Almost-periodic analytic function

An analytic function $f(s)$,, regular in a strip , and expandable into a series

where the are complex and the are real numbers. A real number is called an -almost-period of if for all points of the strip the inequality

holds. An almost-periodic analytic function is an analytic function that is regular in a strip and possesses a relatively-dense set of -almost-periods for every . An almost-periodic analytic function on a closed strip is defined similarly. An almost-periodic analytic function on a strip is a uniformly almost-periodic function of the real variable on every straight line in the strip and it is bounded in , i.e. on any interior strip. If a function , regular in a strip , is a uniformly almost-periodic function on at least one line in the strip, then boundedness of in implies its almost-periodicity on the entire strip . 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). 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

 [1] H. Bohr, "Almost-periodic functions" , Chelsea, reprint (1947) (Translated from German) [2] B.M. Levitan, "Almost-periodic functions" , Moscow (1953) pp. Chapt. 7 (In Russian)