Description of Unconditional Bases Formed by Values of the Dunkl Kernels

Unconditional bases of the form $\{d_\alpha(i\lambda_n t): \lambda_n \in \Lambda\}$ in the space $L_2(-a, a)$ with measure $|x|^\gamma dx$, $\gamma=2\alpha+1$, are described. Here $d_\alpha(ixt)$ is the Dunkl kernel determined by
$$ d_\alpha(z)=2^\alpha\Gamma(\alpha+1)z^{-\alpha}(J_\alpha(z)+iJ_{\alpha+1}(z)), \; \alpha>-1, $$
where $J_\alpha$ is the Bessel function of the first kind.