I am looking for a good way to write down what $H^{-1}(\mathbb{S}^1)$ is. What do I mean by this: Well, certainly one can define this space via charts that map from the sphere into $\mathbb{R}^2$ and then use this definition to give this space some sense, but I have the feeling that this is too abstract in this situation. 


For instance, if I want to describe $H^2(\mathbb{S}^1)$ then I can simply notice that this is the domain of the Laplacian, so 


$$ H^2(\mathbb{S}^1):=D(-\Delta) = \{f \in L^2(\mathbb{S}^1); \sum\_{n=-\infty}^{\infty} |n^2\langle f, e^{-in \cdot} \rangle|^2< \infty \}$$ by the Fourier expansion of the Laplacian, i.e. there is apparently no need for abstract chart definitions in case of $H^2$. So my question is: Is there also a nice characterization of $H^{-1}$ in this case?


Edit: So following Christian Remling's suggestion I assume that the right definition would be something like:


$$\{f \in ?; \sum\_{n=-\infty}^{\infty} |(1+|n|)^{-1}\langle f, e^{-in \cdot} \rangle|^{2}< \infty\}.$$


My question would then be: What kind of objects do we allow for $f$?



Nice way to express $H^{-1}(\mathbb{S}^1)$
I am somewhat confused that, despite saying many true and useful things, no one has said directly that $H^{-1}$ on the circle can be characterized as the set of distributions $\theta$ such that $\sum\_{n\in \mathbb Z}(1+n^2)^{-1}\cdot |\theta(x\to e^{inx})|^2<\infty$. (All distributions on the circle are compactly supported, so can be applied to the exponentials, which are smooth.)


