# almost periodic function (classical definition)

A continuous function $f\colon\mathbb{R}\to\mathbb{R}$ is said to be almost periodic if, for every $\epsilon>0$, there exists an a number $L_{\epsilon}>0$ such that for every interval $I$ of length $L_{\epsilon}$ there exists a number $\omega_{I}\in I$ such that

 $|f(x+\omega_{I})-f(x)|<\epsilon$

whenever $x\in\mathbb{R}$.

Intuition: we want the function to have an ”approximate period”. However, it is easy to write too weak condition. First, we want uniform estimate in $x$. If we allow $\omega$ to be small than the condition degenerates to uniform continuity. If we require a single $\omega$, than the condition still is too weak (it allows pretty wide changes). For periodic function every multiple of a period is still a period. So, if the length of an interval is longer than the period, then the interval contains a period. The definition of almost periodic functions mimics the above property of periodic functions: every sufficiently long interval should contain an approximate period.

It is possible to generalize this notion. The range of the function can be taken to be a normed vector space — in the first definition, we merely need to replace the absolute value with the norm:

 $\|f(x+\omega)-f(x)\|<\epsilon$

In the second definition, interpret uniform convergence as uniform convergence with respect to the norm. A common case of this is the case where the range is the complex numbers. It is worth noting that if the vector space is finite dimensional, a function is almost periodic if and only if each of its components with respect to a basis is almost periodic.

Also the domain may be taken to be a group $G$. A function is called almost periodic iff set of its translates is pre-compact (compact after completion). Equivalently, a continuous function $f$ on a topological group $G$ is almost periodic iff there is a compact group $K$, a continuous function $g$ on $K$ and a (continuous) homomorphism $h$ form $G$ to $K$ such that $f$ is the composition of $g$ and $h$. The classical case described above arises when the group is the additive group of the real number field. Almost periodic functions with respect to groups play a role in the representation theory of non-compact Lie algebras. (In the compact case, they are trivial — all continuous functions are almost periodic.)

The notion of an almost periodic function should not be confused with the notion of quasiperiodic function.

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