Let $g:[0,2\pi]\to \mathbb{C}$ which is $\mathcal{C}^k([0,2\pi],\mathbb{R})$ and periodic. If $\mid f^{(k)}(x) \mid\le 1$ then for each $n\in \mathbb{N}^*$, there exists a trigonometric polynomial $T_{n-1}$ of degree at most $n-1$ such that : $\forall x\in [0,2\pi]$, $\mid g(x)-T_{n-1}(x)\mid \le \frac{\Delta_k}{n^k}$, where $\Delta_k$ depends only on $k$.

I was wondering if someone had references for the proof of this statement. I looked at the book of Achieser "Theory of Aprroximation" but I found that the proof was difficult to follow...

Thanks in advance !


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