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

Two types of classes of entire functions ($W_\alpha$ and $Z_\alpha$), 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 $Z_\alpha$ 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.