A Schwartz-type Space for the $\left(k,\frac{2}{n}\right)-$Generalized Fourier Transform

Abstract

The Schwartz space $\mathcal{S}({\mathbb{R}}^N)$ is not invariant under the $(k,a)$-generalized Fourier transform ${\mathcal{F}}_{k,a}$ unless $a=2$, and in general no such adapted space is known. For $N=1$ and $a=\frac{2}{n}$, $n\in \mathbb{N}$, we construct a tailored Schwartz-type space ${\mathcal{S}}_{k,n}(\mathbb{R})$ defined via seminorms built from natural second-order operators associated with the one-dimensional Dunkl Laplacian ${\Delta }_k$. We prove that ${\mathcal{S}}_{k,n}(\mathbb{R})$ recovers the two basic features of the classical Schwartz space: invariance under the corresponding Fourier-type operator and density in the relevant weighted $L^p-$spaces. To establish these results, we introduce the space ${\mathcal{D}}_{k,n}(\mathbb{R})$ of compactly supported smooth functions, which embeds continuously into ${\mathcal{S}}_{k,n}(\mathbb{R})$ and is dense in the weighted spaces $L^p(d{\mu }_{k,n})$, $1\le p<\infty$. These results provide the first Schwartz-type space for ${\mathcal{F}}_{k,a}$ that simultaneously ensures invariance and $L^p$-density, and admits an $\mathfrak{sl}(2,\mathbb{R})$-based description of the underlying operator structure.

27 pages. Title has changed. Major revision. 3 Figures