The Fourier transform of thick distributions

TL;DR

This paper develops a framework for thick distributions and their Fourier transforms, extending classical distribution theory to include functions with special logarithmic behavior at infinity, and establishes isomorphisms with tempered distributions.

Contribution

It introduces the space of $sl$-thick distributions and constructs Fourier transform isomorphisms extending classical results to these new spaces.

Findings

Defined the space $ ext{W}( ext{R}_c^n)$ with thick expansions at infinity.

Constructed Fourier transform isomorphisms for thick distributions.

Determined Fourier transforms of regularized and thick delta functions.

Abstract

We first construct a space $\mathcal{W}\left( \mathbb{R}_{\text{c}} ^{n}\right) $ whose elements are test functions defined in $\mathbb{R} _{\text{c}}^{n}=\mathbb{R}^{n}\cup\left\{ \mathbf{\infty}\right\} ,$ 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.