Consider $f\in H^{\sigma}(S^1)=W^{\sigma, 2}$ (the usual Sobolev space on the circle) and let $S\_Nf$ be its truncated Fourier series $S\_Nf = \sum\_{|n|\leq N} \hat{f}(n)e^{2\pi i n x}$. I am looking for the theory for inequalities of the following type:
$$\|f-S\_Nf\|\_{H^s} \leq CN^{c(s,\sigma)} \|f\|\_{H^{\sigma}} \, ,$$
What values of $s>0$ work for any given $\sigma >0$? What is known of the constant $C$ and of the rate $c(s,\sigma )$?


*Motivation*: I am familiar with the equivalent theory for orthogonal polynomial expansions (see Canuto and Quarteroni, Math. Comp. 1982), but figured out that the Fourier theory is older, and maybe simpler.



Sobolev convergence of Fourier series
Let us start with pointing out that $f\in H^\sigma$ is equivalent to
$$
(\langle n\rangle^\sigma\hat f(n))\_{n\in \mathbb Z}\in \ell^2(\mathbb Z),
\quad \text{with $\langle n\rangle=\sqrt{1+n^2}$.} 
$$
Then you have for $s<\sigma$
$$
\Vert f-S\_N(f)\Vert\_{H^s}^2=\sum\_{\vert n\vert> N}\langle n\rangle^{2s}\vert\hat f(n)\vert^2=
\sum\_{\vert n\vert> N}\langle n\rangle^{2\sigma}\vert\hat f(n)\vert^2\langle n\rangle^{2s-2\sigma}\le\langle N\rangle^{2s-2\sigma}\Vert f\Vert^2\_{H^\sigma}. 
$$


