A Completeness of a System of Exponential Monomials with Positive Exponents

The work is devoted to studying the problem on the completeness of a system of exponential monomials with positive exponents in the space of analytic functions in a convex domain of the complex plane. Sufficient conditions on the completeness of this system are obtained. They are formulated using simple geometric characteristics of a domain and a sequence (vertical diameter and logarithmic density). A criterion on the completeness of such a system in an arbitrary convex domain is also obtained, provided that the upper and maximum densities of the sequence coincide. This result generalizes the well-known result of B.Ya. Levin and A.F. Leont’ev for measurable positive sequences. Its proof uses the uniqueness theorem for entire functions of exponential type. This theorem is also proved in this paper. The obtained uniqueness theorem generalizes the corresponding results of F. Carlson and L.A. Rubel.