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It is easy to see from the construction of Bernstein polynomials that if $f$ is Lipschitz continuous on $[0,1]$ then the distance of $f$ from the subspace of polynomials of degree at most $n$ is $O(1/n)$ where the implied constant depends on the Lipschitz constant of $f$. More generally any information on the modulus of continuity of $f$ can be used to get some kind of bound on this distance. Now my question is, can one prove any (positive or negative) statement of this kind for arbitrary continuous functions? Is it possible to prove that if a continuous function can be very well approximated by polynomials in some sense then it has to have a property similar to Lipschitz?

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The answer is yes. The results which allow to infer the modulus of continuity of a function from the value of function's best approximation by polynomials are collectively known as Bernstein-type theorems.

Theorem (S.N. Bernstein). Let $E_n(f)$ be the best approximation to the function $f(.) \in C([a, b])$ by algebraic polynomials of degree at most $n$, and let $$E_n(f)=O(n^{-\alpha})$$ with some $0< \alpha \leq1$.

  • If $\alpha < 1$, then $f(.)$ is uniformly Hölder continuous with exponent $\alpha$ on each segment $[a', b'] \subset(a, b)$, i.e. $$\omega(f,\delta)=O(n^{-\alpha}).$$
  • If $\alpha=1$, then $f(.)$ is uniformly almost Lipschitz on each such segment, i.e. $$\omega(f,\delta)=O\left(\delta\ln \frac{1}{\delta}\right).$$

The theorem cannot be improved in the sense that $E_n(f)=O(n^{-1})$ does not imply that $f(.)$ is Lipschitz. This is somewhat easier to see in case of the essentially equivalent problem of the best approximation of periodic functions by trigonometric polynomials on $\mathbb R$. The modulus of continuity of the Weierstrass function $$f_0(x)=\sum_{k=1}^{\infty}\frac{\cos(3^kx)}{3^k} $$ is $M\delta\ln\frac{1}{\delta}$. At the same time, one can show that the best approximation of $f_{0}(.)$ by trigonometric polynomials of degree $\leq n$ is equal to $$E_{n}^{Т}(f_0)=\frac{3}{2}\frac{1}{n}.$$


A very good reference on Bernstein-type theorems and related results is Theory of Approximation by Akhiezer.

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According to Lorentz ("Bernstein polynomials", Chelsea 1986, pages 19 to 23), the best Bernstein polynomial approximation (in uniform norm) for a Liptschitz function is $O(n^{-1/2})$ and can not be improved (he gives the example of $f(x)=|x-1/2|$ at $x=1/2$). However, a theorem of Jackson does provide better approximation using arbitrary polynomials ($O(n^{-1})$ for a Lipschtz function, and $\omega(n^{-1})$ for an arbitrary continuous function, where $\omega$ is the modulus of continuity).

If $f''$ exists, Lorentz shows (following Voronowskaja) that the approximation by Bernstein polynomials is of order precisely $O(n^{-1})$ at any point where the second derivative is non-zero.

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