Chapter 5 - ‘Best’ approximation

Abstract

This chapter discusses the various types of polynomial approximations which are based on minimizing the norm of the error function. The norm of a function belonging to some class of functions C—is a mapping from C to the non-negative real numbers. If f is supposed to be a function in C and some norm on C is chosen where the class of functions C is either C[a, b] or C(X), then this leads to a class of polynomial approximations to f , which are called best approximations. In general, different norms lead to different best approximations. Two classes of approximation problems can be distinguished. The first is approximation to a function on a finite interval. The second class of problems deals with approximation to a function whose values are given at only a finite number of points. Increasing the degree of the approximating polynomial does not always increase the accuracy of the approximation. If the degree is large, the polynomial may have a large number of maxima and minima. It is then possible for the polynomial to fluctuate considerably more than the data, particularly if it contains irregularities due to errors. For this reason it is more appropriate to use different low degree polynomials to approximate to different sections of the data. These are called piecewise approximations, the most commonly used being—spline approximations.