# almost periodic function (classical definition)

A continuous function^{} $f:\mathbb{R}\to \mathbb{R}$ is said to be *almost periodic* if, for every $\u03f5>0$, there exists an a number ${L}_{\u03f5}>0$ such that for every interval $I$ of length ${L}_{\u03f5}$ there exists a number ${\omega}_{I}\in I$ such that

$$ |

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:

$$ |

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 |