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 $f$?

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