Mean value properties of the Laplacian via spectral theory

Let $\phi ({z^2})$ be an even entire function of temperate exponential type, $L$ a selfadjoint realization of $- \Delta + c\,(x)$, where $\Delta$ is the Laplace-Beltrami operator on a Riemannian manifold, and $\phi \,(L)$ the operator given by spectral theory. A Paley-Wiener theorem on the support of $\phi \,(L)$ is proved, and is used to show that $Lu = \lambda u$ on a suitable domain implies $\phi \,(L)\,u = \phi \,(\lambda)\,u$, as well as a generalization of Àsgeirsson's theorem. A concrete realization of the operators $\phi \,(L)$ is given in the case of a compact Lie group or a noncompact symmetric space with complex isometry group.