Construction of optimally conditioned cubic spline wavelets on the interval (Q623377)

The authors construct a primal multiresolution analysis on the interval and a dual multiresolution analysis. The primal scaling functions are B-splines defined on the Schönberg sequence of knots. There are two types of boundary scaling functions. The functions of the first type are defined in order to preserve the full degree of polynomial exactness. The construction of the scaling functions of the second type is a delicate task, because the low condition number and nestedness of the multiresolution spaces have to be preserved. Refinement matrices are computed. Wavelets are constructed by the method of stable completion. It is shown that the constructed set of functions is indeed a Riesz basis for the space \(L^2([0, 1])\) and for the Sobolev space \(H^s ([0, 1])\) for a certain range of \(s\). The primal bases are adapted to homogeneous Dirichlet boundary conditions of the first order and the dual bases to the complementary boundary conditions. Quantitative properties of the constructed bases are presented. The efficiency of an adaptive wavelet scheme is compared for several spline-wavelet bases and the superiority of the authors' construction is shown. Numerical examples are presented for one-dimensional and two-dimensional Poisson equations where the solution has steep gradients.