Remarks on the general Funk-Radon transform and thermoacoustic tomography

The generalized Funk transform is a integral transform acting from densities on a manifold X to functions defined on a family Σ of hypersurfaces in X. The dual operator M is again a Funk transform which is defined for densities on the manifold Σ. We state twoside estimates for the operator M and a description of the range of the Funk transform operatorM and approximation theorem for the kernel of the dual operator. This operator is similar to the integral transform related to a double fibration in the sense of Guillemin [4] and some results can be extracted from his theory. Other results can be generalized for the general double fibration.