The Fourier transform of thick distributions

Abstract

We first construct a space $\mathcal{W}\left({\mathbb{R}}_{\text{c}}^n\right)$ whose elements are test functions defined in ${\mathbb{R}}_{\text{c}}^n={\mathbb{R}}^n\cup \left\{\infty \right\},$ the one point compactification of ${\mathbb{R}}^n,$ that have a thick expansion at infinity of special logarithmic type, and its dual space ${\mathcal{W}}^{\prime }\left({\mathbb{R}}_{\text{c}}^n\right),$ the space of $sl-$thick distributions. We show that there is a canonical projection of ${\mathcal{W}}^{\prime }\left({\mathbb{R}}_{\text{c}}^n\right)$ onto ${\mathcal{S}}^{\prime }\left({\mathbb{R}}^n\right).$ We study several $sl-$thick distributions and consider operations in ${\mathcal{W}}^{\prime }\left({\mathbb{R}}_{\text{c}}^n\right).$ We define and study the Fourier transform of thick test functions of ${\mathcal{S}}_{\ast }\left({\mathbb{R}}^n\right)$ and thick tempered distributions of ${\mathcal{S}}_{\ast }^{\prime }\left({\mathbb{R}}^n\right).$ We construct isomorphisms \[ \mathcal{F}_{\ast}:\mathcal{S}_{\ast}^{\prime}\left( \mathbb{R}^{n}\right) \longrightarrow\mathcal{W}^{\prime}\left( \mathbb{R}_{\text{c}}^{n}\right) \,, \] \[ \mathcal{F}^{\ast}:\mathcal{W}^{\prime}\left( \mathbb{R}_{\text{c}} ^{n}\right) \longrightarrow\mathcal{S}_{\ast}^{\prime}\left( \mathbb{R} ^{n}\right) \,, \] that extend the Fourier transform of tempered distributions, namely, $\Pi {\mathcal{F}}_{\ast }=\mathcal{F}\Pi$ and $\Pi {\mathcal{F}}^{\ast }=\mathcal{F}\Pi ,$ where $\Pi$ are the canonical projections of ${\mathcal{S}}_{\ast }^{\prime }\left({\mathbb{R}}^n\right)$ or ${\mathcal{W}}^{\prime }\left({\mathbb{R}}_{\text{c}}^n\right)$ onto ${\mathcal{S}}^{\prime }\left({\mathbb{R}}^n\right).$ We determine the Fourier transform of several finite part regularizations and of general thick delta functions.

21 pages

References in corpus (2)

• An apparent paradox concerning the field of an ideal dipole
• Stora's fine notion of divergent amplitudes

Cited by in corpus (1)

• Distributions in spaces with thick submanifolds