# Verification report — AMR-155-0008 (Bownik–Rzeszotnik, Open problems in wavelet theory, Problem 8: the functions ψ_b with hat ψ_b = 1_{(−1,−b)∪(b,1)}, 1/8 < b ≤ 1/6)

Verification date: 2026-10-18.

**Verdict.** The note gives a **complete answer** to the question as posed, with complete proofs.
- **Settled.** For every b ∈ (1/8, 1/6], ψ_b is a frame wavelet, with frame bounds 1/54 and 6 (Theorem 1.2).
  Together with the results of Bownik and Weber (2003), ψ_b is a frame wavelet exactly for
  b ∈ (0, 1/6] ∪ [1/3, 1/2] (Corollary 1.3; the ranges outside (1/8, 1/6] are theirs).
- **Not new.** The question, the conventions and the known ranges (Bownik–Weber 2003, restated by
  Bownik–Rzeszotnik); the equivalence of affine and quasi-affine frames (Ron–Shen 1997 under a decay condition;
  Chui–Shi–Stöckler 1998 in general; Bownik 2000; Bownik–Lemvig 2011); the fiberization of shift-invariant systems
  (Ron–Shen 1995; Bownik 2000); the discrete Cheeger inequality (Dodziuk 1984; used in the form of
  Bauer–Keller–Wojciechowski 2015); the Schur test. The non-frame range 1/6 < b < 1/3 is Bownik–Weber's; Remark 4.5
  only explains it in the fiber picture (and gives a short proof for 1/6 < b < 1/5).
- **New, to the best of the searches.** The forest structure of the dual Gramian fibers for 1/8 < b ≤ 1/6
  (Section 4) and the flow argument giving the isoperimetric inequality |F| ≤ 3|∂F| (Section 5).
- **Computer-assisted, not used in the proof.** The lower frame bound 1/16 for all b ∈ [1/8, 1/6] (Remark 6.1),
  from an exact interval dynamic programme of verification run B.
- **Open.** The optimal frame bounds. Numerically (tested, not proved) the optimal lower bound for b = 1/6 lies in
  [1/16, 0.082] and is about 0.07.
- **Checked three times.** Two independent AI-assisted verification runs examined the first written version; a
  third independent AI-assisted verification run examined the note as written (see below). None found a
  mathematical error or a gap.

The note is unrefereed.

## Statement checked
- **Primary source.** M. Bownik, Z. Rzeszotnik, "Open Problems in Wavelet Theory", in: Operator Theory, Operator
  Algebras and Their Interactions with Geometry and Topology (Operator Theory: Advances and Applications),
  Springer, Cham, 2020, pp. 77–100, doi:10.1007/978-3-030-43380-2_4 (bibliographic data from Crossref);
  arXiv:1905.08230 (v1, the only version). Read in the arXiv LaTeX source of v1, where the question is Problem 2.7
  (Section 2, subsection "A simple question that nobody has bothered to answer"); the corpus calls it Problem 8.
  - Conventions of the source: hat f(ξ) = ∫ f(x) e^{−2πixξ} dx; ψ_{j,k}(x) = 2^{j/2} ψ(2^j x − k); frame wavelet =
    the affine system is a frame for L²(R) (display (fr) of the source).
  - The question, for ψ_b defined by hat ψ_b = 1_{(−1,−b)∪(b,1)} (0 < b < 1): "For what values of 1/8 < b ≤ 1/6,
    is ψ_b a frame wavelet?" The table of known ranges is attributed to Bownik–Weber.
- **Original source of the question.** M. Bownik, E. Weber, "Affine frames, GMRA's, and the canonical dual",
  Studia Math. 159 (2003) 453–479, doi:10.4064/sm159-3-8, Example 2 (pp. 473–477) and Remark 1 (p. 468), read in
  the publisher's open-access PDF. Normalisation there: Fourier transform ∫ f(x) e^{−ixξ} dx (p. 456), affine
  system {2^{j/2}ψ(2^j x − k)}, hat ψ_b = 1_{[−2π,−b]∪[b,2π]}, thresholds π/4, π/3, 2π/3, π. Their ψ_b is therefore
  the function ψ_{b/(2π)} of the note, and dividing by 2π gives the survey's table. Their perturbation argument covers b ≤ π/4 and fails for
  π/4 < b ≤ π/3; they believe ψ_b to be a frame wavelet there on heuristic evidence and call b = π/3 the hardest
  case (p. 474); Case II proves the non-frame property for π/3 < b < 2π/3.
- **Corpus record.** UnsolvedMath v1.8.0 (snapshot d371ee9), record AMR-155-0008: the LaTeX of the problem,
  verbatim, status "open"; the record calls it Problem 8 (the eighth numbered item of Section 2 of the survey if
  the conjecture of that section is counted).

## Readings
- The Fourier normalisation is fixed by the source; b = 1/2 gives the Shannon wavelet only with the e^{−2πixξ}
  convention, consistent with the table.
- Open or closed intervals in hat ψ_b change ψ_b only on a null set of frequencies; irrelevant.
- Dilation 2 and translation lattice Z, as in the source. The endpoint b = 1/6 belongs to the question, b = 1/8
  does not (b ≤ 1/8 is in Bownik–Weber's frame range).
- "Frame wavelet" means a frame for all of L²(R), not for a subspace. A complete answer is the set of b in
  (1/8, 1/6] for which ψ_b is a frame wavelet; the note shows it is the whole interval.
- Not asked: affine duals of ψ_b, tightness, Riesz property, other dilations.

## Results in the paper
- Theorem 1.2: for 1/8 < b ≤ 1/6, (1/54)‖f‖² ≤ Σ_{j,k} |⟨f, ψ_{b;j,k}⟩|² ≤ 6‖f‖².
- Corollary 1.3: for 0 ≤ b < 1, ψ_b is a frame wavelet iff b ∈ (0, 1/6] ∪ [1/3, 1/2].
- Proposition 2.1 (affine ⇔ quasi-affine, same bounds) and Proposition 2.2 (fiberization) are cited, with the
  decay condition (4.6) of Ron–Shen verified for ψ_b.
- Lemmas 3.1–3.4: the dual Gramian of the quasi-affine system is G(ξ) = D(ξ) + A(ξ), with d(y) ∈ {2, 3} and A the
  adjacency matrix of a graph on ξ + Z (edge rule (3)); after the sign change v_y = sgn(y) w_y the quadratic form is
  a graph Laplacian plus the surplus S = d − deg ≥ 0; ‖G(ξ)‖ ≤ 6 by the Schur test.
- Lemmas 4.1–4.4: for ξ outside a countable set, the graph is a forest; each tree has one root, with S ≥ 1, and
  S = 0 elsewhere; children are c_1 always and c_2 iff τ > 4b; a vertex with one child never has a child with one
  child (the only use of b ≤ 1/6).
- Lemmas 5.1–5.3: a flow with loads ≤ 3 (≤ 2 on one-child vertices) gives |F| ≤ 3|∂F| with a ghost edge at the
  root; the Cheeger argument with maximal degree 3 gives Q_T ≥ (1/54)‖v‖².
- Remark 4.5: why the argument stops at b = 1/6 (the interval [2 − 8b, 4b] of one-child positions is invariant
  under τ ↦ 1 − τ/2 for b > 1/6).
- Remark 6.1 (computer-assisted lower bound 1/16), Table 1 and Remark 6.2 (numerical evidence), Section 7 (frame
  wavelet sets).

## Computations (sanity checks; programs and outputs in reproducibility/)
- `check_note.py` (standard library, 16 s, all checks passed): constants; Lemma 3.1 against the literal generator
  sums (1152 entries, max error 1.7·10⁻¹⁵); on 88 fibers (b = 1/8 and nine values in (1/8, 1/6]; 60 random and
  28 within 10⁻⁹ of exceptional configurations), with compressions of G(ξ) to 0 < |y| ≤ 128 computed from the entry
  formula for all pairs in exact arithmetic: Lemma 3.2, Lemmas 4.1–4.4, the row-sum bound, the flow bounds (maximum
  9/4), Lemma 5.2 on 3520 random sets (minimum |∂F|/|F| = 8/11), and positive definiteness of G_W − I/54 and
  6I − G_W by exact elimination; Remark 4.5 (exact tent quotients 3/14, 1/20); Section 7 (P(1−t/2) = P(1−t/4) = t;
  escape times near 2/3). A test with b = 1/5 and b = 1/10 (outside the range) makes the checks fail, as it should.
- Table 1 (numerical illustration): smallest eigenvalues of window compressions at ξ = 67/100 from
  `verification_run_B/window_seq.py`; identical to the first version's values (computed with its own programs).
- Remark 6.2: the single-tree compression to depth 2^40 (3,145,727 vertices) of verification run A, smallest
  eigenvalue 0.08190; extrapolation about 0.075; floating-point dynamic programme 0.0697.
- Remark 6.1: the exact interval dynamic programme of verification run B (M = 3072, 128 b-intervals covering
  [1/8, 1/6], λ = 1/16 and 1/54 certified in exact rational arithmetic). Re-run for the note from a copy:
  identical output except the time field (414 s against 458 s).

## Independent verification runs
The result was first obtained with proofs and with the programs in `reproducibility/original/`. Two independent
verification runs, both AI-assisted, followed (2026-10-18); each re-derived the proofs, wrote its own programs and
examined the sources. The note was then written; its program `check_note.py` re-checks the finite parts.

| Item | Run A | Run B |
|---|---|---|
| Statement, conventions, table against the sources | CONFIRMED (arXiv LaTeX source; Bownik–Weber PDF) | CONFIRMED (arXiv PDF; Bownik–Weber PDF) |
| Proposition 2.1 (affine ⇔ quasi-affine) | CONFIRMED_WITH_FIXES: Ron–Shen Thm 5.5 needs decay condition (4.6); cite the unconditional version | CONFIRMED_WITH_FIXES: cite Bownik–Lemvig 2011, Prop. 3.10 / Thm 1.1 and Bownik–Weber Remark 1 |
| Proposition 2.2 (fiberization) | CONFIRMED (Bownik 2000, Thm 2.3(i); Ron–Shen 1995, Thm 3.3.5) | CONFIRMED (Bownik 2000, Thms 2.3(i), 2.5(ii)) |
| Lemmas 3.1–3.4 | CONFIRMED (re-derived; generator sums vs rule 7·10⁻¹⁵); wording of the Schur bound | CONFIRMED (re-derived; literal sums, brute force) |
| Lemmas 4.1–4.4 | CONFIRMED (re-derived; 442 fibers); misprint j = −1 for j = 1 | CONFIRMED (re-derived; 470 fibers from the definition; failures above 1/6 as expected); same misprint |
| Lemmas 5.1–5.3, Theorem 1.2 | CONFIRMED; induction should be on depth | CONFIRMED |
| Independent confirmations | affine quadratic form computed without quasi-affine systems or fibers (Shannon: 1; b = 2/5: (3 ∓ √5)/2; agreement with fibers to 12 digits) | exact interval dynamic programme: lower bound ≥ 1/16 for b ∈ [1/8, 1/6] |
| Sharpness statement | "of order 0.1" to be replaced: ≤ 0.0819 at depth 40, extrapolation ~0.075 | float DP ≈ 0.0697 near 1/6 |
| Description of the first version's programs | — | to be corrected (edge rule in `fiber_sparse.py`; power-of-two differences in `fiber_gramian.py`; `n_roots`) |
| Literature | no resolution found | no resolution found; frame-wavelet-set literature to be added; three items not readable |
| Bibliography | — | CHECKED against Crossref; survey pages 77–100 missing |

Neither run found a mathematical gap.

**Corrections required by the two runs, all applied in the note:**
1. (Run A, fixes 1 and 6; run B, fix 1.) Proposition 2.1 is cited as Ron–Shen 1997, Theorem 5.5(b), with its decay condition (4.6) stated
   and verified for ψ_b (hat ψ_b bounded with compact support), and in the unconditional form of
   Chui–Shi–Stöckler 1998 as stated in Bownik 2000 (ACHA), Theorem 1.3(ii), and Bownik–Weber 2003, Remark 1, and as
   contained in Bownik–Lemvig 2011, Theorem 1.1 and Proposition 3.10. "Quoted from memory" is gone; the theorem
   numbers of Ron–Shen refer to the authors' preprint, which is said in the note.
2. (Run A, fix 2; run B, fix 2.) The misprint j = −1 (for j = 1) concerned the last sentence of the first version's central-vertex
   lemma; that sentence is not needed in the note (Lemma 4.4 uses only the first version's part (a)) and was
   removed; the exceptional set of the note is stated for all j ∈ Z.
3. (Run A, fix 3.) Remark 6.2: the optimal lower bound for b = 1/6 is at most 0.0819 (depth-40 truncation), extrapolation
   about 0.075; "of order 0.1" removed.
4. (Run A, fix 4.) Lemma 3.4: the row sum d(y) + deg(y) ≤ 6 is what the Schur test uses.
5. (Run A, fix 5.) Lemma 5.1: the induction is on the depth (Lemma 4.3(b) shows that depths are finite).
6. (Run B, fix 3.) Section 8 and the README describe the first version's programs correctly, and the structural claims are
   checked against the definition by `check_note.py` and by run B's programs.
7. (Run B, fix 4.) Bibliographic data from Crossref (survey pages 77–100); Bownik 2000 (JFA), Theorem 2.5(ii) is cited for
   Proposition 2.2.
8. (Run B, fix 5.) Section 7: frame wavelet sets (Dai–Diao–Gu 2001; Dai–Diao 2005, Problem 2.12; Dai–Diao–Gu 2004); the
   sufficient conditions of Dai–Diao 2005, Theorems 2.5 and 2.11, are shown not to apply to 2πE_b; Li 2003, Guo
   2013 and Yadav–Kumar 2023 are named as not readable beyond titles and abstracts (the third run corrected the
   author data of the first: Li–Cao 2003).
9. (When the note was written.) Cheeger's inequality: Dodziuk 1984 (not accessible: the AMS site refused the request) is
   credited; the statement and the constant used are those of Bauer–Keller–Wojciechowski, Lemma 3.5 (arXiv v1),
   read and checked; the note also gives the short proof.

### Third verification run (the note as written; programs in `reproducibility/independent_run_2/`)
After the note was written, a third independent verification run, also AI-assisted, examined its text
(2026-10-18). It downloaded and read the sources again: the survey (arXiv v1, LaTeX source; Crossref record of the
published chapter), Bownik–Weber 2003 (pp. 453–456, 468, 473–477), Ron–Shen 1997 (authors' preprint: Section 1,
condition (4.6), Theorem 5.5), Bownik 2000 (JFA: Proposition 1.2, Definition 2.1, Theorems 2.3 and 2.5), Bownik 2000
(ACHA: pp. 204–205, Theorem 1.3), Bownik–Lemvig 2011 (Theorem 1.1, Definitions 2.2 and 3.1, Proposition 3.10),
Bauer–Keller–Wojciechowski (arXiv v1: Sections 2 and 3, Lemma 3.5 with its setting) and Dai–Diao 2005 (Section 2).
It re-derived every proof of Sections 2–5, Corollary 1.3, Remark 4.5 and the claims of Section 7 line by line.

| Item | Third run |
|---|---|
| Statement, conventions, correspondence with Bownik–Weber | CONFIRMED; their Fourier transform (e^{−ixξ}) makes their ψ_b equal to ψ_{b/(2π)} — now stated in the note; Example 2 ends on p. 477 |
| Propositions 2.1, 2.2 and their hypotheses | CONFIRMED against the texts read; (4.6) holds for ψ_b; the dual Gramian entries are defined (Σ_φ \|φ̂(ξ+m)\|² = d(y) ≤ 3) |
| Lemmas 3.1–3.4, 4.1–4.4, 5.1–5.3, Theorem 1.2, Corollary 1.3 | CONFIRMED (re-derived); the abstract's "no vertex with one child has a child with one child" also for the central vertex — one sentence added to the proof of Lemma 4.4 |
| Bauer–Keller–Wojciechowski, Lemma 3.5 | CONFIRMED (countably infinite graph without isolated vertices, m ≡ 1, path metric with edge length 3^{−1/2} intrinsic since degrees ≤ 3; α²/2 ≥ 1/54); wording of the metric corrected |
| Own programs `run2_structure.py` | 111,276 checks, 0 failed: Lemma 3.1 against the generator sums (16 fibers, 2.4·10⁻¹⁴); 132 fibers (14 values of b, random and within 10⁻⁹ of exceptional configurations), all pairs of windows with 512 vertices from the entry formula: Lemma 3.2 literally, Lemmas 3.4 and 4.1–4.4, Lemma 5.1 (max p = 21/8), **Lemma 5.2 for all finite F of the inner part** (dynamic programme; sup \|F\|/\|∂F\| = 5/3), Lemma 5.3, spectra in [0.2307, 5.609]; exact LDL of G_W − I/54 and 6I − G_W (12 fibers); Remark 4.5 (b = 7/40, 9/50, 19/100, 39/200); Section 7 |
| Own program `run2_spectra.py` | 2,566 checks, 0 failed: Table 1 to all printed digits by two eigenvalue methods; largest eigenvalue 5.763129 (b = 1/7, N = 18); tree of y_c to modulus 2^40: 3,145,727 vertices, smallest eigenvalue 0.0818984838; on it p → 3 (simple: → 2), sup \|F\|/\|∂F\| ≈ 2, bottom of Q_T 0.0831 |
| Re-run of the package from `source.zip` | all recorded outputs reproduced: `check_note.py`; the certificate of Remark 6.1 (all 128 intervals at 1/54 and 1/16, min float DP λ 0.06971); all programs of run B; `original/` (`flow_check`, `scan` both modes, `structure_compare`, `nonframe_check`, `fiber_gramian --allpairs`, `fiber_sparse`); run A (`vA_fiber`, `vA_affine` line by line; `vA_sparse`, `vA_tree` to 3·10⁻¹⁶) — identical up to timing fields and the last digits of iterative eigenvalue computations |
| Literature | no answer to the question found (arXiv API, zbMATH Open, Crossref, one web search; OpenAlex could not be queried); [Li03] has two authors (Weihua Li, Huaixin Cao; J. Shaanxi Normal Univ. Nat. Sci. Ed. 31 (2003) 12–15, Zbl 1023.42023); the zbMATH summaries of Li–Cao 2003 and Yadav–Kumar 2023 announce necessary and sufficient conditions for frame wavelet sets, which could not be checked against 2πE_b |
| Bibliography | all 15 DOIs CHECKED against Crossref |

It found no mathematical error and no gap. **Corrections required by the third run, all applied in the note:**
1. Li 2003 → Li–Cao 2003 with volume, issue, pages and Zbl number; text adjusted.
2. Section 1.1 states the Fourier normalisation and affine system of Bownik–Weber and the correspondence
   b ↦ b/(2π), from which Corollary 1.3 follows.
3. Bownik–Weber, Example 2: pp. 473–477.
4. Section 8: `check_note.py` tests Lemma 5.2 on random sets F.
5. Proof of Lemma 4.4: the central vertex (if y_c has one child, it is z₀, which is branching); not used in the proof.
6. Citation of Bauer–Keller–Wojciechowski: the path metric with edge length 3^{−1/2}.
7. Section 7 and "Scope and priority": the zbMATH summaries of Li–Cao and Yadav–Kumar announce necessary and
   sufficient conditions; whether they decide 2πE_b is not known.
8. "Scope and priority": dual frame wavelets of ψ_b are not considered.
9. The Verification paragraph, the search paragraph and Section 8 record the third run.
10. Remark 6.2: the lower end 1/16 is the computer-assisted bound of Remark 6.1.
11. Section 1.1: the record calls the question Problem 8.

## Relation to the literature, novelty and scope
- **Read.** The survey (arXiv v1, LaTeX source); Bownik–Weber 2003, Example 2 and Remark 1; Ron–Shen 1997
  (authors' preprint), Theorems 1.1 and 5.5 and condition (4.6); Ron–Shen 1995, Theorem 3.3.5; Bownik 2000 (JFA),
  Definition 2.1, Theorems 2.3 and 2.5; Bownik 2000 (ACHA), Section 1; Bownik–Lemvig 2011, Theorem 1.1,
  Section 3.1, Proposition 3.10; Bauer–Keller–Wojciechowski (arXiv v1), Sections 1–3; Dai–Diao 2005, Section 2.
  Not read: Chui–Shi–Stöckler 1998 (closed access), Dodziuk 1984 (refused), Dai–Diao–Gu 2001 and 2004, Schur 1911;
  what the note says about them comes from the sources read. The third verification run read the sources of the
  first item again (see above).
- **Searched (2026-10-18).** arXiv API (authors; "frame wavelet" with characteristic functions; frame wavelet sets;
  quasi-affine), Crossref, zbMATH Open, OpenAlex and Semantic Scholar lists of works citing Bownik–Weber 2003
  (28 and 31) and the survey (2 and 7), arXiv listing of M. Bownik, OpenAlex works citing Dai–Diao–Gu 2001, and a
  few web searches. The third verification run repeated arXiv API queries of the same kinds and the query for
  M. Bownik, searched zbMATH Open for frame wavelet sets (titles, and the summaries of Li–Cao 2003, Guo 2013,
  Yadav–Kumar 2023 and three further works), checked all DOIs with Crossref and made one web search; OpenAlex
  could not be queried, so the counts of citing works were not checked again. No printed answer to the
  question was found in these sources. A search that finds nothing is
  not a proof of novelty; no priority is claimed.
- **Caveats.** The proofs of the cited theorems were not checked. The journal version of Ron–Shen 1997 was not
  compared with the preprint. Three papers on frame wavelet sets could not be read (Li–Cao 2003, Guo 2013,
  Yadav–Kumar 2023); according to their zbMATH summaries, Li–Cao and Yadav–Kumar give necessary and sufficient
  conditions for frame wavelet sets, and it is not known whether these decide 2πE_b.
- **Scope.** The note answers the question completely; it does not determine the optimal frame bounds and does not
  treat dual frame wavelets of ψ_b (the third column of the survey's table). In the UnsolvedMath collection the record is listed as open; any change of its status
  is for the maintainers of the collection to decide.

## 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/