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 Quanteroni, Math. Comp. 1982), but figured out that the Fourier theory is older, and maybe simpler.

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.