View source for Conditionally-periodic function

A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024540/c0245401.png" /> which is the composition of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024540/c0245402.png" />-periodic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024540/c0245403.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024540/c0245404.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024540/c0245405.png" />-dimensional torus, and a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024540/c0245406.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024540/c0245407.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024540/c0245408.png" /> 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

<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/c/c024/c024540/c0245409.png" /></td> </tr></table>

where

<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/c/c024/c024540/c02454010.png" /></td> </tr></table>

<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/c/c024/c024540/c02454011.png" /></td> </tr></table>

If a conditionally-periodic function is continuous, then it coincides with a [[Quasi-periodic function|quasi-periodic function]] with periods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024540/c02454012.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.I. Arnol'd,   "Chapitres supplémentaires de la théorie des équations différentielles ordinaires" , MIR  (1980)  (Translated from Russian)</TD></TR></table>



====Comments====
A conditionally-periodic function is almost periodic, cf. [[Almost-periodic function|Almost-periodic function]].