Let $f:[-\pi, \pi] \to \mathbb{R}$ be the step function $f(x) = -1$ if $x<0$, $f(x) = 1$ if $x>0$. The Fourier coefficient of $f(x)$ is given as $\widehat{f}(n) = -\frac{2i}{\pi n}$ if $n$ is odd and $ \widehat{f}(n) = 0$ if $n$ is even.
The problem is to find a function $g$ whose Fourier coefficients are given by $\widehat{g}(n) = \frac{1}{(in)^{2}} \widehat{f}(n)$ if $n$ is odd, $\widehat{g}(n) = 0$ if $n$ is even. I realized that the function $f$ is not $2 \pi$-periodic and I tried to integrate $f$ twice to get $g$ but did not get the right thing.