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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.

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  • $\begingroup$ This kind of questions is core for the branch of mathematics called "approximation theory". This site is one of its headquarters. (Look also here). Your question is surely well-known to them. $\endgroup$ – Giuseppe Negro May 25 '17 at 9:03
  • $\begingroup$ Grazie per i miravigliosi riferimenti ai due siti Internet, caro @Giuseppe! $\endgroup$ – Georges Elencwajg May 25 '17 at 10:23
  • $\begingroup$ @GeorgesElencwajg: Prego, mi fa piacere essere stato utile, caro Georges. $\endgroup$ – Giuseppe Negro May 25 '17 at 11:05
  • $\begingroup$ This is a question of similar type: math.stackexchange.com/questions/271013/… $\endgroup$ – ploosu2 May 25 '17 at 11:09
  • $\begingroup$ In a paper that was linked in the other question's answer (m-hikari.com/imf-password2008/29-32-2008/…) the following (Theorem 1.2) is achieved: The Berstein polynomials of $f$ give the bound $\epsilon(n) = O(\omega(1/\sqrt n))$, where $\omega$ is the modulus of $f$, (see the paper how it's defined). I don't think you can get rid of the dependence on $f$, since $f$ can be rapidly oscillating and you need to have bigger $n$ depending on $f$. $\endgroup$ – ploosu2 May 25 '17 at 11:20
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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.

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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.

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