A Symbolic Algorithm to Obtain Low or High Degree Splines from Discrete Fourier Transforms

Obtaining high-degree splines with the use of traditional spline interpolation methods is not an easy task; therefore, traditional spline interpolation is typically limited to cubic splines. In this paper, we present a symbolic and numeric algorithm to obtain splines of any degree, while providing detailed procedures and examples of how to use this algorithm so that it is immediately useful for an interested user. This method, which was initially developed by Beaudoin and Beauchemin [2, 3], works for splines of any degree and yields very accurate results when the boundary conditions are chosen wisely. It also provides approximations of higher order derivatives, something that is not available with the use of cubic splines. This paper presents formulas that can be used in a straightforward manner to obtain interpolation splines of first degree (linear splines), second degree (parabolic splines) and third degree (cubic splines). For splines of higher degree, a short but complete symbolic algorithm to compute the formulas is presented. The resulting formulas can be used in the same manner as those presented for splines of lower degree. A complete numerical example is included to show how the results are obtained and a link to the complete Maple source code is given.