Spectral analysis for radial sections of some homogeneous vector bundles on certain noncompact Riemannian symmetric spaces (Q636040)

Schwartz's theorem on \(\mathbb R\) states that if \(f\in C^\infty(\mathbb R)\), then the closure of the set \(\{W*f\mid W\in C^\infty(\mathbb R)^\prime\}\) contains the function \(x\mapsto e^{i\lambda x}\) for some \(\lambda\in \mathbb C\). An analogue of Schwartz's theorem for \(K\)-biinvariant functions of a rank one symmetric space was obtained by \textit{S. C. Bagchi} and \textit{A. Sitaram} [Pac. J. Math. 84, 241--250 (1979; Zbl 0442.43017)] by replacing \(x\mapsto e^{i\lambda x}\) with zonal spherical functions. In this paper S. Pusti and P. Sarkar obtain Schwartz's theorem for \(\tau\)-radial functions on \(G=\text{Spin}_0(n,1)\). By using the slice projection theorem for the Abel transform and the Paley-Wiener theorem for \(\tau\)-radial compactly supported distributions on \(G\), they reduce the proof to the Euclidean case. As applications, they show that \(\tau\)-radial \(C^\infty\) functions in Lorentz spaces on \(G\) are not mean periodic and a closed \(\tau\)-radial-translation invariant subspace of Lorentz spaces does not contain \(\tau\)-spherical functions.