha256 as before); Han 2003 (Question 4, the gloss, Theorems 1.2, 1.3, Corollary 3.1, the definitions) and Han 2004 (copies of the publishers' versions, read as text); Bownik–Jasper–Speegle in arXiv v1 and the journal version (same numbering; Theorem 4.1 with its converse, Remark 4.1, §1 and §4; no example, no proof of properness, no citation of Han 2003 or Frank 2004); Christensen arXiv v1 §2.3; Bownik–Lemvig §1; the corpus record (faithful paraphrase of Frank's sentence; it does not name Han's Question 4) |
| Proofs | CORRECT, no gap | every proof re-derived, in particular the closed form ⟨Ue_μ, e_n⟩ = u_{n−μ} − u_{n+μ+2} (Lemma 3.3), d(ε)² = 1 − u_0² − u_1² (Proposition 4.1; d = ε/2 − ε³/32 + O(ε⁵)), Remark 5.3 (identity and bound), Remark 7.2 (D′ = D^{−1}, T′ = T_1 satisfy the Baumslag–Solitar relation; the converse of BJS Theorem 4.1 applies), Proposition 1.3 and Lemma 5.1 with the proof of Theorem 1.1(d) for cZ and D^r |
| Computations | CONFIRMED | four own programs (see the Computations section above), all passing; every program of the package re-run from `source.zip` with byte-identical outputs |
| Novelty and credit | NO PRIOR PROOF OR EXAMPLE FOUND | arXiv API (13 searches), Crossref (5 bibliographic searches and all 20 DOIs of the bibliography), zbMATH Open (the 5 documents citing BJS 2011; "local commutant" & wavelet: 9 documents), OpenAlex (refused, HTTP 429), one web search, arXiv:1901.08024 (listed with the question in the corpus): nothing beyond the assertion of BJS 2011 |
| Presentation | MINOR FIXES | all pages rendered and inspected; house style respected |

Corrections required by the third run, all made (the note now has 14 pages):
1. The Verification paragraph and Section 8 record this run; the package has four parts.
2. Abstract and "Scope and priority": Bownik, Jasper and Speegle *asserted*, without an example or a proof, that the
   property fails for some frame wavelets (as Section 1.3 already said).
3. Abstract: the example is credited to "(Daubechies; Chui; Chui and Shi)", as in "Scope and priority"; Christensen
   names Daubechies and Chui's tutorial volume, Bownik–Lemvig name Daubechies and Chui–Shi.
4. Section 1.3: Bownik and Lemvig count the result of Daubechies and of Chui and Shi among "the first results" on
   canonical and alternate duals (the earlier wording "the first wavelet frames" said more than their text).
5. Section 1.1: the definition of a Riesz wavelet is cited from Han's §3 as well as §1.
6. Section 1.3: a reason is given why U_{D,T} is not group-like (T_{1/2} = DT_1D^{−1} lies in the group it generates
   and is no scalar multiple of any D^nT_ℓ).
7. "Scope and priority": the searches of this run (zbMATH Open citing list of BJS 2011, OpenAlex refused, ten web
   searches in all) and arXiv:1901.08024 are recorded.
8. Remark 7.2: the auxiliary Hilbert space is called H′ (K is the frequency set of Section 6).

No mathematical correction was needed and nothing was removed.

## Relation to the literature, novelty and scope
- **Searches (17 October 2026, all stages):** arXiv API, Crossref, zbMATH Open; works citing Han 2003 (zbMATH Open
  19, Semantic Scholar 28), citing Frank's abstract (Semantic Scholar 3), citing Bownik–Jasper–Speegle 2011 (Semantic
  Scholar 5; zbMATH Open 5: Naimark complementary frames 2018, dual g-frames 2019, Guo–Chen 2020 on dilations of
  frame generators, dual Hilbert–Schmidt frames 2022, Krein spaces 2025 — none on the question), citing
  Daubechies–Han 2002, Bownik–Weber 2003, Bownik–Lemvig 2007 (Semantic Scholar); titles and available abstracts read;
  ten web searches in all. OpenAlex returned HTTP 429 for every list of citing works and was not used.
- **Found:** the question in Han 2003 and Frank 2004; Han's positive semi-orthogonal theorem; the classical example
  for canonical duals; the assertion of Bownik–Jasper–Speegle 2011 (§1: dilation possible "only for a proper subset
  of frame wavelets"; §4 and Remark 4.1), which by their Theorem 4.1 asserts that S^{-1/2} ∈ C_ψ can fail. That paper
  gives no example or proof of the assertion and does not cite Han 2003 or Frank 2004 (author copy of the journal
  version and arXiv v1 read).
- **Read:** Han 2003 and Han 2004 completely (copies of the publishers' versions); Frank's abstract and p. 488 of the
  report; Bownik–Jasper–Speegle 2011; Bownik–Lemvig 2007; Bownik–Weber 2003; Christensen's survey (arXiv v1); the
  Frank–Paulsen–Tiballi preprint; Bownik–Rzeszotnik 2019, two surveys of D. Larson and arXiv:1901.08024
  (Gómez-Cubillo–Villullas 2019, listed with the question in the problem corpus) (searched for the question).
- **Not read:** Han–Larson 2009 (Acta Appl. Math.; not accessible, abstract only), Han–Larson 2011 (abstract only),
  Han's SPIE 2001 paper (full text behind an automated-access check; repository abstract only), Han–Larson Memoir
  2000, the books of Daubechies (1992), Chui (ed., 1992) and Christensen (2016), and the papers Daubechies 1990 and
  Chui–Shi 1993 (their content is cited as described by Christensen and by Bownik–Lemvig).
- **Novelty.** Apart from the assertion of Bownik–Jasper–Speegle, no printed answer, example or proof was found. What
  the note adds: the exact computation of the polar factor for the classical example and the proofs that it destroys
  wavelet structure (Theorem 1.1(c), (d)), Proposition 4.1, Proposition 5.2 and Proposition 1.3. A search that finds
  nothing is not a proof of novelty; no priority is claimed.

## Public release and license
This paper, its source files, and this verification report are licensed under the Creative Commons Attribution
4.0 International License (CC BY 4.0): https://creativecommons.org/licenses/by/4.0/