Dirichlet series for an analytic almost-periodic function

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, , .

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)