Almost periodic function

Remarks

Equivalently, every sequence of translates $f_{a_1}, f_{a_2}, \dots$ has a subsequence that converges uniformly on $G$. This is Bochner's form of the definition. For $G = \R$ it is equivalent to Bohr's definition through relatively dense sets of almost periods, see Bohr almost periodic function. For a locally compact abelian group $G$, the almost periodic functions are exactly the functions $F \circ b$, where $b \colon G \to bG$ is the Bohr compactification and $F$ is continuous on $bG$.

Example: on $\R$, let $f(x) = \cos x + \cos(\sqrt{2} x)$. The translates of $\cos$ are $\cos(x + a) = \cos a \cos x - \sin a \sin x$, the image of the circle $\set{(\cos a, \sin a)}$ under a continuous map into $C_b(\R)$, so they form a compact set. The same holds for $\cos(\sqrt{2} x)$, and the translates of $f$ lie in the sum of these two compact sets, which is compact. So $f$ is almost periodic.

Non-example: $f(x) = \arctan x$ on $\R$ is bounded and continuous, but not almost periodic. The translates $f_n$, $n = 1, 2, \dots$, converge pointwise to the constant $\pi/2$. A uniformly convergent subsequence would have the same limit, but $\abs{f_n(-n) - \pi/2} = \pi/2$ for every $n$.