In my research I would like to approximate uniformely a function $f: [a,b] \xrightarrow{} \mathbb{R}$ by polynomials (where $f$ is typically $C^1$). If the Stone-Weierstrass gives a density result for arbitrarly big polynomials, for my problem I would in fact need a bound on the error at a finite step (when the polynomials available can't be too big).
To be more concret I would like to have an inequality of the type $$\underset{(a_k)_{k\in \mathbb{N}},\sum_{k=0}^\infty |k a_k| < \alpha}{\inf}||f-\sum_{k=0}^\infty a_k X^k||_{\infty}\leq \frac{C}{\alpha^n}$$ (where $n>0$ and $C$ is a constant that depend of $||f||_\infty$ and $||\nabla f||_\infty$ but not of $\alpha$).
So I would like to know if there is a theorem which proove this kind of result (or a close one) in the literature.