Convolution in frequency domain, what is the output bandwidth
Following a time-domain convolution analysis which relates the output signal's domain of support to the input signals' domain of supports, we could reach a similar conclusion:

Given two frequency domain signals with domain of supports (for positive frequencies) such that $X(\omega) \neq 0$ for $\omega_1 &lt; \omega &lt; \omega_2$ and $H(\omega) \neq 0$ for $\omega_3 &lt; \omega &lt; \omega_4$, then their convolution $Y(\omega) = X(\omega) \star H(\omega)$ will have the domain of support as $Y(\omega) \neq 0$ for $$ \omega_1 + \omega_3 &lt; \omega &lt; \omega_2 + \omega_4 $$

Therefore the bandwidth of $Y(\omega)$ is related to the bandwidths of $X(\omega)$ and $H(\omega)$ as:

$$ BW_Y =  (\omega_2 + \omega_4) - (\omega_1 + \omega_3) $$
$$ BW_Y =  (\omega_4 - \omega_3) + (\omega_2 - \omega_1) $$
$$ BW_Y =  BW_H + BW_X $$

This assumes that the domains were compact and finite. If they are either noncompact or non-finite the result should be modified accordingly.