Frequency response of marginally stable LTI systems The frequency response of a system is defined as:
$$\int_0^\infty{h(t)e^{-j\omega t}dt}$$
where $h(t)$ is the impulse response. But in marginally stable systems, $h(t)$ does not decay so the integral doesn't converge. Does the frequency response "exist" in this case?
As long as the system is linear and time-invariant, it can be described using the frequency response / transfer function $H(j\omega)$. For marginally stable systems, $H(j\omega)$ does exist, but cannot be expressed in terms of the Fourier-transformed impulse response $h(t)$.

The Form $H(j\omega) = \frac{Y(j\omega)}{X(j\omega)}$ however is still valid for systems not stable in the more narrow sense simply because they are both linear and time-invariant.