# Quasi-periodic function

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)

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=48391
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