Motivated by problems in control theory concerning decay rates for the damped wave equation \(\begin{aligned} w_{tt}(x,t) + \gamma (x) w_t(x,t) + (-\Delta + 1)^{s/2} w(x,t) = 0, \end{aligned}\) we consider an analogue of the classical Paneah-Logvinenko-Sereda theorem for the Fourier Bessel transform. In particular, if \(E \subset \mathbb {R}^+\) is \(\mu _\alpha \) -relatively dense (where \(d\mu _\alpha (x) \approx x^{2\alpha +1}\, dx\) ) for \(\alpha > -1/2\) , and \(\operatorname {supp} \mathcal {F}_\alpha (f) \subset [R,R+1]\) , then we show \(\begin{aligned} \Vert f\Vert _{L^2_\alpha (\mathbb {R}^+)} \lesssim \Vert f\Vert _{L^2_\alpha (E)}, \end{aligned}\) for all \(f\in L^2_\alpha (\mathbb {R}^+)\) , where the constants in \(\lesssim \) do not depend on \(R > 0\) . Previous results on PLS theorems for the Fourier-Bessel transform by Ghobber and Jaming (2012) provide bounds that depend on R. In contrast, our techniques yield bounds that are independent of R, offering a new perspective on such results. This result is applied to derive decay rates of radial solutions of the damped wave equation.
