Geodesic
Classes of entire functions that are rapidly decreasing on the real axis: theory and applications
Sbornik. Mathematics, Tome 199 (2008) no. 1, pp. 131-157

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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. 
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     title = {Classes of entire functions that are rapidly decreasing on the real axis: theory and applications},
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A. M. Sedletskii. Classes of entire functions that are rapidly decreasing on the real axis: theory and applications. Sbornik. Mathematics, Tome 199 (2008) no. 1, pp. 131-157. http://geodesic.mathdoc.fr/item/SM_2008_199_1_a5/