Irregular multiresolution analysis and associated wavelet (Q471302)

Let \(F\) be a closed subspace of the Hilbert space \(H\). The sequence \((F_j)_{j \in Z} \subset H\) is an outer frame of \(F\) if there are constants \(0 < m \leq M < \infty\) such that, for every \(f \in F\), \(m||f||^2 \leq \sum_{j \in Z} |\langle f, f_j \rangle |^2 \leq M ||f||^2\). If \(X=(X^j)_{j \in Z}\) where \(X^j =(x_k^j)_{k \in Z}\) and \(x_k^j \in \mathbb R^d\), and \({\mathcal V}=(V_j)_{j \in Z}\) is a set of closed subspaces of \(L^2(\mathbb R^d)\), the author defines the pair \(({\mathcal V}, X)\) to be an \textit{Irregular Generalized Multiresolution Analysis} (IGMRA) in \(L^2(\mathbb R^d)\), if the following four conditions are satisfied: (a)\ \(V_k \subset V_{k+1}\) for \(k \in Z\); (b)\ \(\;\bigcup_{k \in Z} V_k\) is dense in \(L^2(\mathbb R^d)\); (c)\ \(\bigcap_{k \in Z} V_k= \{0\}\); (d)\ there is a sequence of functions \((\phi_j)_{j \in Z}\) in \(L^2(\mathbb R^d)\) such that, for each \(j\), \(\{\phi_j(x-x_k^j)\}_{k \in Z}\) is an outer frame of \(V_j\). She also defines a \textit{Generalized Wavelet} in \(L^2(\mathbb R^d)\) to be a sequence of pairs \((\psi_j, X^j)_{j \in Z}\) such that for each \(j\in Z\) the sequence \(\{\psi(\cdot - x_k^j)\}_{k \in Z}\) is a frame in \(L^2(\mathbb R^d)\). After defining the concept of wavelet associated to an IGMRA the author proves the existence of IGMRA's and of wavelets associated to an IGMRA for any \(L^2(\mathbb R^d)\), and gives several examples of wavelets associated to an IGMRA both with and without good localization, for \(d=1\) and \(d=2\).