# Functional analysis. What are the almost periodic functions in the complex plane?

The almost periodic functions in the real line can be characterized as uniform limits of trigonometric functions. I was wondering if there is a similar definition in the complex plane (a locally compact group under addition).

In particular, I'm trying to find out if there is a quasi-periodic, non-constant function $$f$$ in $$mathbb {C}$$ such that $$f$$ it is invariant under rotations that is, $$f (tz) = f (z)$$ for all $$t in mathbb {T}$$, $$z in mathbb {C}$$.

Any help is really appreciated.