Sobolev convergence of Fourier series

Question

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.

Answer 1

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 $s<\sigma$ $$ \Vert f-S_N(f)\Vert_{H^s}^2=\sum_{\vert n\vert> N}\langle n\rangle^{2s}\vert\hat f(n)\vert^2= \sum_{\vert n\vert> N}\langle n\rangle^{2\sigma}\vert\hat f(n)\vert^2\langle n\rangle^{2s-2\sigma}\le\langle N\rangle^{2s-2\sigma}\Vert f\Vert^2_{H^\sigma}. $$