This is not exactly a harmonic analysis problem but, I hope,

it is OK to ask somewhat "off-topic" questions here now and then.

Assume that $f_n$ ($-\infty<n<+\infty$)is a sequence of analytic

in the unit disk and continuous up to the boundary functions.

Suppose that the sequence $f_n(z)$ is almost periodic for every

$z$ on the unit circumference. Does it imply that the sequence

$f_n(0)$ is almost periodic as well?

Fedja.