Theorems of Chernoff and Ingham for certain eigenfunction expansions

We prove an uncertainty principle for certain eigenfunction expansions on $L^2(\mathbb{R}^+,w(r)dr)$ and use it to prove analogues of theorems of Chernoff and Ingham for Laplace-Beltrami operators on compact symmetric spaces, special Hermite operator on $\mathbb{C}^n$ and Hermite operator on $\mathbb{R}^n.$