Real Paley-Wiener theorems for the inverse fourier transform on a Riemannian symmetric space

We prove real Paley-Wiener theorems for the inverse Fourier transform on a semisimple Riemannian symmetric space G/K of the noncompact type. The functions on G/K whose Fourier transform has compact support are characterised by a L-2 growth condition. We also obtain real Paley-Wiener theorems for the inverse spherical transform.