Ring-like structures of frequency domains of wavelets
The authors study orthonormal wavelets, i.e. sets \(\{\psi_\mu\}_{\mu=1}^{2^d-1}\) of functions in \(L^2(\mathbb R^d)\) such that their integer translations and dyadic dilations form an orthonormal basis, so the union \(\Omega\) of the supports of the Fourier transforms \(\hat {\psi_\mu}\) (the frequency domain) is a compact set. The set \(\Omega\) can then be written as \(\Omega = S \setminus S_*\) with \(0\in S_*\subset S\) and the authors give a characterization of these sets \(S\) and \(S_*\). As consequence of this characterization is that \(S\) and \(S_*\) cannot be balls. Reverse results on what sets can be the frequency domain of orthonormal wavelets. A number of examples are given as well.
scientific article; zbMATH DE number 5757857