Biorthogonal wavelets in \(H^m(\mathbb{R})\)
The Sobolev space \(H^m(\mathbb R)\) \((m\in \mathbb N)\) consists of the functions \(f\) for which \(\int_\mathbb R (1+x^2)^m |\widehat{f}(x)|^2 dx < \infty\). Given functions \(\{\psi^{(j)}\}_{j\in \mathbb Z}\), define \(\psi_{j,k}(x):= 2^{j/2} \psi^{(j)}(2^jx-k)\), \(k\in \mathbb Z\), \(x\in \mathbb R\). Given two classes of functions \(\{\psi^{(j)}\}_{j\in \mathbb Z}\), \(\{\widetilde{\psi}^{(j)}\}_{j\in \mathbb Z} \), the families \(\{\psi_{j,k}\}_{j,k\in \mathbb Z}\), \(\{\widetilde{\psi}_{j,k}\}_{j,k\in \mathbb Z}\) are said to be biorthogonal wavelets in \(H^m(\mathbb R)\) if \(\{\psi_{j,k}\}_{j,k\in \mathbb Z}\) and \( \{\widetilde{\psi}_{j,k}\}_{j,k\in \mathbb Z}\) are both Riesz bases for \(H^m(\mathbb R)\) and \(\langle \psi_{j,k}, \widetilde{\psi}_{j^\prime,k^\prime}\rangle_{H^m(\mathbb R)} = \delta_{j j^\prime} \delta_{kk^\prime}\). (The usual wavelets for \(L^2(\mathbb R)\) correspond to the case where \(\psi^{(j)}\) and \( \widetilde{\psi}^{(j)}\) are independent of \(j\).) In the present paper biorthogonal wavelets with prescribed (finite) regularity and compact support are constructed. Also, biorthogonal bases of exponential splines are constructed. They are semi-orthogonal, in the sense that for \(j \neq j^\prime\), \(\text{span} \{\psi_{j,k} \}_{k\in \mathbb Z} \bot \text{span} \{\psi_{j^\prime,k} \}_{k\in \mathbb Z}.\)
scientific article; zbMATH DE number 1269683
scientific article