Zeros of holomorphic almost periodic functions (Q5947092)

Let \(S\) be a horizontal strip of finite or infinite width in the complex plane. A discrete measure \(\mu\) in \(S\) is called a divisor if the mass of \(\mu\) at each point of its support is integer. The divisor \(\mu _f\) of a holomorphic function \(f\) is the discrete measure whose value at any point is equal to the multiplicity of zero of \(f\) at this point. A divisor \(\mu\) in \(S\) is called almost-periodic (a.p.) if for any continuous function \(\chi\) compactly supported in \(S\) the convolution \((\mu \ast \chi)(t) = \sum _{a_n \in \operatorname{supp} \mu } \chi (a_n + t) \mu (a_n)\) is an a.p. function. The union of the spectra of \(\mu \ast \chi\) over all \(\chi\) is called the spectrum of \(\mu\). It is known (H. Tornehave, 1988, 1989) that the divisor of any holomorphic a.p. function \(f\) in \(S\) is a.p., but not every a.p. divisor is generated by the zeros of a holomorphic a.p. function. In this deep article S. Favorov completely characterizes zeros of holomorphic a.p. functions in strips. Let \(G\) be an additive subgroup of \(R\), \(H^2 (\widehat{G}, Z)\) be the cohomology group of its Pontryagin dual group \(\widehat{G}\), \(AP(S,G)\) and \(DP(S,G)\) stand for the sets of all holomorphic a.p. functions and a.p. divisors in \(S\) with spectra in \(G\). Main Result. There exists a homomorphism \(\gamma : DP(S,G) \rightarrow H^2 (\widehat{G}, Z)\) such that its kernel consists of the divisors of functions from \(AP(S,G)\).