The Functions ψ_b of Bownik and Weber Are Frame Wavelets for 1/8 < b ≤ 1/6
Year: 2026
Type: preprint
Abstract: For 0 < b < 1 let ψ_b ∈ L²(R) be given by hat ψ_b = 1_{(−1,−b)∪(b,1)}. Bownik and Weber (Studia Math. 159 (2003)) determined for which b the dyadic affine system of ψ_b is a frame, except for 1/8 < b ≤ 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 ψ_b is a frame wavelet for every b ∈ (1/8, 1/6], with frame bounds 1/54 and 6 (not sharp). With the results of Bownik and Weber, ψ_b is a frame wavelet exactly for b ∈ (0, 1/6] ∪ [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 ≤ 1/6 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. 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: AMR-155-0008 (Bownik-Rzeszotnik, Open problems in wavelet theory, Problem 8; question of Bownik-Weber, Studia Math. 159 (2003)).
Author: Ferudun, Alper
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7222878193
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