k^s |c_k| < \infty  \right\} & \| c \|_{s} =&  \sup_{k \in{\mathbb{N}}} k^s | c_k|.\end{aligned}$$ The smoother a function is the faster its Fourier coefficients will decay; if a function is $s$–times continuously differentiable, then its Fourier coefficients will be in $ {\Omega}^s$. Since periodic solutions to are real analytic [@wright1955non; @nussbaum-analytic], it follows that their Fourier coefficients will be in $ {\Omega}^s$ for all $ s \geq 0$.

If $y$ is a solution to Wright’s equation, then by substituting into we obtain: $$\sum_{k \in {\mathbb{Z}}} i \omega k c_k e^{ i \omega k t} = - \alpha \left(\sum_{k \in {\mathbb{Z}}}  c_k e^{-i \omega k}  e^{ i \omega k t}\right) \left( 1 + \sum_{k \in {\mathbb{Z}}}  c_k e^{ i \omega k t}\right).$$ By matching the $ e^{ i \omega k t}$ terms, subtracting the RHS, and dividing through by $\alpha$, we obtain the following sequence of equations for $k \in {\mathbb{Z}}$ below: $$\begin{aligned}
[F(\alpha,\omega,c)]_k=&\,
\left( i \tfrac{\omega}{\alpha} k +  e^{ - i \omega k} \right) c_k +  \sum_{\substack{k_1,k_2\in{\mathbb{Z}}\\ k_1 + k_2 = k}} e^{- i \omega k_1} c_{k_1} c_{k_2} 
\label{eq:FourierSequenceEquation} \\
  =&\,
\left( i \tfrac{\omega}{\alpha}  k + e^{ - i \omega k }\right) c_k +   \sum_{j=1}^{k-1} e^{-i \omega j } 