Multi-resolution analysis with frontal decomposition (Q1892137)

The dual wavelets (scaling functions) of the Chui-Wang wavelets possess infinite masks such that the wavelet decomposition of functions via the pyramid algorithm requires the truncation of the masks. The authors deal with linear Chui-Wang wavelets. They suggest to replace the decomposition of functions via the pyramid scheme by a `frontal technique' which can increase the computational efficiency of the decomposition. The frontal method works as follows: Let \(P_{j+ 1}\), \(P_j\) denote the orthogonal projectors onto the consecutive function spaces \(V_j\) of the multiresolution, i.e. \(P_j f+ \sum^N_{k= 0} c_{jk} \varphi_{jk}\). Minimize \(\int [(P_{j+ 1}- P_j) f]^2 dx\) by the direct solution of the corresponding tridiagonal system \((P_j f, \varphi_{jk})= (P_{j+ 1} f, \varphi_{jk})\) \((k= 0,\dots, N)\). (Find \(c_{jk}\) for given \(c_{j+ 1, k}\)).