Denote by $f * g$ the convolution of two functions, and by $\widehat{f}$ the Fourier transform, i.e.,

$$[f * g](x)=\int_{-\infty}^{\infty} f(t) g(x-t) d t, \quad \widehat{f}(\lambda)=\int_{-\infty}^{\infty} f(x) e^{-i \lambda x} d x$$

(a) Show that, for suitable functions $f$ and $g$, the Fourier transform $\widehat{F}$of the convolution $F=f * g$ is given by $\widehat{F}=\widehat{f} \cdot \widehat{g}$.

(b) Let

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

and let $f_{2}=f_{1} * f_{1}$ be the convolution of $f_{1}$ with itself. Find the Fourier transforms of $f_{1}$ and $f_{2}$, and, by applying Parseval's theorem, determine the value of the integral

$$\int_{-\infty}^{\infty}\left(\frac{\sin y}{y}\right)^{4} d y$$