View source for Bohl almost-periodic functions

A class of functions the typical property of which is that they can be uniformly approximated on the whole real axis by generalized trigonometric polynomials of the form

<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/b/b016/b016760/b0167601.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016760/b0167602.png" /> are arbitrary integers, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016760/b0167603.png" /> are given real numbers. This class of functions contains the class of continuous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016760/b0167604.png" />-periodic functions and is contained in the class of [[Bohr almost-periodic functions|Bohr almost-periodic functions]]. P. Bohl specified several necessary and sufficient conditions for a function to be almost-periodic. In particular, any function of the type

<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/b/b016/b016760/b0167605.png" /></td> </tr></table>

where each one of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016760/b0167606.png" /> is continuous and periodic (with possibly different periods), is a Bohl almost-periodic function.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P. Bohl,   "Über die Darstellung von Funktionen einer Variabeln durch trigonometrische Reihen mit mehreren einer Variabeln proportionalen Argumenten" , Dorpat  (1893)  (Thesis)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P. Bohl,   "Ueber eine Differentialgleichung der Störungstheorie"  ''J. Reine Angew. Math.'' , '''131'''  (1906)  pp. 268–321</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B.M. Levitan,   "Almost-periodic functions" , Moscow  (1953)  (In Russian)</TD></TR></table>



====Comments====
A very well-known reference for this kind of topic is [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Bohr,   "Almost periodic functions" , Chelsea, reprint  (1947)  (Translated from German)</TD></TR></table>