Conditionally-periodic function

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.