Spectral analysis for radial sections of some homogeneous vector bundles on certain noncompact Riemannian symmetric spaces

We prove an analogue of Schwartz’s theorem on spectral analysis for radial sections of some homogeneous vector bundles on noncompact Riemannian symmetric spaces. We include some results and observations regarding mean periodic functions in this case. We also observe failure of spectral analysis for various Lorentz spaces and Lebesgue spaces of radial sections and relate it with the failure of the Wiener-Tauberian theorems in this setup.