On the spline-based wavelet differentiation matrix Original Research Article

The differentiation matrix for a spline-based wavelet basis without decomposition at multiple scales is constructed. It is shown that regardless of the properties that one imposes on a spline basis, the differentiation matrix is always the one encountered when the basis is simply the B-spline basis. The order of accuracy of this differentiation matrix is shown to display superconvergence. Furthermore, it is shown that spline-based bases generate a class of compact finite difference schemes. With this knowledge boundary conditions of the proper order of accuracy can be imposed.