PlanetMath: almost periodic function (classical definition)

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almost periodic function (classical definition)

(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 of length $L_\epsilon$ there exists a number $\omega_I \in I$ such that

$\displaystyle \vert f(x + \omega_I) - f(x) \vert < \epsilon$

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 . 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:

$\displaystyle \Vert f(x + \omega) - f(x) \Vert < \epsilon$

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 . A function is called almost periodic iff set of its translates is pre-compact (compact after completion). Equivalently, a continuous function on a topological group is almost periodic iff there is a compact group , a continuous function on and a (continuous) homomorphism form to such that is the composition of and . 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.)