Fourier transform over the spaces \({\mathcal S}_ k'\) (Q1910851)

In [\textit{J. Horváth}, ``Topological vector spaces and distributions'', Vol. I (1966; Zbl 0143.15101)] the space \({\mathcal S}_k\), \(k\in\mathbb{Z}\), was introduced by J. Horváth. \({\mathcal S}_k'\) denotes as usual the dual space of \({\mathcal S}_k\). In this paper, the authors prove a representation theorem for the usual distributional Fourier transform over the spaces \({\mathcal S}_k\), \(k\in\mathbb{Z}\), \(k<0\). An inversion formula is also obtained by which it is proved that \({\mathcal S}_k'\) is a commutative convolution algebra with unit element. Furthermore, a uniqueness theorem is also proved. Lastly, the authors prove that \({\mathcal S}_k'\) is closed under convolution and that the convolution is associative and commutative with unit element.