Real Paley-wiener Theorems for the Inverse Fourier Transform on a Riemannian Symmetric Space

The classical Fourier transform Fcl is an isomorphism of the Schwartz space S(Rk) onto itself. The space C∞ c (Rk) of smooth functions with compact support is dense in S(Rk), and the classical Paley-Wiener theorem characterises the image of C∞ c (R k) under Fcl as rapidly decreasing functions having an holomorphic extension to Ck of exponential type. Since Rk is self-dual, the same theorem also applies to the inverse Fourier transform. Let G be a noncompact semisimple Lie group and K a maximal compact subgroup of G. The Fourier transform F on the Riemannian symmetric space X = G/K is an analogue of the classical Fourier transform on Rk. A Paley-Wiener theorem for the Fourier transform F , which characterises the image of C∞ c (X) under F in terms of holomorphic extensions and growth behaviour, as in the classical case, was proved by Helgason, see [7]. Furthermore, the L2-Schwartz space S2(X) contains C∞ c (X) as a dense subspace and F is an isomorphism of S2(X) onto some generalised Schwartz space, see [4]. Unlike the classical case, however, we can not use a duality argument to deduce a Paley-Wiener theorem for the inverse Fourier transform. So how can we characterise the functions whose Fourier transform F has compact support? The Fourier transform on X reduces to the spherical transform H on G when restricted to K-invariant functions. The paper [8] provides an answer to the above question for the spherical transform on Schwartz functions in the rank one and complex cases. The characterisation is in analogy with the classical Paley-Wiener theorem given in terms of meromorphic extensions and growth conditions.