# A quantitative version of the Weierstrass' Approximation Theorem

Assume that $f\in\mathcal{C}^0([0,1])$. By using Chebyshev Polynomials, it is possible to show that there exists a sequence of polynomials $\{p_n(x)\}_{n\in\mathbb{N}}$ such that: $$\max_{x\in[0,1]}|p_n(x)-f(x)|=O\left(\frac{1}{\partial p_n}\right),$$ where $\partial p_n$ is the degree of $p_n$. My question is: is it possible to do better? I.e.: given a generic $f\in\mathcal{C}^0([0,1])$, is it possible to find a sequence of polynomials $\{p_n(x)\}_{n\in\mathbb{N}}$ such that $$\max_{x\in[0,1]}|p_n(x)-f(x)|=O\left(\frac{1}{(\partial p_n)^{1+\alpha}}\right),$$ for a certain $\alpha>0$?

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Are Jackson's Theorem and Bernstein's Theorem (see en.wikipedia.org/wiki/Jackson%27s_inequality and en.wikipedia.org/wiki/…) helpful? – Johannes Kloos Nov 2 '12 at 12:09
Not too much. I know that, if $f$ is a regular function, say $\mathcal{C}^k([0,1])$ there exist a polynomial approximation with error $O(\partial p_n^{-k})$, but i'm interested in the very basic case where $f\in\mathcal{C}^0$ but is not differentiable, just like $f(x)=|x-1/2|$. – Jack D'Aurizio Nov 2 '12 at 15:07

Your first statement is wrong. In fact, if we have an approximating sequence of polynomials $p_n$ of degree $n$ with $|p_n - f| \le C n^{-\alpha}$ for some $\alpha \in (0,1)$, then $f$ is $\alpha$-Hölder continuous, by Bernstein's theorem.
Bernstein's theorem completely solves the problem. My first statement is wrong: I was thinking to Lipschitz functions over $[-1,1]$ just as $|x|$ rather then continous, non-differentiable functions. Thanks for the answer. – Jack D'Aurizio Nov 3 '12 at 14:24