Let $\theta(t) = (1-|t|)^+$ for $t \in \mathbb{R}$. In order to show that \begin{align} \int_{-\infty}^\infty \theta(t) e^{-i \xi t}\ dt = \bigg(\frac{sin\big(\frac{\xi}{2}\big)}{\frac{\xi}{2}}\bigg)^2, \qquad (*) \end{align} I want to compute the Fourier transform of $\phi(t) = \mathbb{1}_{\{-1/2 \leq t \leq 1/2\}}$ and relate this to $\theta$.
It could be shown that the Fourier transform $\hat{f}$ of $\phi$ equals \begin{align} \hat{f}(\xi) = \frac{\sin(\pi \xi)}{\pi \xi}. \qquad (**) \end{align} Making the substitution $u = \frac{1}{2}t$ in $(*)$ gives \begin{align} \int_{-\infty}^\infty \theta(t) e^{-i \xi t}\ dt = 2 \int_{-1/2}^{1/2} (1-|2u|)^+ e^{-2i \xi u}\ du \qquad (***) \end{align} and could be worked out. However, I do not see how $(**)$ could be related to $(***)$. Any help is appreciated!
You can write $\theta$ as $1_I\ast1_I$ for an interval $I$. Say $I=[-a,a]$. Then you have to compute $$ \int_{-a}^a1_{[-a,a]}(x-t)dt. $$