This is the proof that $S_i$ is fourier coefficients of a distribution iff $\sum_{i\in Z}(1+i^2)^{-k}S_i<\infty$.
Let $S_i$ be fourier coefficients of a distribution $S$. Then $\exists C,\forall j\in Z, |<S,exp(-i jx)>|=|S_i|\leq C|exp(-ijx)|_l$ where $|f|_l=\sum_{a\leq l}max\{|(\frac{d}{dx})^af|\}$ where I have assumed distribution lives on circle $S^1$.
Then the book says $|S_i|\leq C'(1+i^2)^{\frac{1}{2}}$.(This conclusion has no dependence on $l$.)
$\textbf{Q:}$ How did the book reach the conclusion $|S_i|\leq C'(1+i^2)^{\frac{1}{2}}$? I reached a different conclusion but it has the same effect. $|exp(-ijx)|_l=\sum_{a\leq l}|j^a|\leq(1+|j|)^l\leq (1+|j|^2)^l$ by $j\in Z$. The other option is to cast $|-|_l$ into Sobolev norm. One can use $|-|_l\leq |-|_{2l}$ on smooth functions and this needs to bump up the sobolev norm from $l$ to $2l+[n/2]+1$ to contain $|-|_{2l}$ estimation.
Ref. Kodaira's Complex Manifolds and Deformation of Complex Structure Appendix Sec 1, Thm 1.6 on pg 373