Almost periodic divisors in a strip
We omit here all ``almost periodic'' definitions, referring the reader to the paper under review. The theory of continuous and holomorphic almost periodic functions enjoyed intensive study in the twenties-forties. In particular, \textit{M. G. Krejn} and \textit{B. Y. Levin} [Dokl. Akad. Nauk SSSR, n. Ser. 64, 285-287 (1949; Zbl 0035.17401)] [see also \textit{B. Ya. Levin's} monograph ``Distribution of zeros of entire functions'' (Russian, 1956; Zbl 0111.07401); English translation, 1964; Zbl 0152.06703)] have proved that the zero divisor of some special entire almost periodic functions is almost periodic itself. \textit{H. Tornehave} [Mat.-Fys. Medd., K. Dan. Vidensk. Selsk. 42, No. 3, 125-142 (1989; Zbl 0687.42013)] has extended this result to arbitrary holomorphic almost periodic functions, but the converse is not true. In this deep article, the authors continue at first Tornehave's research. Theorem 1. Let \(Z\) be an almost periodic divisor in a horizontal strip \(S\) in the complex plane, whose support does not intersect some internal strip \(S^0\subset S\). Then there exists a holomorphic almost periodic function in \(S\) such that \(Z\) is its zero divisor. Theorem 4. In order for a divisor \(Z\subset S\) to be almost periodic, it is necessary and sufficient that there exists a holomorphic function \(f\) in \(S\) such that \(|f(z)|\) is an almost periodic function in \(S\) whose zero divisor coincides with \(Z\). Next, in the proofs the authors use the concept of an almost periodic distribution, which raises a natural question on extending all these results to subharmonic almost periodic functions. In particular, they prove Theorem 6. Let \(\mu\) be an almost periodic positive measure in a strip \(S\). Then there exists an almost periodic (in the sense of distributions) subharmonic function \(u\) in \(S\) such that its Riesz measure \(\mu_u= \mu\).
scientific article; zbMATH DE number 1187478