Dirichlet series for an analytic almost-periodic function

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).

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 [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

[1]	H. Bohr, "Almost periodic functions" , Chelsea, reprint (1947) (Translated from German)
[2]	B.M. Levitan, "Almost-periodic functions" , Moscow (1953) (In Russian)

References

[a1]	A.S. Besicovitch, "Almost periodic functions" , Cambridge Univ. Press (1932)
[a2]	C. Corduneanu, "Almost periodic functions" , Wiley (1968) (Translated from Rumanian)