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A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024540/c0245401.png" /> which is the composition of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024540/c0245402.png" />-periodic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024540/c0245403.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024540/c0245404.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024540/c0245405.png" />-dimensional torus, and a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024540/c0245406.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024540/c0245407.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024540/c0245408.png" /> 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
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024540/c0245409.png" /></td> </tr></table>
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A function  $  A \circ \phi $
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which is the composition of a  $  2 \pi $-
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periodic function  $  A: T ^ { n } \rightarrow \mathbf C $,
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where  $  T ^ { n } $
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is the  $  n $-
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dimensional torus, and a function  $  \phi : \mathbf R \rightarrow \mathbf R  ^ {n} $
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such that  $  \dot \phi  = \omega $,
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where  $  \omega = ( \omega _ {1} \dots \omega _ {n} ) $
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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
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$$
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\sum _ {i = 1 } ^ { n }
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[ A _ {i}  \sin ( \omega _ {i} t + \psi _ {i} ) +
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B _ {i}  \cos ( \omega _ {i} t + \psi _ {i} )],
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$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024540/c02454010.png" /></td> </tr></table>
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$$
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= A ( \phi _ {1} \dots \phi _ {n} )  = \
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\sum _ {i = 1 } ^ { n }
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[ A _ {i}  \sin  \phi _ {i} +
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B _ {i}  \cos  \phi _ {i} ],
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024540/c02454011.png" /></td> </tr></table>
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$$
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\phi  = ( \phi _ {1} ( t) \dots \phi _ {n} ( t))  = ( \omega _ {1} t + \psi _ {1} \dots \omega _ {n} t + \psi _ {n} ).
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$$
  
If a conditionally-periodic function is continuous, then it coincides with a [[Quasi-periodic function|quasi-periodic function]] with periods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024540/c02454012.png" />.
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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====
 
====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>
 
<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====
 
====Comments====
 
A conditionally-periodic function is almost periodic, cf. [[Almost-periodic function|Almost-periodic function]].
 
A conditionally-periodic function is almost periodic, cf. [[Almost-periodic function|Almost-periodic function]].

Latest revision as of 17:46, 4 June 2020


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 with periods $ \omega _ {1} \dots \omega _ {n} $.

References

[1] V.I. Arnol'd, "Chapitres supplémentaires de la théorie des équations différentielles ordinaires" , MIR (1980) (Translated from Russian)

Comments

A conditionally-periodic function is almost periodic, cf. Almost-periodic function.

How to Cite This Entry:
Conditionally-periodic function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conditionally-periodic_function&oldid=11212
This article was adapted from an original article by Yu.V. Komlenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article