Inverse Harish-Chandra Transform and Difference Operators

In the paper we calculate the images of the operators of multiplication by Laurent polynomials with respect to the Harish-Chandra transform and its non-symmetric generalization due to Opdam. It readily leads to a new simple proof of the Harish-Chandra inversion theorem in the zonal case (see [HC,He1]) and the corresponding theorem from [O1]. We assume that k > 0 and restrict ourselves to compactly supported functions, borrowing the growth estimates from [O1]