Generalized Dirichlet classes in a half-plane and their application to approximations

We introduce generalized Dirichlet classes of analytic functions in a disc and a half-plane. We establish a relationship between these classes and their zero sets. A precise sufficient condition for a zero subset of a generalized Dirichlet class in a half-plane is obtained. Using this condition, we prove a necessary condition (which is also precise) for a system of exponential functions to be complete in the space L-2 on a half-line with regularly varying weight of order alpha (sic) [-1, 0].