On the ultradistribution of Mellin transform (Q1847561)

Using the well-known relationship between Mellin and Fourier transforms, the authors obtain a necessary and sufficient condition for the space \({\mathcal Z}\) of functions whose Mellin transforms are elements of the usual test function space \({\mathcal D}\) of smooth functions having a compact support in the real line. These results are similar to the description of the Zemanian test function space whose elements are Fourier transforms of elements of \({\mathcal D}\). The authors also extend the Mellin transform to the dual \({\mathcal Z}\) using the Parseval relation and investigate their properties.