The almost periodic functions on the real line can be characterized as uniform limits of trigonometric functions. I was wondering whether a similar definition exists on the complex plane (a locally compact group under addition).
In particular, I am trying to figure out if there exists a non-constant almost periodic function $f$ on $\mathbb{C}$ such that $f$ is invariant under rotations i.e. f(tz) = f(z)$ for all $t\in \mathbb{T}$, $z\in \mathbb{C}$$f(tz) = f(z)$ for all $t\in \mathbb{T}$, $z\in \mathbb{C}$.
Any help is much appreciated.
