### 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.