Abstract
A series expansion for Heckman–Opdam hypergeometric functions φλ is obtained for all λ∈aC⁎. As a consequence, estimates for φλ 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 Lp-theory for the hypergeometric Fourier transform is developed for 0<p<2. In particular, an inversion formula is proved when 1⩽p<2.
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