I was curious if there is an effective way to compute (the asymptotic of) the Fourier coefficients of $$ F(x)= \log\log\left(\frac{1}{\left\lvert x\right\rvert}\right) \cdot \chi\left(\left\lvert x\right\rvert\right) $$ defined on $[-\pi,\pi]$ where $\chi(\cdot)$ is some smooth bump function which equals $1$ on $[-1/4,1/4]$ and $0$ outside of $[-1/2,1/2]$.
My goal is to check whether this function belongs to certain fractional Sobolev spaces, more specifically $H^s([-\pi,\pi])$ for $0 < s<1/2$.