High order interpolation and differentiation using B-splines

We present a methodology of high order accuracy that constructs in a systematic way functions which can be used for the accurate interpolation and differentiation of scattered data. The functions are based on linear combination of polynomials (herein B-splines are used). The technique is applied to one-dimensional datasets but can be extended as needed for multidimensional interpolation and differentiation. The methodology can also construct one-sided functions for high-order interpolation and differentiation. The constructed functions possess compact support. The penalty for the high order of accuracy is the need to solve a system of L×L equations where L is the order of the approximation. In order to have a robust solution of the L×L system the singular value decomposition technique was adopted. The proposed technique can also be applied in the context of other methods, in order to increase their accuracy. The main novel features of the technique are that no grid-based information (connectivity) is necessary and a minimum number of samples are required to achieve the desired order of approximation. The order of the approximation is not affected when more samples than the minimum necessary are added in the domain of influence.