View source for Dirichlet series for an analytic almost-periodic function

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A series of the type

$$ \tag{* }
f ( s)  \sim  \sum _ { n } A _ {n} e ^ {\Lambda _ {n} \tau }
e ^ {i \Lambda _ {n} t }  = \ 
\sum _ { n } A _ {n} e ^ {\Lambda _ {n} s } ,\  \alpha < \tau < \beta ,
$$

representing in the strip  $  ( \alpha , \beta ) $, 
$  - \infty \leq  \alpha < \beta \leq  + \infty $, 
the complete Fourier series of the analytic, regular almost-periodic function  $  f ( s) = f ( \tau + it ) $, 
defined on the union of straight lines  $   \mathop{\rm Re} ( s) = \tau $(
cf. [[Almost-periodic analytic function|Almost-periodic analytic function]]).

To two different almost-periodic functions in the same strip correspond two different Dirichlet series. In the case of a  $  2 \pi $-
periodic function the series (*) becomes a Laurent series. The numbers  $  A _ {n} $
and  $  \Lambda _ {n} $
are known, respectively, as the Dirichlet coefficients and exponents. Unlike for classical Dirichlet series, the set of real exponents  $  \Lambda _ {n} $
in (*) may have finite limit points and may even be everywhere dense. If all Dirichlet exponents have the same sign, for example, if  $  f ( s) $
is an almost-periodic function in a strip  $  ( \alpha , \beta ) $
and if in (*)  $  \Lambda _ {n} < 0 $, 
then  $  f ( s) $
is an almost-periodic function in the strip  $  ( \alpha , + \infty ) $, 
and  $  \lim\limits _ {\tau \rightarrow + \infty }   f ( s) = 0 $
uniformly with respect to  $  t $. 
A similar theorem is valid for positive Dirichlet exponents [[#References|[2]]]. If  $  f ( s) $
is an almost-periodic function in a strip  $  [ \alpha , \beta ] $
and if the indefinite integral of  $  f ( s) $
in the strip  $  [ \alpha , \beta ] $
is bounded, then the series

$$ 
\sum _ {\Lambda _ {n} < 0 } A _ {n} e ^ {\Lambda _ {n} s } ,\ \ 
\sum _ {\Lambda _ {n} \geq  0 } A _ {n} e ^ {\Lambda _ {n} s }
$$

are the Dirichlet series of two functions  $  f _ {1} ( s) $
and  $  f _ {2} ( s) $
which are almost-periodic in every strip  $  [ \alpha _ {1} , + \infty ) $, 
$  \alpha _ {1} > \alpha $
or, respectively,  $  ( - \infty , \beta _ {1} ] $, 
$  \beta _ {1} < \beta $.

====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)  (In Russian)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.S. Besicovitch,   "Almost periodic functions" , Cambridge Univ. Press  (1932)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C. Corduneanu,   "Almost periodic functions" , Wiley  (1968)  (Translated from Rumanian)</TD></TR></table>