Ja n 20 07 Almost periodicity in complex analysis ∗

This is a brief survey of up-to-date results on holomorphic almost periodic functions and mappings in one and several complex variables, mainly due to the Kharkov mathematical school. While the notion of almost periodic function on R (or R m) seems to be quite understood, it is not the case for holomorphic almost periodic functions on a strip in C (or, more generally, on a tube domain in C m). Investigation of the zero sets of holomorphic almost periodic functions leads to such objects as almost periodic divisors and holomorphic chains. The central problem is to determine if a divisor (holomorphic chain) is generated by a holomorphic almost periodic function (mapping). Partial results in the one-dimensional situation ([14], [15], [30], [31]) indicate that the problem is highly non-trivial. We present here a brief survey of recent results in this direction, together with some related topics, mainly due to the Kharkov mathematical school. Definitions and basic properties. We start with standard notions of the theory of almost periodic functions. Definition 1 A continuous mapping f from R m to a metric space X is called almost periodic if its orbit {T t f } t∈R m is a relatively compact set in C(R m , X) with respect to the topology of uniform convergence on R m , where T t is the translation by t ∈ R m : (T t f)(x) = f (x + t). Definition 2 A continuous mapping f from a tube domain T Ω := T m + iΩ = {z = x + iy : x ∈ R m , y ∈ Ω} to a metric space X is called almost periodic on T Ω if its orbit {T t f } t∈R m is a relatively compact subset of C(T Ω , X) with respect to the topology of uniform convergence on each tube subdomain T Ω ′ , Ω ′ ⊂⊂ Ω.