almost periodic function (classical definition)

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

whenever $x\in ℝ$.

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

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 $G$ isalmost periodic iff there is a compact group $K$, a continuousfunction $g$ 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.