I need some help understanding the relationship between distributions $\mathcal{D}^\prime(\mathbb{T})$, where $\mathbb{T}=\mathbb{R}/\mathbb{Z}$, and the subset of distributions that are invariant under shifts in $\mathbb{Z}$. Is there a one-to-one correspondence?
To clarify notation: For me $\mathcal{D}^\prime(\mathbb{R})$ is the dual of $\mathcal{C}^\infty_c(\mathbb{R})$, the dual of Schwartz functions would be $\mathcal{S}^\prime(\mathbb{R})$.
If I have a periodic distribution, I can always get a real distribution that is invariant under shifts via periodic extension. What about the other direction? I'd guess you have to use support properties of the Fourier transform (so it would only work with $\mathcal{S}^\prime$?), is that true and is there a more direct approach? Perhaps I can mollify, integrate over one copy of the torus, and take the limit?
If I work with periodic and non-periodic Besov spaces, can I get an isomorphism from $\mathcal{B}_{p,q}^\alpha(\mathbb{T})$ to some subset of $\mathcal{B}_{p,q}^\alpha(\mathbb{R})$ by localizing the distribution that I get from the continuous extension (with a partition of unity)?
