Let us assume that a real function $f\in C^k(\mathbb R)$ is $2\pi-$periodic. Is it true that, denotig $\Pi_n$ the set of trigonometric polynomials of degree $n$, we have
$$\inf_{g\in \Pi_n} \|f-g\|_\infty = O(n^{-k})?$$
The reason I am asking the question is that a similar result is well known when $\Pi_n$ is instead the class of standard polynomials of degree $n$. I have found some paper which deal with the topic of best approximation by trigonometric polynomials, like this, but for what I have understood no such result is present.