The Horv\'ath's spaces $\mathscr{S}_k^{\prime}$ and the Fourier transform

Abstract

In this paper we establish new properties for the Fourier transform over the space of distributions $\mathscr{S}_k^{\prime}$ introduced by Horv\'ath. We prove Abelian theorems for the Fourier transform over the space $\mathscr{S}^{\prime}_k$, $k\in\mathbb{Z}$, $k<0$.
Continuity properties and some results concerning regular distributions are studied. We also prove that the Fourier transform is an injection from $\mathscr{S}^{\prime}_k$, $k\in\mathbb{Z}$, $k<0$, into $\mathscr{O}_C^{-2k-1}$, where this space denotes the union of the spaces $\mathscr{S}^{-2k-1}_{k^\ast}$, as $k^{\ast}$ varies in $\mathbb{Z}$, which have been given by  Horv\'ath. The convolution over $\mathscr{S}_k^{\prime}$ for certain regular distributions and its relation with the usual convolution product of functions is exhibited. Finally, some illustrative examples are considered.