Nice way to express $H^{-1}(\mathbb{S}^1)$ 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 $f$?
 A: 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.)
A: Note first of all that $H^{-1}$   will contain  objects that are not functions, such as the Dirac $\delta$ concentrated at a point. (Think that $2\delta_0=\frac{d^2}{dx^2}|x|$.)
The correct definition is that $H^{-1}$ is the completion of $C^\infty(S^1)$ with respect to the norm $\newcommand{\ii}{\boldsymbol{i}}$
$$\Vert f\Vert^2=\sum_{n\in\mathbb{Z}}\bigl(1+n^2\bigr)^{-1} \bigl\vert f_n\bigr\vert^2,\;\;f_n=\int_{S^1} f(\theta)e^{-n\ii\theta} d\theta. $$
Equivalently, $H^{-1}$ consists of distributions $\alpha$  on $S^1$ with the property that there exists $C>0$ such that
$$\big\vert\; \alpha(f)\;\bigl| \leq C\Vert f\Vert_{H^1},\;\;\forall f\in C^\infty(S^1). $$
