I recently came across this post stating that the continuous ideal LPF is BIBO-unstable since the impulse response is not absolutely integrable, and this post stating some examples.
I have been trying to understand this using the frequency response. For example, take a system with bounded input $x(t) = \sin(2\pi B t)u(t)$ where $u(t)$ indicates the Heaviside unit step function and the impulse response of the ideal LPF $h(t) = \frac{\sin(2\pi B t)}{2\pi B t}$. If we find the convolution of these two functions $y(t) = x(t) * h(t)$ for $t = 0$, we would find that the integral diverges: the output is unbounded due to $h(t)$ not being absolutely summable. If we would use frequency-domain multiplication instead of time-domain convolution for this system, would we find the same result? And maybe more importantly: does the convolution property of the Fourier Transform always hold if the Fourier Transforms of the two signals exist?
Thanks in advance for any help, this has been bugging me for the last couple of days. I feel that there may be something larger to this problem which I don't quite grasp yet.