View source for Almost-periodic analytic function

An analytic function $f(s)$,<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011960/a0119602.png" />, regular in a strip <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011960/a0119603.png" />, and expandable into a 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/a/a011/a011960/a0119604.png" /></td> </tr></table>

where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011960/a0119605.png" /> are complex and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011960/a0119606.png" /> are real numbers. A real number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011960/a0119607.png" /> is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011960/a0119609.png" />-almost-period of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011960/a01196010.png" /> if for all points of the strip <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011960/a01196011.png" /> the inequality

<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/a/a011/a011960/a01196012.png" /></td> </tr></table>

holds. An almost-periodic analytic function is an analytic function that is regular in a strip <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011960/a01196013.png" /> and possesses a relatively-dense set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011960/a01196015.png" />-almost-periods for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011960/a01196016.png" />. An almost-periodic analytic function on a closed strip <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011960/a01196017.png" /> is defined similarly. An almost-periodic analytic function on a strip <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011960/a01196018.png" /> is a uniformly almost-periodic function of the real variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011960/a01196019.png" /> on every straight line in the strip and it is bounded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011960/a01196020.png" />, i.e. on any interior strip. If a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011960/a01196021.png" />, regular in a strip <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011960/a01196022.png" />, is a uniformly almost-periodic function on at least one line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011960/a01196023.png" /> in the strip, then boundedness of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011960/a01196024.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011960/a01196025.png" /> implies its almost-periodicity on the entire strip <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011960/a01196026.png" />. Consequently, the theory of almost-periodic analytic functions turns out to be a theory analogous to that of almost-periodic functions of a real variable (cf. [[Almost-periodic function|almost-periodic function]]). Therefore, many important results of the latter theory can be easily carried over to almost-periodic analytic functions: the uniqueness theorem, Parseval's equality, rules of operation with Dirichlet series, the approximation theorem, and several other theorems.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Bohr,   "Almost-periodic functions" , Chelsea, reprint  (1947)  (Translated from German)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.M. Levitan,   "Almost-periodic functions" , Moscow  (1953)  pp. Chapt. 7  (In Russian)</TD></TR></table>



====Comments====
The hyphen between almost and periodic is sometimes dropped.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Corduneanu,   "Almost periodic functions" , Interscience  (1961)  pp. Chapt. 3</TD></TR></table>