A Schwartz-type Space for the $\left(k,\frac{2}{n}\right)-$Generalized Fourier Transform
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 $\displaystyle a=\frac{2}{n}$, $n\in\mathbb{N}$, 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.