Z transform and non-$\ell^{1}$ causal sequences

Question

Consider a complex sequence $h[n]$, such that:

• if $n < 0 \Rightarrow h[n] = 0 $
• $\sum_{n=0}^{+\infty} \left|h[n]\right|$ diverges
• $H(z) = \sum_{n=0}^{+\infty} h[n]z^{-n}$ converges uniformly $\forall z \in \mathbb{C}: |z| = 1$.

My questions are:

1. Does it exist? (If not: why?)
2. If yes, which is an example or a set of such a sequence?
3. Can I state that the lower bound of the region of convergence of $H(z)$ is $1$?

Answer 1

It is a fundamental fact that every Laurent series converges uniformly and absolutely in its region of convergence.

If $H(z)$ admits such a Laurent series expansion for $|z|=1$, then $$\sum^\infty_{n=0}|h[n]z^{-n}|= \sum^\infty_{n=0}|h[n] |$$ necessarily converges, which leads to a contradiction.