View source for Conditionally-periodic function

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A function  $  A \circ \phi $
which is the composition of a  $  2 \pi $-
periodic function  $  A:  T ^ { n } \rightarrow \mathbf C $, 
where  $  T ^ { n } $
is the  $  n $-
dimensional torus, and a function  $  \phi :  \mathbf R \rightarrow \mathbf R  ^ {n} $
such that  $  \dot \phi  = \omega $, 
where  $  \omega = ( \omega _ {1} \dots \omega _ {n} ) $
is a constant vector whose components are linearly independent over the rational numbers. Examples of conditionally-periodic functions are given by partial sums of Fourier series

$$ 
\sum _ {i = 1 } ^ { n } 
[ A _ {i}  \sin ( \omega _ {i} t + \psi _ {i} ) +
B _ {i}  \cos ( \omega _ {i} t + \psi _ {i} )],
$$

where

$$ 
A  =  A ( \phi _ {1} \dots \phi _ {n} )  = \ 
\sum _ {i = 1 } ^ { n } 
[ A _ {i}  \sin  \phi _ {i} +
B _ {i}  \cos  \phi _ {i} ],
$$

$$ 
\phi  =  ( \phi _ {1} ( t) \dots \phi _ {n} ( t))  =  ( \omega _ {1} t + \psi _ {1} \dots \omega _ {n} t + \psi _ {n} ).
$$

If a conditionally-periodic function is continuous, then it coincides with a [[Quasi-periodic function|quasi-periodic function]] with periods  $  \omega _ {1} \dots \omega _ {n} $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.I. Arnol'd,   "Chapitres supplémentaires de la théorie des équations différentielles ordinaires" , MIR  (1980)  (Translated from Russian)</TD></TR></table>

====Comments====
A conditionally-periodic function is almost periodic, cf. [[Almost-periodic function|Almost-periodic function]].