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''with periods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076630/q0766301.png" />''
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A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076630/q0766302.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076630/q0766303.png" /> for some continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076630/q0766304.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076630/q0766305.png" /> variables that is periodic with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076630/q0766306.png" /> with periods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076630/q0766307.png" />, respectively. All the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076630/q0766308.png" /> are required to be strictly positive and their reciprocals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076630/q0766309.png" /> have to be rationally linearly independent. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076630/q07663010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076630/q07663011.png" /> are continuous periodic functions with periods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076630/q07663012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076630/q07663013.png" />, respectively, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076630/q07663014.png" /> is irrational, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076630/q07663015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076630/q07663016.png" /> are quasi-periodic functions.
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''with periods  $  \omega _ {1} \dots \omega _ {n} $''
 +
 
 +
A function $  f $
 +
such that $  f ( t) = F ( t \dots t ) $
 +
for some continuous function $  F ( t _ {1} \dots t _ {n} ) $
 +
of $  n $
 +
variables that is periodic with respect to $  t _ {1} \dots t _ {n} $
 +
with periods $  \omega _ {1} \dots \omega _ {n} $,  
 +
respectively. All the $  \omega _ {1} \dots \omega _ {n} $
 +
are required to be strictly positive and their reciprocals $  p _ {1} \dots p _ {n} $
 +
have to be rationally linearly independent. If $  f _ {1} $
 +
and $  f _ {2} $
 +
are continuous periodic functions with periods $  \omega _ {1} $
 +
and $  \omega _ {2} $,  
 +
respectively, where $  \omega _ {1} / \omega _ {2} $
 +
is irrational, then $  g = f _ {1} + f _ {2} $
 +
and $  h = \max \{ f _ {1} , f _ {2} \} $
 +
are quasi-periodic functions.
  
 
The theory of quasi-periodic functions serves as a basis for the creation of the theory of almost-periodic functions (cf. [[Almost-periodic function|Almost-periodic function]]). In the case of continuous functions, quasi-periodic functions are a generalization of periodic functions, but a particular case of almost-periodic functions.
 
The theory of quasi-periodic functions serves as a basis for the creation of the theory of almost-periodic functions (cf. [[Almost-periodic function|Almost-periodic function]]). In the case of continuous functions, quasi-periodic functions are a generalization of periodic functions, but a particular case of almost-periodic functions.
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Quasi-periodic functions have a representation
 
Quasi-periodic functions have a representation
  
<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/q/q076/q076630/q07663017.png" /></td> </tr></table>
+
$$
 +
f ( t)  = \
 +
\sum c _ {k _ {1}  \dots k _ {n} }
 +
e ^ {i ( k _ {1} p _ {1} + \dots + k _ {n} p _ {n} ) t } ,
 +
$$
  
where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076630/q07663018.png" /> are such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076630/q07663019.png" />. Quasi-periodic functions possess the following properties: addition and multiplication of quasi-periodic functions yield quasi-periodic functions; a sequence of quasi-periodic functions that is uniformly convergent for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076630/q07663020.png" /> gives in the limit an almost-periodic function; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076630/q07663021.png" /> is an almost-periodic function and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076630/q07663022.png" />, then there exists a quasi-periodic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076630/q07663023.png" /> such that
+
where the $  c _ {k _ {1}  \dots k _ {n} } = c _ {k} $
 +
are such that $  \sum | c _ {k} |  ^ {2} < \infty $.  
 +
Quasi-periodic functions possess the following properties: addition and multiplication of quasi-periodic functions yield quasi-periodic functions; a sequence of quasi-periodic functions that is uniformly convergent for $  t \in \mathbf R $
 +
gives in the limit an almost-periodic function; if $  g $
 +
is an almost-periodic function and $  \epsilon > 0 $,  
 +
then there exists a quasi-periodic function $  f $
 +
such that
  
<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/q/q076/q076630/q07663024.png" /></td> </tr></table>
+
$$
 +
| f ( t) - g ( t) |  < \epsilon \  \textrm{ for }  t \in \mathbf R .
 +
$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P. Bohl,  "Über die Darstellung von Funktionen einer Variabeln durch trigonometrische Reihen mit mehreren einer Variabeln proportionalen Argumenten" , Dorpat  (1893)  (Thesis)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.Kh. Kharasakhal,  "Almost-periodic solutions of ordinary differential equations" , Alma-Ata  (1970)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P. Bohl,  "Über die Darstellung von Funktionen einer Variabeln durch trigonometrische Reihen mit mehreren einer Variabeln proportionalen Argumenten" , Dorpat  (1893)  (Thesis)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.Kh. Kharasakhal,  "Almost-periodic solutions of ordinary differential equations" , Alma-Ata  (1970)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
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Consider Hill's differential equation
 
Consider Hill's differential equation
  
<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/q/q076/q076630/q07663025.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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$$ \tag{a1 }
 +
 
 +
\frac{d  ^ {2} u }{dx  ^ {2} }
 +
+
 +
F( x) u  = 0
 +
$$
 +
 
 +
with periodic  $  F $,
 +
$  F( x+ 2 \pi )= F( x) $.
 +
A particular case is Mathieu's differential equation
  
with periodic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076630/q07663026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076630/q07663027.png" />. A particular case is Mathieu's differential equation
+
$$ \tag{a2 }
  
<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/q/q076/q076630/q07663028.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
\frac{d  ^ {2} u }{dx  ^ {2} }
 +
= ( a- 2q  \cos  2z) u  = 0 .
 +
$$
  
A solution of (a1) need not be periodic. However, there is always a particular solution of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076630/q07663029.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076630/q07663030.png" /> periodic (Floquet's theorem; cf. [[#References|[a1]]] for a more precise statement). If the characteristic exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076630/q07663031.png" /> is real, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076630/q07663032.png" /> is a quasi-periodic function.
+
A solution of (a1) need not be periodic. However, there is always a particular solution of the form $  u( x) = e ^ {i \mu x } \phi ( x) $
 +
with $  \phi ( x) $
 +
periodic (Floquet's theorem; cf. [[#References|[a1]]] for a more precise statement). If the characteristic exponent $  \mu $
 +
is real, $  u( x) $
 +
is a quasi-periodic function.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.I. Arnol'd,  "Mathematical methods of classical mechanics" , Springer  (1978)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.G. Bohl,  "Ueber eine Differentialgleichung der Störungstheorie"  ''Crelles J.'' , '''131'''  (1906)  pp. 268–321</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  B.M. Levitan,  V.V. Zhikov,  "Almost periodic functions and differential equations" , Cambridge Univ. Press  (1984)  pp. 47–48  (Translated from Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  W. Magnus,  S. Winkler,  "Hill's equation" , Dover, reprint  (1979)  pp. 4ff</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.I. Arnol'd,  "Mathematical methods of classical mechanics" , Springer  (1978)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.G. Bohl,  "Ueber eine Differentialgleichung der Störungstheorie"  ''Crelles J.'' , '''131'''  (1906)  pp. 268–321</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  B.M. Levitan,  V.V. Zhikov,  "Almost periodic functions and differential equations" , Cambridge Univ. Press  (1984)  pp. 47–48  (Translated from Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  W. Magnus,  S. Winkler,  "Hill's equation" , Dover, reprint  (1979)  pp. 4ff</TD></TR></table>

Latest revision as of 08:09, 6 June 2020


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

A function $ f $ such that $ f ( t) = F ( t \dots t ) $ for some continuous function $ F ( t _ {1} \dots t _ {n} ) $ of $ n $ variables that is periodic with respect to $ t _ {1} \dots t _ {n} $ with periods $ \omega _ {1} \dots \omega _ {n} $, respectively. All the $ \omega _ {1} \dots \omega _ {n} $ are required to be strictly positive and their reciprocals $ p _ {1} \dots p _ {n} $ have to be rationally linearly independent. If $ f _ {1} $ and $ f _ {2} $ are continuous periodic functions with periods $ \omega _ {1} $ and $ \omega _ {2} $, respectively, where $ \omega _ {1} / \omega _ {2} $ is irrational, then $ g = f _ {1} + f _ {2} $ and $ h = \max \{ f _ {1} , f _ {2} \} $ are quasi-periodic functions.

The theory of quasi-periodic functions serves as a basis for the creation of the theory of almost-periodic functions (cf. Almost-periodic function). In the case of continuous functions, quasi-periodic functions are a generalization of periodic functions, but a particular case of almost-periodic functions.

Quasi-periodic functions have a representation

$$ f ( t) = \ \sum c _ {k _ {1} \dots k _ {n} } e ^ {i ( k _ {1} p _ {1} + \dots + k _ {n} p _ {n} ) t } , $$

where the $ c _ {k _ {1} \dots k _ {n} } = c _ {k} $ are such that $ \sum | c _ {k} | ^ {2} < \infty $. Quasi-periodic functions possess the following properties: addition and multiplication of quasi-periodic functions yield quasi-periodic functions; a sequence of quasi-periodic functions that is uniformly convergent for $ t \in \mathbf R $ gives in the limit an almost-periodic function; if $ g $ is an almost-periodic function and $ \epsilon > 0 $, then there exists a quasi-periodic function $ f $ such that

$$ | f ( t) - g ( t) | < \epsilon \ \textrm{ for } t \in \mathbf R . $$

References

[1] P. Bohl, "Über die Darstellung von Funktionen einer Variabeln durch trigonometrische Reihen mit mehreren einer Variabeln proportionalen Argumenten" , Dorpat (1893) (Thesis)
[2] V.Kh. Kharasakhal, "Almost-periodic solutions of ordinary differential equations" , Alma-Ata (1970) (In Russian)

Comments

Quasi-periodic functions of time occur naturally in Hamiltonian mechanics to describe multi-periodic motions of integrable systems (see [a1] and Quasi-periodic motion).

Consider Hill's differential equation

$$ \tag{a1 } \frac{d ^ {2} u }{dx ^ {2} } + F( x) u = 0 $$

with periodic $ F $, $ F( x+ 2 \pi )= F( x) $. A particular case is Mathieu's differential equation

$$ \tag{a2 } \frac{d ^ {2} u }{dx ^ {2} } = ( a- 2q \cos 2z) u = 0 . $$

A solution of (a1) need not be periodic. However, there is always a particular solution of the form $ u( x) = e ^ {i \mu x } \phi ( x) $ with $ \phi ( x) $ periodic (Floquet's theorem; cf. [a1] for a more precise statement). If the characteristic exponent $ \mu $ is real, $ u( x) $ is a quasi-periodic function.

References

[a1] V.I. Arnol'd, "Mathematical methods of classical mechanics" , Springer (1978) (Translated from Russian)
[a2] P.G. Bohl, "Ueber eine Differentialgleichung der Störungstheorie" Crelles J. , 131 (1906) pp. 268–321
[a3] B.M. Levitan, V.V. Zhikov, "Almost periodic functions and differential equations" , Cambridge Univ. Press (1984) pp. 47–48 (Translated from Russian)
[a4] W. Magnus, S. Winkler, "Hill's equation" , Dover, reprint (1979) pp. 4ff
How to Cite This Entry:
Quasi-periodic function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-periodic_function&oldid=14617
This article was adapted from an original article by Yu.V. KomlenkoE.L. Tonkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article