A quantitative version of the Weierstrass' Approximation Theorem

Question

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 $\alpha>0$?

Answer 1

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.