algebra, which is to say that $\| a * b \|_{\ell^1} \leq \|a \|_{\ell^1}   \| b\|_{\ell^1} $ for all $ a , b \in \ell^1$. While $ {\Omega}^s$ is not a Banach algebra *per se*, if $ s \geq 2$ then there exists a constant $B \geq 0$ such that $ \| a * b \|_s \leq B \| a \|_s   \| b \|_s$ for all $a ,b \in \Omega^s$ (see [@lessard2010recent; @berg2008chaotic]). Lastly, we define a linear operator ${\mathcal{K}}: \Omega^{s} \to \Omega^{s+1}$ and a continuous family of linear operators $U_\omega :  \Omega^{s }\to \Omega^{s-1}$ as below: $$\begin{aligned}
[{\mathcal{K}}c ]_k &:=  c_k  /k ,&&&
[ U_\omega c ]_k &:= e^{-i k \omega} c_k .\end{aligned}$$ The loss of regularity in the range of $U_\omega$ is necessary for its continuity, as $\frac{\partial}{\partial \omega} U_{\omega} = - i {\mathcal{K}}^{-1} U_{\omega}$. We may extend $ U_\omega$ to act on bi-infinite sequences $\{c_k\}_{k \in {\mathbb{Z}}}$ using the same component-wise definition. Additionally, this extension is compatible with our definition of the discrete convolution, as $[U_\omega c]_k^* = [U_\omega c]_{-k}$ whenever $c_{k}^* = c_{-k}$. In Definition \[def:Functional\] we rewrite in operator notation and list several propositions, the proofs of which are left to the reader.

\[def:Functional\] Define 