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

Abstract

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.

Wiener–Tauberian type theorems for radial sections of homogenous vector bundles on certain rank one Riemannian symmetric spaces of noncompact type

“…An alternative approach would be to adapt the method of Anker (see [2]) who proves the isomorphism of L p -Schwartz spaces for K -biinvariant functions using the Paley-Wiener theorem and the Abel transform along with the slice projection theorem. In [39,Chapter 5] this method was used for the spinor bundle. We mention here a few examples where isomorphism of Schwartz spaces is either available in the literature or can be obtained through one of these methods and consequently, proofs of all the theorems (Theorems 6.1, 6.2, 7.1, 8.1) can be readily extended.…”

“…But it can be verified that the closed ideal generated by f in L p,1 τ (G) does not contain the function g and hence is a proper ideal. (See[40] for details.) This construction works also when n is odd if we define f ∈ C p τ (G) by f (σ + , λ) = e − cosh λα .…”