The Fourier transform of thick distributions (1909.03945v2)

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.