almost periodic function (classical definition)
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 to be small than the condition degenerates to uniform continuity. If we require a single , 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.
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.)
The notion of an almost periodic function should not be confused with the notion of quasiperiodic function.
|Title||almost periodic function (classical definition)|
|Date of creation||2013-03-22 14:53:14|
|Last modified on||2013-03-22 14:53:14|
|Last modified by||drini (3)|
|Synonym||almost periodic function|