Do there exist a sequence $e_n:\ [-1,1]\to [-1,1]$ of functions such that, for every function $f\in C^s([-1,1])$ we have the following approximation property:
$$\inf_{g\in \text{span}(e_1,\dots e_n)}\|f-g\|_\infty \le C(f)n^{-s-\varepsilon}?$$
(where $\varepsilon>0$). My question is motivated by the case where $e_n(x)=x^{n-1}$, which falls under the name of Polynomial of best approximation, in which the bound with $C(f)n^{-s}$ has been proved, and I conjecture it is not possble to do strictly better.