2.1 Best polynomial approximation

Polynomials are the only functions that a computer can exactly evaluate and so it is natural to try to approximate any function by a polynomial. When considering spectral methods, we use global polynomials on a few domains. This is to be contrasted with finite difference schemes, for instance, where only local polynomials are considered.

In this particular section, real functions of are considered. A theorem due to Weierstrass (see for instance [65]) states that the set of all polynomials is a dense subspace of all the continuous functions on , with the norm . This maximum norm is defined as

This means that, for any continuous function of , there exists a sequence of polynomials that converges uniformly towards :

This theorem shows that it is probably a good idea to approximate continuous functions by polynomials.

Given a continuous function , the best polynomial approximation of degree , is the polynomial that minimizes the norm of the difference between and itself:

Chebyshev alternate theorem states that for any continuous function , is unique (theorem 9.1 of [178] and theorem 23 of [149]). There exist points such that the error is exactly attained at those points in an alternate manner:

where or . An example of a function and its best polynomial approximation is shown in Figure 1.