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

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. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: OWR-720-011 (D. Han, Approximations for Gabor and wavelet frames, Trans. Amer. Math. Soc. 355 (2003), Question 4; M. Frank, Oberwolfach Rep. 1 (2004), Report 10/2004, p. 531).