The inverse Abel transform forSU(p, q)
Hence, a function fEKCc(G) g" is equal to the inverse spherical transform of ~ In the case of SU(p, q) there is an explicit formula for the spherical functions, due to Berezin and Karpelevich, and so one can write out an explicit formula for the inverse spherical transform. This involves a product of inverse Jacobi transforms, one for each of the dim (A) variables describing coordinates in a*, applied to the Fourier transform of a /-//<j (O~-O~)Ff. Here Oi means partial differentiation with respect to the ith coordinate on a. The final step uses a result of Koornwinder,