Let $A:=\{ u \in H^k(0,2\pi): u^{(j)}(0)=u^{(j)}(2\pi) \mbox{ for } j=0,1,\ldots, k-1\}$, where $H^{k}(0,2\pi)\subseteq L^2(0, 2 \pi)$ is the Sobolev space of order $k$ on $(0, 2 \pi)$. Can we say that $u \in A$ iff $$\sum_{n=-\infty}^\infty (1+n^2)^k |\hat{u}(n)|^2<\infty?$$ In the above series $\hat{u}(n)$ are the Fourier coefficients of $u$. We think that the answer to this question is affirmative because maybe we can identify $A$ with the Sobolev space of the torus $H^k(\mathbb{T})$, and use this result.
Do you know any reference for a characterization of $A$ with Fourier series?
Thank you for any help you can provide us.