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)

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