# Sobolev convergence of Fourier series

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.

• See Lemma A. 43 in [A. Kirsch: An Introduction to the Mathematical Theory of Inverse Problems. Springer, New York, 1996]. Mar 23, 2020 at 4:06
• Thanks! Is this tight, in terms of spaces, rates, and constants? Mar 23, 2020 at 14:50
• @YidongLuo my comment above Mar 23, 2020 at 15:00
• The rates is tight for the periodic Sobolev space $H^r, r \in \mathbb{R}$ in terms of the regular estimate proof. Mar 24, 2020 at 3:33

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