On the kernel of the -Generalized fourier transform

Abstract

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.