The Regularized Spline (R-Spline) Method for Function Approximation

Abstract

Many constructions of cubic splines are described in the literature. Most of the methods focus on cubic splines of defect 1, i.e., cubic splines that are continuous together with their first and second derivative. However, many applications do not require continuity of the second derivative. The Hermitian cubic spline is used for such problems. For the construction of a Hermitian spline we have to assume that both the values of the interpolant function and the values of its derivative on the grid are known. The derivative values are not always observable in practice, and they are accordingly replaced with difference derivatives, and so on. In the present article, we construct a C¹ cubic spline so that its derivative has a minimum norm in L₂ . The evaluation of the first derivative on a grid thus reduces to the minimization of the first-derivative norm over the sought values.