For the step-function

$$F(x)= \begin{cases}1, & |x| \leqslant 1 / 2 \\ 0, & \text { otherwise }\end{cases}$$

its convolution with itself is the hat-function

$$G(x)=[F * F](x)= \begin{cases}1-|x|, & |x| \leqslant 1 \\ 0, & \text { otherwise }\end{cases}$$

Find the Fourier transforms of $F$ and $G$, and hence find the values of the integrals

$$I_{1}=\int_{-\infty}^{\infty} \frac{\sin ^{2} y}{y^{2}} d y, \quad I_{2}=\int_{-\infty}^{\infty} \frac{\sin ^{4} y}{y^{4}} d y$$