# Almost-periodic analytic function

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An analytic function $f(s)$, $s=\sigma+i\tau$, regular in a strip $-\infty\leqslant\alpha<\sigma<\beta\leqslant+\infty$, and expandable into a series $$\sum a_ne^{i\lambda_ns},$$

where the $a_n$ are complex and the $\lambda_n$ are real numbers. A real number $\tau$ is called an $\varepsilon$-almost-period of $f(s)$ if for all points of the strip $(\alpha, \beta)$ the inequality

$$|f(s+i\tau) - f(s)|<\varepsilon$$

holds. An almost-periodic analytic function is an analytic function that is regular in a strip $(\alpha, \beta)$ and possesses a relatively-dense set of $\varepsilon$-almost-periods for every $\varepsilon>0$. An almost-periodic analytic function on a closed strip $\alpha\leqslant\sigma\leqslant\beta$ is defined similarly. An almost-periodic analytic function on a strip $[\alpha, \beta]$ is a uniformly almost-periodic function of the real variable $\tau$ on every straight line in the strip and it is bounded in $[\alpha, \beta]$, i.e. on any interior strip. If a function $f(s)$, regular in a strip $(\alpha, \beta)$, is a uniformly almost-periodic function on at least one line $\sigma=\sigma_0$ in the strip, then boundedness of $f(s)$ in $[\alpha, \beta]$ implies its almost-periodicity on the entire strip $[\alpha, \beta]$. 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 functio ns: 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)