Classes of entire functions that are rapidly decreasing on the real axis: theory and applications

Two types of classes of entire functions (Wα and Zα), which are rapidly decreasing on the real axis are considered. Conditions to ensure that these classes are non-trivial are found and the classes of the corresponding Fourier transforms are described. Results on the classes Zα are applied to the question of whether a rapidly decreasing function with rapidly decreasing Fourier transform is trivial. This yields not just an extension of Morgan's well-known theorem, but also its converse. Bibliography: 18 titles