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

Abstract: In this study, we introduce the generalized Schwartz space $\mathcal{S}{k,n}(\mathbb{R})$, designed specifically for the $\left(k,\frac{2}{n}\right)-$generalized Fourier transform $\mathcal{F}{k,n}:=\mathcal{F}{k,\frac{2}{n}}$. We establish that $\mathcal{S}{k,n}(\mathbb{R})$ is invariant under $\mathcal{F}{k,n}$ and explore its principal properties in detail. Furthermore, by defining the subspace $\mathcal{D}{k,n}(\mathbb{R})$, consisting of functions with bounded support, we show that the subspace is dense in $L^p(d\mu_{k,n})$ for $1 \leq p < \infty$. Lastly, we address a conjecture regarding the higher-dimensional scenario.