One-dimensional $(k,a)$-generalized Fourier transform

We study the two-parametric $(k,a)$-generalized Fourier transform $\mathcal{F}_{k,a}$, $k,a>0$, on the line. For $a\neq 2$ it has deformation properties and, in particular, for a function $f$ from the Schwartz space $\mathcal{S}(\mathbb{R})$, $\mathcal{F}_{k,a}(f)$ may be not infinitely differentiable or rapidly decreasing at infinity. It is proved that the invariant set for the generalized Fourier transform $\mathcal{F}_{k,a}$ and differential-difference operator $|x|^{2-a}\Delta_kf(x)$, where $\Delta_k$ is the Dunkl Laplacian, is the class
$$ \mathcal{S}_{a}(\mathbb{R})=\{f(x)=F_1(|x|^{a/2})+xF_2(|x|^{a/2})\colon F_1,F_2\in\mathcal{S}(\mathbb{R}),\,\, F_1,F_2 - \text{are even}\}.$$
For $a=1/r$, $r\in\mathbb{N}$, we consider two generalized translation operators $\tau^{y}$ and $T^y=(\tau^{y}+\tau^ {-y})/2$. Simple integral representations are proposed for them, which make it possible to prove their $L^{p}$-boundedness as $1\le p\le\infty$ for $\lambda=r(2k-1)>-1/2$. For $\lambda\ge 0$ the generalized translation operator $T^y$ is positive and its norm is equal to one. Two convolutions are defined and Young's theorem is proved for them. For generalized means defined using convolutions, a sufficient $L^{p}$-convergence condition is established. The generalized analogues of the Gauss–Weierstrass, Poisson, and Bochner–Riesz means are studied.