Support of the convolution of two rectangular signals
We have rectangular pulses

$$H_1 (\omega) = u \left(\omega + \frac{\pi}{5}\right) - u \left(\omega - \frac{\pi}{5}\right)$$

and

$$H_2 (\omega) = u \left(\omega + \frac{\pi}{10}\right) - u \left(\omega - \frac{\pi}{10}\right)$$

and would like to convolve them. When convolving piecewise constant functions, a useful "trick" is to differentiate, convolve and then integrate. Differentiating $H_1$,

$$H_1 ' (\omega) = \delta \left(\omega + \frac{\pi}{5}\right) - \delta \left(\omega - \frac{\pi}{5}\right)$$

and, convolving with $H_2$,

$$(H_1' * H_2) (\omega) = H_2 \left(\omega + \frac{\pi}{5}\right) - H_2 \left(\omega - \frac{\pi}{5}\right) = \begin{cases} \,~~ 1, &amp; \text{if } \omega \in \left[ -\frac{3 \pi}{10}, -\frac{\pi}{10} \right]\\ -1, &amp; \text{if } \omega \in \left[ \frac{\pi}{10}, \frac{3\pi}{10} \right]\\ \,~~0, &amp; \text{otherwise}\end{cases}$$

Finally, integrating,

$$(H_1 * H_2) (\omega) = \begin{cases} \, \frac{3 \pi}{10} + \omega, &amp; \text{if } \omega \in \left[ -\frac{3 \pi}{10}, -\frac{\pi}{10} \right]\\ \qquad\frac{\pi}{5}, &amp; \text{if } \omega \in \left[ -\frac{\pi}{10}, \frac{\pi}{10} \right]\\ \, \frac{3 \pi}{10} - \omega, &amp; \text{if } \omega \in \left[ \frac{\pi}{10}, \frac{3\pi}{10} \right]\\ \,\, \qquad 0, &amp; \text{otherwise}\end{cases}$$