the function $F:{\mathbb{R}}^2 \times  \Omega^{s} \to  \Omega^{s-1}$ as: $$F(\alpha,\omega,c) := ( i \tfrac{\omega}{\alpha} {\mathcal{K}}^{-1} +  U_{\omega}) c +  (U_{\omega} c) * c.$$

\[prop:Equivalence\] Let $\alpha, \omega>0$. If $ c \in \ell^1$ solves $F(\alpha,\omega,c) =0$, then $y(t)$, given by  with $ c_0=0$ and $ c_{-k} = c^*_k$, is a periodic solution of  with period $2\pi/\omega$. Vice versa, if $y(t)$ is a periodic solution of  with period $2\pi/\omega$, then its Fourier coefficients satisfy $c_0 = 0 $, $c_{-k} =c_{k}^*$, $\{c_k\}_{k=1}^\infty \in \ell^1 $ and solve $F(\alpha,\omega,\{c_k\}_{k=1}^\infty) =0$.

\[prop:Frechet\] For each $\alpha > 0$ and $ s \geq 2 $ the function $F:{\mathbb{R}}^2 \times  \Omega^{s} \to    \Omega^{s-1}$ is Frechet differentiable, with partial derivatives given as: $$\begin{aligned}
                \frac{\partial}{\partial \omega} 
    F(\alpha,\omega,c) &= i {\mathcal{K}}^{-1} (\alpha^{-1}  I - U_{\omega}) c - i  ({\mathcal{K}}^{-1} U_{\omega} c) * c  \label{eq:dFdW} \\
            \frac{\partial }{\partial c} F(\alpha,\omega,c) \cdot h &= ( i \tfrac{\omega}{\alpha} {\mathcal{K}}^{-1} + U_\omega) h + (U_\omega c) * h + (U_\omega h )*c, \label{eq:dFdC} 
    \end{aligned}$$ where $ h \in \Omega^{s}$.

\[prop:DerivativeComponent\] Define $\gamma_1(k,n) := e^{-i \omega(n+k)} + e^{i \omega n} $ and $ \gamma_2(k,n) := e^{-i \omega n} + e^{i \omega (n-k) } $. Writing $ c_k = a_k + i b_k$, the component-wise derivatives of $ 