A series expansion for Heckman-Opdam hypergeometric functions ϕ λ is obtained for all λ ∈ a * C . As a consequence, estimates for ϕ λ away from the walls of a Weyl chamber are established. We also characterize the bounded hypergeometric functions and thus prove an analogue of the celebrated theorem of Helgason and Johnson on the bounded spherical functions on a Riemannian symmetric space of the noncompact type. The L ptheory for the hypergeometric Fourier transform is developed for 0 < p < 2. In particular, an inversion formula is proved when 1 ≤ p < 2.
We prove a genuine analogue of the Wiener Tauberian theorem for L1false(G//Kfalse), where G is a real rank one noncompact, connected, semisimple Lie group with finite centre. This generalizes the corresponding result on the automorphism group of the unit disk by Y. Ben Natan, Y. Benyamini, H. Hedenmalm, and Y. Weit. We extend this result for hypergeometric transforms and as an application we prove an analogue of Furstenberg theorem on harmonic functions for hypergeometric transforms.
We will show that an uniform treatment yields Wiener-Tauberian type results for various Banach algebras and modules consisting of radial sections of some homogenous vector bundles on rank one Riemannian symmetric spaces G/K of noncompact type. One example of such a vector bundle is the spinor bundle. The algebras and modules we consider are natural generalizations of the commutative Banach algebra of integrable radial functions on G/K . The first set of them are Beurling algebras with analytic weights, while the second set arises due to Kunze-Stein phenomenon for noncompact semisimple Lie groups.
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