ABSTRACT
Polynomials have long been the functions most widely used to approximate other functions, mainly because of their simple mathematical structures. However, it is a well-known observation called Runge-phenomenon (cf. C. Runge [1901], C. Runge, F.A. Willers [1915]) that a polynomial of moderately high degree fitted to a fairly large number of given data points tends to exhibit numerous and severe undulations (cf. Example 1.5 and Table 1.2). There is now considerable evidence that in many circumstances a spline function is a more adaptable approximating function than a polynomial. This conclusion is based in part on actual numerical experience, and in part on mathematical demonstrations that the solutions of a variety of problems of “best approximation” actually turn out to be spline functions.
