Observability Inequalities and the Logvinenko--Sereda Theorem for the Dunkl Transform

Let $R$ be a normalized root system in $\mathbb{R}^d$ with reflection group $G$, let $k$ be a $G$-invariant multiplicity function, and let $\mathcal{F}_k$ be the associated Dunkl transform. We write $k_α=k(α)>0$ and $\mathrm{d}μ_k(x)=w(x)\,\mathrm{d}x$, where $w(x)=\prod_{α\in R_+}|\langle x,α\rangl