How well can continuous functions on $[0,1]$ be approximated by polynomials up to a given degree?

Question

Let $f:[0,1] \to \mathbb{R}$ be continuous. The Stone–Weierstrass theorem tells us that there is a sequence $p_n$ of polynomials defined on $[0,1]$ such that $$\lim_{n \to +\infty} ||p_n - f||_{\infty} = 0$$

A consequence of this theorem is that

$$\lim_{n \to +\infty} \inf \{||p-f||_{\infty} : p \ \text{is a polynomial of degree} \leq n\} =0$$

This is the quantity I'm interested in. Namely, $$\epsilon(n) \stackrel{\rm def}{=}\inf \{||p-f||_{\infty} : p \ \text{is a polynomial of degree} \leq n\}$$

This roughly tells us the following: if we want to uniformly approximate $f$ by a polynomial of degree $\leq n$, then we should expect an error of at least $\epsilon(n)$.

Is there an explicit upper bound on $\epsilon(n)$ (e.g., something like $\epsilon(n) \leq \frac{15}{\log n})$? Is there an asymptotic bound on $\epsilon(n)$ (e.g., $\epsilon(n) = O(n^{-2})$?

The bounds should be as tight as possible.

Answer 1

There's no reasonable answer without restricting the class of functions $f$ you're considering. The error may be $O(r^{-n})$ if $f$ has an analytic continuation to an ellipse with foci $0, 1$ in the complex plain, it may be just $O(1/n)$ if you only know it's differentiable (with bounded derivative) on $[0,1]$, or it can be still worse, if $f$ is just continuous, but not smooth.

Answer 2

First of all, the quantity $\epsilon(n)$ is given by a certain polinomial $\tilde p$, so that $$\epsilon(n)=||f-\tilde p||_{\infty}$$ A bound of the error is given by Jackson's theorems. If $f$ is only continuous you can tell that $$ \epsilon(n) \le M \omega(f, \frac{b-a}{n}) $$ where the modulus of continuity is defined as $$ \omega(f,\delta) := \sup \{|f(x)-f(y)| : x,y \in [a,b],|x-y|\le \delta \}$$ and $M$ is a real constant (in your case the domain is $[a,b]=[0,1] $). If $f$ is increasingly regual, like lipschitz continuous, $C^p$, $C^{\infty}$ analytic and analytic on all the complex plane, the bounds become more severe and the approximation is much better.