Best Approximation

Let \(||\cdot||_\infty\) denote the uniform norm (sup norm), namely, \(||f||_\infty = ||f||_{\infty, S} = \text{sup}\{|f(x)|:s\in S\}\) where \(S\) is a set. 

Problem: is there a polynomial \(p^\ast\) of specific degree \(n\) that is the best approximation to a given continuous function \(f\) in the sense of minimizing the uniform norm of the difference on an interval?

It is known that \(p^\ast\) exists and is unique. Caution:  best approximations in the uniform norm are not always as useful as one might imagine.

Weierstrass Approximation Theorem#

Theorem. Let \(f\in C[a,b]\), then for every \(\epsilon >0\) there is a polynomial \(p\) such that \(||f-p||<\epsilon\).

Equioscillation#

Equioscillation property: the error curve attains its extreme magnitude with alternating signs at a succession of values of \(x\).

Theorem. A continuous function \(f\) on \([-1,1]\) has a unique best approximation \(p^\ast \in\mathcal{P}_n\). If \(f\) is real, then \(p^\ast\) is real too, and in this case a polynomial \(p\in\mathcal{P}_n\) is equal to \(p^\ast\) iff \(f-p\) equioscillates in at least \(n+2\) extreme points.