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

∥f ∥∞ = max |f (x)|. (15 ) x∈[−1,1]

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:

li→im∞ ∥f − pi∥∞ = 0. (16 )
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:

∥f − p⋆ ∥ = min {∥f − p∥ ,p ∈ ℙ } . (17 ) N ∞ ∞ N

Chebyshev alternate theorem states that for any continuous function f, p⋆N is unique (theorem 9.1 of [178Jump To The Next Citation Point] and theorem 23 of [149]). There exist N + 2 points x ∈ [− 1,1] i such that the error is exactly attained at those points in an alternate manner:

⋆ i+δ ⋆ f (xi) − pN (xi) = (− 1) ∥f − pN ∥∞ , (18 )
where δ = 0 or δ = 1. An example of a function and its best polynomial approximation is shown in Figure 1View Image.
View Image

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.

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