Chapter 2 - One-dimensional linear and cubic splines

Abstract

This chapter provides an overview of one-dimensional linear and cubic splines. The cases of linear and cubic splines and, in general, splines of odd polynomial degree are very similar. The most important feature of these odd-order splines is that the data points and the knot points coincide. The situation with splines of degree 2, the so-called parabolic splines, is completely different. It has been explained that it is very inappropriate to choose knot points that coincide with the data points. In general, all splines of even polynomial degree satisfy that principle—namely, that the knot points have to differ from the data points. Otherwise the error available in the data does not decay exponentially. The linear splines are simpler and may be considered as the one-dimensional analog to the so-called harmonic polysplines. The focus in this chapter is on cubic splines, linear splines, and variational (Holladay) property of the odd-degree splines. It also gives details about existence and uniqueness of interpolation odd-degree splines. Basic concepts related to the Holladay theorem are also explained in this chapter.