Paley-Wiener theorems on groups of split rank one

Let G be a linear semisimple Lie group of split rank one with K a maximal compact subgroup. In this paper, we consider the space C_c^∞(G:F) of all functions in C_c^∞(G) whose left and right K-translates span a finite-dimensional space. Using the analytic continuation of the principal series to define the Fourier transform, we give a characterization of the Fourier transform of the space C_c^∞(G:F). This gives an analog of the classical Paley-Wiener theorem which gives a characterization of the Fourier transform of the space C_c^∞(Rⁿ).