View source for Quasi-periodic function

''with periods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076630/q0766301.png" />''

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.

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.

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>

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

<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>

====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>



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

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>

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

<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>

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.

====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>