Splines

Abstract

A spline is a function which is piecewise defined on intervals such that the pieces are joined together smoothly. The terminology was introduced by I. J. Schoenberg [1946], although these kinds of functions had been used earlier by several other authors. For example, the Euler method for constructing a piecewise polynomial approximation to the solution of an initial-value problem for ordinary differential equations (and which is often used to establish the Peano Theorem on the existence of solutions of such problems) can be regarded as a simple application of splines. In this regard we should also mention the papers of C. Runge [1901], W. Quade and L. Collatz [1938], J. Favard [1940] and R. Courant [1943], among others. The theory of splines is a good example of an area in mathematics which was developed in response to practical needs. One of the early problems which gave impetus to the development of splines was the need for usuable methods for constructing smooth approximations on the basis of tabulated data arising in ballistics. The subject has steadily developed over the past thirty years, and at present there are several thousand research papers on splines and their applications. In view of this large literature, it is clear that within the framework of this book, we will only be able to give an introduction to a part of the theory. Our discussion will focus on splines constructed from polynomial pieces.