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 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:

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.

Title	almost periodic function (classical definition)
Canonical name	AlmostPeriodicFunctionclassicalDefinition
Date of creation	2013-03-22 14:53:14
Last modified on	2013-03-22 14:53:14
Owner	drini (3)
Last modified by	drini (3)
Numerical id	13
Author	drini (3)
Entry type	Definition
Classification	msc 42A75
Synonym	almost periodic function
Related topic	ExampleOfNonSeperableHilbertSpace
Defines	almost periodic