The Functions ψ_b of Bownik and Weber Are Frame Wavelets for 1/8 < b ≤ 1/6

For \(0<b<1\) let \(\psi_b\in L^2(\mathbb{R})\) be given by \(\widehat{\psi_b}=\mathbf{1}_{(-1,-b)\cup(b,1)}\). Bownik and Weber (Studia Math. 159 (2003)) determined for which \(b\) the dyadic affine system of \(\psi_b\) is a frame, except for \(1/8<b\le 1/6\), where they believed it to be a frame and called \(b=1/6\) the hardest case; Bownik and Rzeszotnik list this question as open in their survey of open problems in wavelet theory. We prove that \(\psi_b\) is a frame wavelet for every \(b\in(1/8,1/6]\), with frame bounds \(1/54\) and \(6\) (not sharp). With the results of Bownik and Weber, \(\psi_b\) is a frame wavelet exactly for \(b\in(0,1/6]\cup[1/3,1/2]\). The proof uses the equivalence of affine and quasi-affine frames (Ron and Shen; Chui, Shi and Stöckler) and the fiberization of shift-invariant systems (Ron and Shen; Bownik). Every fiber of the dual Gramian is, after a change of signs, the Laplacian of a forest plus a nonnegative diagonal term; for \(1/8<b\le1/6\) no vertex with one child has a child with one child, a flow gives an isoperimetric inequality, and the discrete Cheeger inequality gives the lower bound. A computer-assisted computation from one of the verification runs improves it to \(1/16\). No earlier answer was found in the sources we searched. This is an unrefereed note.