Best polynomial approximation

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 $[− 1,1]$ are considered. A theorem due to Weierstrass (see for instance [65 ]) states that the set $ℙ$ of all polynomials is a dense subspace of all the continuous functions on $[− 1,1]$ , with the norm $∥⋅∥ ∞$ . This maximum norm is defined as

This means that, for any continuous function $f$ of $[− 1,1]$ , there exists a sequence of polynomials $(pi),i ∈ ℕ$ that converges uniformly towards $f$ :

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

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

Chebyshev alternate theorem states that for any continuous function $f$ , $p⋆N$ is unique (theorem 9.1 of [179 ] and theorem 23 of [150 ]). There exist $N + 2$ points $x ∈ [− 1,1] i$ such that the error is exactly attained at those points in an alternate manner:

where $δ = 0$ or $δ = 1$ . An example of a function and its best polynomial approximation is shown in Figure 1

Figure 1: Function $f = cos3(πx ∕2) + (x + 1)3 ∕8$ (black curve) and its best approximation of degree 2 (red curve). The blue arrows denote the four points where the maximum error is attained.