View source for Fourier coefficients of an almost-periodic function

The coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041020/f0410201.png" /> of the Fourier series (cf. [[Fourier series of an almost-periodic function|Fourier series of an almost-periodic function]]) corresponding to the given almost-periodic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041020/f0410202.png" />:

<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/f/f041/f041020/f0410203.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/f/f041/f041020/f0410204.png" /></td> </tr></table>

The coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041020/f0410205.png" /> are completely determined by the theorem on the existence of the mean value

<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/f/f041/f041020/f0410206.png" /></td> </tr></table>

which is non-zero only for the countable set of values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041020/f0410207.png" />.



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.S. Besicovitch,   "Almost periodic functions" , Cambridge Univ. Press  (1932)  pp. Chapt. I</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N. Wiener,   "The Fourier integral and certain of its applications" , Dover, reprint  (1933)  pp. Chapt. II</TD></TR></table>