Recall that the definition of a frequency response $H(\omega)$ is given by the summation (in DT):
$$ H(\omega) = \sum \limits_{n \in \Z}h(n)e^{-i\omega n}  $$

Notice that if we take the absolute value of the terms in the sum, we have
$$\sum \limits_{n \in \Z}\abs{h(n)e^{-i\omega n}} = \sum \limits_{n \in \Z}\abs{h(n)}  $$
Therefore, if $\sum \abs{h(n)}$ is divergent, then the frequency response has issues.
\begin{enumerate}
\item $\sum \abs{h(n)} \lt \infty \Rightarrow H(\omega) $ is nice and smooth.
\item $\sum \abs{h(n)} = \infty \wedge \sum \abs{h(n)}^2 \lt \infty \Rightarrow H(\omega) $ is bounded but will have discontinuities. Note that $\sum \abs{h(n)}^2$ represents the energy of the system, thus $\sum \abs{h(n)}^2 \lt \infty$ means that the system has finite energy.
\item $\sum \abs{h(n)} = \infty \wedge \sum \abs{h(n)}^2 = \infty \Rightarrow$ "all bets are off". Either $H(\omega)$ has impulses, or doesn't exist. The system may not have a frequency response.
\end{enumerate}

Let $BIBO$ be the statement that the system produces a bounded output for every bounded input.

Let $FE$ be the statement that the system has finite energy ($\sum \abs{h(n)}^2 \lt \infty$)

Let $F$ be the statement that the system has a frequency response.


Then we can sum up the above statements regarding frequency response:
\begin{align*}
BIBO &\Rightarrow F \\
\lnot BIBO \wedge FE &\Rightarrow F \\
\lnot BIBO \wedge \lnot FE &\Rightarrow F \vee \lnot F \\
\end{align*}


Implications of BIBO for Frequency Response