About nonstationary multiresolution analysis and wavelets
Let \(\psi^{(j)} \in L^2(\mathbb R)\) and \(\psi_{j,k} = 2^{j/2}\,\psi^{(j)}(2^j\cdot -k)\) for \(j,k\in \mathbb Z\) be given. In the nonstationary case, \(\psi^{(j)}\) are called mother wavelets, if \(\{\psi_{j,k};\,j,k\in \mathbb Z\}\) is an orthonormal basis of \(L^2(\mathbb R)\). In the stationary case, where \(\psi^{(j)}=\psi\) for all \(j\in \mathbb Z\), the following characterization of a wavelet is known: The wavelet \(\psi\in L^2(\mathbb R)\) can be obtained from a multiresolution analysis if and only if \[ \sum_{n=1}^{\infty} \sum_{k=-\infty}^{\infty} |{\hat \psi}(2^n(\cdot + 2k\pi))|^2 = 1 \quad a.e.\,, \] where \(\hat \psi\) is the Fourier transform of \(\psi\) (see [\textit{E. Hernández} and \textit{G. Weiss}, A first course on wavelets. Studies in Advanced Mathematics. Boca Raton, FL: CRC Press (1996; Zbl 0885.42018)]). In the paper under review, the authors generalize this and related results to the nonstationary case.
scientific article; zbMATH DE number 6135261
scientific article