Wavelets on irregular meshes
The paper deals with the construction of wavelets on specific locally refined meshes \(\Gamma\) satisfying the following conditions: \(\Gamma = \bigcup_{j \geq 0} \Gamma_j\), where (1) \(\Gamma_0 := Z\), \(\Gamma_j \subset 2^{-j} Z\); (2) if \(\Lambda_j := \Gamma_{j + 1} \setminus \Gamma_j\) then \(\Lambda_j \subset 2^{-j - 1} Z \setminus 2^{-j} Z\); (3) cone condition: There exists \(\rho \in N\) such that, if \(\gamma = (2k + 1) 2^{-j - 1} \in \Lambda_j\) then all points \(l2^{-j}\) with \(|l - k |\leq \rho\) belong to \(\Gamma_j\). The new sample space \(V_j\) on \(\Gamma_j\) are appropriate subspaces of given classical sample spaces \({\mathcal V}_j\) generating a multiresolution analysis on the uniform mesh \(2^{-j} Z\). With the help of the ``scale associated to a point \(\gamma \in \Gamma_j\)'', \[ a_j(\gamma) = \inf_{\mu \in \Gamma_j, \mu \neq \gamma} |\gamma - \mu| \] a weighted scalar product on \(V_j\), \((f,g)_j = \sum_{\gamma \in \Gamma_j} a_j(\gamma) f(\gamma) \overline{g(\gamma)}\) is introduced, and the corresponding norm is shown to be equivalent with the classical \(L^2\)-norm. The wavelet spaces \(X_j\) are then constructed as orthogonal complements of \(V_j\) in \(V_{j + 1}\) with regard to this scalar product \((\cdot, \cdot)_j\). Corresponding fast analysis algorithms and synthesis algorithms are established. The stability of the analysis algorithm is still an open problem.
scientific article; zbMATH DE number 842127
scientific article