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

D. Han asked in 2003 (Trans. Amer. Math. Soc. 355, Question 4) whether \(S^{-1/2}\psi\) is always a wavelet frame when \(\psi\) is a wavelet frame with frame operator \(S\); M. Frank restated the question in 2004 (Oberwolfach Rep. 1): does the canonical normalized tight frame \(\{S^{-1/2}D^jT_k\psi\}\) 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 \(\psi\) of \(L^2(\mathbb{R})\) and \(0<\varepsilon<1\), the function \(\eta=\psi+\varepsilon D\psi\) generates a Riesz wavelet basis with optimal frame bounds \((1-\varepsilon)^2\) and \((1+\varepsilon)^2\); 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 \(\{U\psi_{j,k}\}\), where \(U\) is the unitary polar factor of \(I+\varepsilon B\) and \(B\psi_{j,k}=\psi_{j+1,2k}\). We compute \(U\) exactly: \(B\) is a bilateral shift on the span of the \(\psi_{j,0}\), where \(U\) is multiplication by \((1+\varepsilon z)/|1+\varepsilon z|\), and a unilateral shift on the other invariant subspaces, where the matrix of \(U\) is \((u_{n-m}-u_{n+m+2})_{n,m\ge0}\) with explicit binomial series \(u_i\), all nonzero and of known sign. Consequently \(\|S^{-1/2}T_1\eta-T_1S^{-1/2}\eta\|\ge\varepsilon(\frac12-\frac38\varepsilon^2)\), so \(S^{-1/2}\) is not in Han's local commutant, and the set \(\{S^{-1/2}D^jT_k\eta\}\) is not of the form \(\{D^{rj}T_{ck}\phi_l\}\) for any integer \(r\ge1\), nonzero dyadic rational \(c\) and family \((\phi_l)\). For the Shannon wavelet and \(1/\sqrt2\le\varepsilon<1\), \(S^{-1/2}\eta\) 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.