An analogue of Bochner's theorem for Damek-Ricci spaces
  We characterize the image of radial positive measures $\theta$'s on a harmonic $NA$ group $S$ which satisfies $\int_S\phi_0(x)\,d\theta(x)<\infty$ under the spherical transform, where $\phi_0$ is the elementary spherical function. 
Sanjoy Pusti
math.FA