On quantum tomography on locally compact groups

We introduce quantum tomography on locally compact Abelian groups $G$. A linear map from the set of states $C^*$-algebra $A(G)$ generated by projective unitary representation $G$ to space characteristic functions is constructed. The dual determining symbols observables derived. Given a function state tomogram consisting probability distributions introduced. provide three examples in which $G={\mathbb R}$ (the optical tomography), Z}_n$ (corresponding measurements mutually unbiased bases) and T}$ phase). As an application we have calculated for output Weyl channels.