The Dirchlet conditions state that if the signal is absolutely summable then it the DTFT of the signal definitely exists. This is a sufficient condition but not necessary condition.
There are systems like Ideal Low Pass Filter, which are not absolutely summable. The condition of absolutely summability is a necessary and sufficient condition for stability. Therefore, I can find a bounded input for which the system produces unbounded input.
For such systems, which are not absolutely summable, but are absolutely square summable, we define the DTFT in the same manner and argue that even though non-zero error exists, the energy of the error is 0, which essentially gives rise to Gibbs Phenomenon.
But the condition of $y[n] = x[n] * h[n]$ is independent of stability, and simply depends on Linear and shift invariance property. From this definition, I can find a bounded input which will produce unbounded output. But the relation, $Y(w) = X(w)H(w)$ is also independent of stability. Even if I take into account Gibbs Phenomenon, I am unable to deduce the existence of a bounded input which will produce and unbounded output, and this essentially suggests the fact that bounded input will give bounded output.
Where am I going wrong?