Spline functions

When approximating functions for interpolation or for fitting measured data, it is necessary to have classes of functions which have enough flexibility to adapt to the given data, and which, at the same time, can be easily evaluated on a computer. Traditionally polynomials have been used for this purpose. These have some flexibility and can be computed easily. However, for rapidly changing values of the function to be approximated the degree of the polynomial has to be increased, and the result is often a function exhibiting wild oscillations. The situation changes dramatically when the basic interval is divided into subintervals, and the approximating or fitting function is taken to be a piecewise polynomial. That is, the function is represented by a different polynomial over each subinterval. The polynomials are joined together at the interval endpoints (knots) in such a way that a certain degree of smoothness (differentiability) of the resulting function is guaranteed. If the degree of the polynomials is k, and the number of subintervals is n+1 the resulting function is called a (polynomial) spline function of degree k (order k+1) with n knots.

Splines are highly recommended for function approximation or data fitting whenever there is no particular reason for using a single polynomial or other elementary functions such as sine, cosine or exponential functions.

For practical problems, spline functions have the following useful properties. They are:

• - smooth and flexible,
• - easy to store and manipulate on a computer,
• - easy to evaluate, along with their derivatives and integrals,
• - easy to generalize to higher dimensions.

The name spline function was introduced by Schönberg in 1946. The real explosion in the theory, and in practical applications, began in the early 1960s. Spline functions are used in many applications such as interpolation, data fitting, numerical solution of ordinary and partial differential equations (finite element method), and in curve and surface fitting.

An early book about splines with programs is Boor78, a more recent publication is Press95. also Flowers95. An application of spline functions to track fitting is given in Wind74.