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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 $t\in \mathbb{T}$, $z\in \mathbb{C}$.

Any help is much appreciated.

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  • $\begingroup$ If $f$ is almost periodic and $f(tz) = f(z)$ then the Fourier--Bohr coefficients are invariant under rotations. But the Fourier--Bohr coefficients are zero outside a countable set, therefore only the FB coefficient at 0 can be non-zero, i.e. $f$ is constant. $\endgroup$
    – Nick S
    Commented Aug 26, 2020 at 4:17

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Yes to the first question: the almost periodic functions on any LCA group are the uniform limits of linear combinations of characters. In the case of $\mathbb{R}^2$ these are the functions $e^{i(ax + by)}$.

I don't see how that immediately answers your second question, but there is an easy negative answer straight from the definition. Suppose $f$ is a rotationally invariant nonconstant almost periodic function. WLOG $f(0,0) =0$ and $f(1,0) = 1$. So $f$ is constantly $1$ on the unit circle. Now find $t > 1$ such that $f$ and its shift by $(t, 0)$ are uniformly at most $1/3$ apart. Then $f(t,0)$ is within $1/3$ of $0$, so the same must be true at any point on the circle of radius $t$ about the origin. But at the same time, $f$ must be within $1/3$ of $1$ on the circle of radius $1$ about $(t,0)$, and that is contradictory.

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