### 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.

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.

#### Keywords

Fourier transform; order of a distribution; Abelian theorems; regular distributions; injection; convolution.

#### Full Text:

PDFISSN: 1331-0623 (Print), 1848-8013 (Online)