Fourier Series estimation

Question

I know that the Fourier coefficient of $t\mapsto \frac{1}{\sqrt{\vert t\vert}}$ are given by some Fresnel integral, and behave like $O(n^\frac{-1}{2})$. Reciprocally, if I get a Fourier Series whose coefficients are $c_n= \frac{\epsilon_n}{n^\frac{1}{2}}$ with $\epsilon_n\in\{-1,1\}$, does there is any chance that $$\left\vert \sum_{n\in \mathbb{Z}} c_n e^{in\pi t} \right\vert \leq \frac{C} {\sqrt{\vert t\vert}},$$ for some $C>0$ and $t\in[-1,1]\setminus{0}? Thx

Answer 1

Suppose that $$ c_n=\left\{\begin{array}{cl} \hphantom{-}|n|^{-1/2}&\text{if }n\equiv0,1\pmod{4}\\ -|n|^{-1/2}&\text{if }n\equiv2,3\pmod{4} \end{array}\right. $$ Then $$ \sum_{n\in\mathbb{Z}}c_ne^{in\pi/2} $$ diverges ($t=1/2$).