Define the convolution $f * g$ of two functions $f$ and $g$. Defining the Fourier transform $\tilde{f}$ of $f$ by

$$\tilde{f}(k)=\int_{-\infty}^{\infty} \mathrm{e}^{-\mathrm{i} k x} f(x) \mathrm{d} x$$

show that

$$\widehat{f * g}(k)=\tilde{f}(k) \tilde{g}(k) .$$

Given that the Fourier transform of $f(x)=1 / x$ is

$$\tilde{f}(k)=-\mathrm{i} \pi \operatorname{sgn}(k),$$

find the Fourier transform of $\sin (x) / x^{2}$.