# 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.

How to Cite This Entry:
Almost-periodic analytic function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Almost-periodic_analytic_function&oldid=29514
This article was adapted from an original article by E.A. Bredikhina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article