Inverse Fourier Transform of piecewise function
So you know that in the frequency domain the function
$$s(t)=\frac{\sin(2t)}{\pi t}\tag{1}$$
is a rectangle with height $1$ for $|\omega|&lt;2$. Now think graphically about convolving $S(j\omega)$ with itself. Can your result be right? You found that $X(j\omega)$ is only non-zero for positive frequencies. That would correspond to a complex-valued time domain function. So check that result again. The result of the convolution of two rectangles is basic and you should be able to see the result graphically.
After having figured out the convolution of $S(j\omega)$ with itself, it should be easy to obtain the bandwidth, i.e. the frequency $\omega_c$ above which the spectrum of the result is zero. With that knowledge it's straightforward to apply $H(j\omega)$ if you know the time-shifting property of the Fourier transform.