Zeroes of holomorphic functions with almost – periodic modulus Favorov

We give necessary and sufficient conditions for a divisor in a tube domain to be the divisor of a holomorphic function with almost–periodic modulus. Zero distribution for various classes of holomorphic almost–periodic functions in a strip was studied by many authors (cf. The notion of almost–periodic discrete set appeared in [9] and [17] in connection with these investigations. Its generalization to several complex variables was the notion of almost–periodic divisor, introduced by L. I. Ronkin (cf. [14]) and studied in his works and works of his disciples (cf. [5], [6], [15]). But these notions are not sufficient for a complete description of zero sets of holomorphic almost–periodic functions (cf. [18]): in addition, one needs some topological characteristic, namely, Chern class of the special (generated by an almost–periodic set or a divisor) line bundle over Bohr's compact set (cf. [2], [3]). On the other hand, the class of zero sets of holomorphic functions with almost–periodic modulus in a strip is just the class of almost–periodic discrete sets (cf. [4]). That's why it is natural to obtain a description of zeroes of holomorphic functions with the almost–periodic modulus for several complex variables without using topological terms. This problem is just solved in our paper. By T S denote a tube set {z = x + iy : x ∈ R m , y ∈ S}, where the base S is a subset of R m. Definition 1. A continuous function f on T S is called almost–periodic, if for each sequence {f (z +h n)} hn∈R m of shifts there exists a uniformly convergent on T S subsequence. In particular, for S = {0} we obtain the definition of an almost–periodic function on R m. It follows easily that any almost–periodic function on a tube set with a compact base is bounded. 1 This definition is equivalent to another one that makes use of the notion of an ε-almost period; for m = 1 see, for example, [12], the extension to m > 1 is trivial.