|
|
|
Viewing Version
5
of
'almost periodic function'
|
[ view 'almost periodic function'
|
back to history
]
| Title of object: |
almost periodic function |
| Canonical Name: |
AlmostPeriodicFunction |
| Type: |
Definition |
| Created on: |
2004-12-12 22:29:00 |
| Modified on: |
2005-07-10 01:42:44 |
| Classification: |
msc:42A75 |
| Defines: |
almost periodic |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here |
Content:
A continuous function $f \colon \mathbb{R} \to \mathbb{R}$ is said to be \emph{almost periodic} if, for every $\epsilon > 0$, there exists an a number $L_\epsilon > 0$ such that for every interval $I$ of length $L_\epsilon$ there exists a number $\omega_I \in I$ such that
$$| f(x + \omega_I) - f(x) | < \epsilon$$
whenever $x \in \mathbb{R}$.
Also, there is an equivalent definition: A function $f \colon \mathbb{R} \to \mathbb{R}$ is almost periodic if every sequence of translates of $f$ has a uniformly convergent subsequence.
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:
$$\| f(x + \omega) - f(x) \| < \epsilon$$
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 $G$. 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. |
|
|
|
|
|
