A series of the type
representing in the strip
,
,
the complete Fourier series of the analytic, regular almost-periodic function
,
defined on the union of straight lines
(cf.
Almost-periodic analytic function).
To two different almost-periodic functions in the same strip correspond two different
Dirichlet series.
In the case of a
-periodic
function the series
(*)
becomes a
Laurent series.
The numbers
and
are known, respectively, as the
Dirichlet coefficients
and
exponents.
Unlike for classical Dirichlet series, the set of real exponents
in
(*)
may have finite limit points and may even be
everywhere dense. If all Dirichlet exponents have the same sign, for example, if
is an almost-periodic function in a strip
and if in
(*)
,
then
is an almost-periodic function in the strip
,
and
uniformly with respect to
.
A similar theorem is valid for positive Dirichlet exponents
[2].
If
is an almost-periodic function in a strip
and if the indefinite integral of
in the strip
is bounded, then the series
are the Dirichlet series of two functions
and
which are almost-periodic in every strip
,
or, respectively,
,
.