For the kernel
$B_{\kappa ,a}(x,y)$ of the
$(\kappa ,a)$-generalized Fourier transform
$\mathcal {F}_{\kappa ,a}$, acting in
$L^{2}(\mathbb {R}^{d})$ with the weight
$|x|^{a-2}v_{\kappa }(x)$, where
$v_{\kappa }$ is the Dunkl weight, we study the important question of when
$\|B_{\kappa ,a}\|_{\infty }=B_{\kappa ,a}(0,0)=1$. The positive answer was known for
$d\ge 2$ and
$\frac {2}{a}\in \mathbb {N}$. We investigate the case
$d=1$ and
$\frac {2}{a}\in \mathbb {N}$. Moreover, we give sufficient conditions on parameters for
$\|B_{\kappa ,a}\|_{\infty }>1$ to hold with
$d\ge 1$ and any a.
We also study the image of the Schwartz space under the
$\mathcal {F}_{\kappa ,a}$ transform. In particular, we obtain that
$\mathcal {F}_{\kappa ,a}(\mathcal {S}(\mathbb {R}^d))=\mathcal {S}(\mathbb {R}^d)$ only if
$a=2$. Finally, extending the Dunkl transform, we introduce nondeformed transforms generated by
$\mathcal {F}_{\kappa ,a}$ and study their main properties.