I have a complex-valued function defined as $$\psi(z) = \sum_{j\in\Bbb Z} \psi_jz^j$$
We of course know that $\sum_j\lvert\psi_j\rvert < \infty$ implies $\psi(z)$ is well-defined (finite) on the unit circle.
What I want to know is whether the converse is true. In other words, if $\psi(z)$ is finite for any $z$ on the unit circle, does it mean that $\sum_j\lvert\psi_j\rvert < \infty$? If yes, can someone post a proof of this?
As pointed out by @user1952009 this is not true. Now I strengthen the argument a bit. Suppose $\psi(z)$ is analytic on an annulus containing the unit circle. How do I then show the absolute summability above?